monoidal topology - sweetsweet.ua.pt/dirk/artigos/hst14_monoidal_topology_a...monoidal topology...

Dirk HofmannGavin J. SealWalter Tholen (Editors)

Monoidal TopologyA Categorical Approach to Order, Metric and Topology

Contributing Authors:

Maria Manuel ClementinoEva ColebundersDirk HofmannRobert LowenRory Lucyshyn-WrightGavin J. SealWalter Tholen

August 3, 2014

To Horst Herrlich

Summary of Contents

Preface ix

I IntroductionRobert Lowen, Walter Tholen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

II Monoidal StructuresGavin J. Seal, Walter Tholen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

III Lax AlgebrasDirk Hofmann, Gavin J. Seal, Walter Tholen . . . . . . . . . . . . . . . . . . . 145

IV Kleisli MonoidsDirk Hofmann, Robert Lowen, Rory Lucyshyn-Wright, Gavin J. Seal . . . . . . 281

V Lax Algebras as SpacesMaria Manuel Clementino, Eva Colebunders, Walter Tholen . . . . . . . . . . . 369

Bibliography 463

Tables 477

Index 478

v

Preface

Monoidal Topology describes an active research area that, after many proposals throughoutthe past century on how to axiomatize “spaces” in terms of convergence, started to emergeat the beginning of the millennium. It provides a powerful unifying framework and theoryfor fundamental ordered, metric and topological structures. Inspired by the topologicalconcept of filter convergence, its methods are lax-algebraic and categorical, with generalizednotions of monoid recurring frequently as the fundamental building blocks of its key notions.Since the main components of this new area have to date been available only in a scatteredarray of research articles, the authors of this book hope that a self-contained and consistentintroduction to the theory will serve a broad range of mathematicians, scientists and theirgraduate students with an interest in a modern treatment of the mathematical structures inquestion. With all essential elements from order and category theory provided in the book,it is assumed that the reader will appreciate a framework which highlights the power ofequationally-defined algebraic structures as particularly important elements of the broaderlax-algebraic context which, roughly speaking, replaces equalities by inequalities.There are two principal roots to the theory presented in this book: Barr’s 1970 relational pre-sentation of topological spaces which naturally extends Manes’ 1969 equational presentationof compact Hausdorff spaces as the Eilenberg–Moore algebras of the ultrafilter monad, andLawvere’s 1973 description of metric spaces as (small individual) categories enriched overthe extended non-negative real half-line. In hindsight it seems surprising that it took somethirty years until the two general parameters at play here were combined in a compatiblefashion, given by a monad T replacing the ultrafilter monad and a quantale (or, moregenerally, a monoidal closed category) V replacing the half-line. Of course, when consideredseparately, these two pivotal papers triggered numerous important developments. Lawvere’ssurprising discovery quickly became a cornerstone of enriched category theory, with hischaracterization of Cauchy completeness in purely enriched-categorical terms enjoying mostof the attention, and Barr’s paper was followed by at least two major but quite distinctattempts to develop a general topologically-inspired theory using a lax-algebraic monadapproach, by Manes [1974] and Burroni [1971]. However, the up-take of these articlesin terms of follow-up work remained sporadic, perhaps because not many strikingly newapplications beyond Barr’s work came to the fore, with one prominent exception: theinclusion of Lambek’s 1969 multicategories in addition to Barr’s topological spaces providesa powerful motivation for Burroni’s elegant setting.In 2000 Bill Lawvere was the first to suggest (in a private communication to Walter Tholen)

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viii PREFACE

that, in the same way as topological spaces generalize ordered sets, Lowen’s 1989 approachspaces should be describable as generalized metric spaces “using V-multicategories in a goodway” instead of just V-categories, thus implicitly envisioning a merger of the parameters Tand V. At about the same time, following a suggestion by George Janelidze, Clementinoand Hofmann [2003] gave a lax-algebraic description of approach spaces using a “numericalextension” of the ultrafilter monad. Both suggestions set the stage for Clementino andTholen [2003] to develop a setting that combines the two parameters efficiently, especiallywhen the monoidal-closed category V is just a quantale. As emphasized in [Clementino,Hofmann, and Tholen, 2004b], this setting suffices to capture ordered, metric and topologicalstructures. In a slightly relaxed form as presented in [Seal, 2005] it also permits to replaceultrafilter convergence by filter convergence (and its “approach generalization”) for its keyapplications, and it is this setting that has been adopted in this book.When following a meeting in Barisciano (Italy) in 2006 the authors of this book began toembark decisively on a project to give a self-contained presentation of the emerging theory,the heterogeneous make-up of the group itself made it necessary to clearly document allneeded ingredients in a coherent fashion. Hence, this book contains:

• a “crash course” on order and category theory that highlights many aspects not readilyavailable in existing texts and of interest beyond its use for order, metric and topology;

• an in-depth presentation of the syntactical framework involving the monad T and thequantale V needed for a unified treatment of the principal target categories;

• and some novel applications leading to new insights even in the context of ordinarytopological spaces, with ample directions to additional or subsequent work that couldnot be included in this book.

In acknowledging the valuable advice and contributions received from many colleagues,we should highlight first some theses written on subjects pertaining to this book and tovarious degrees influencing its development, including the PhD theses of Van Olmen [2005],Schubert [2006], Cruttwell [2008] and Reis [2013], and the Master’s theses of Akhvlediani[2008] and Lucyshyn-Wright [2009]. We are grateful especially to Christoph Schubertand Andrei Akhvlediani, who respectively helped to transform Walter Tholen’s lecturenotes for courses given at the University of Bremen (Germany) in 2003 and at a workshoporganized by Francis Borceux at Haute Bodeux (Belgium) in 2007 into something legibleand digestible. Christoph was also an active contributor to the various meetings that thegroup of authors held at the University of Antwerp until 2009, generously organized byEva Colebunders and Robert Lowen.The long but surely incomplete list of names of colleagues who offered helpful comments atvarious stages includes those of Bernhard Banaschewski, Francis Borceux, Franck van Breugel,Marcel Erne, Cosimo Guido, Eraldo Giuli, Horst Herrlich, Kathryn Hess, George Janelidze,Bill Lawvere, Frederic Mynard, Robert Pare, Hans Porst, Sergejs Solovjovs, Isar Stubbe,Pawe l Waszkiewicz and Richard Wood; we thank them all. We also appreciate the help inproofreading provided by Luca Hunkeler, Valentin Mercier and Eiichi Piguet.

PREFACE ix

The authors of this book gratefully acknowledge the financial assistance received fromvarious sources while working on this book project, including the Research FoundationFlanders (FWO) research project G.0244.05, the Centre for Mathematics of the Universityof Coimbra, the European Regional Development Fund through the program COMPETE,the Portuguese Government through the Fundacao para a Ciencia e a Tecnologia, theCenter for Research and Development in Mathematics and Applications of the Universityof Aveiro, the European Union through a Marie-Curie International Reintegration Grant,the Swiss National Science Foundation through an Advanced Researcher Fellowship, andthe Natural Sciences and Engineering Council of Canada through Discovery Grants. Wealso thank Roger Astley of Cambridge University Press for having been receptive and opento our wishes regarding the publication of this book.We dedicate this book to Horst Herrlich whose work and dedication to mathematics havehad formative influence on all authors of this book.

D.H., G.J.S. and W.T.Aveiro, Lausanne and Toronto—August 3, 2014

Contents

Preface vii

I Introduction 11 The ubiquity of monoids and their actions . . . . . . . . . . . . . . . . . . . 1

1.1 Monoids and their actions in algebra . . . . . . . . . . . . . . . . . . . . 11.2 Orders and metrics as monoids and lax algebras . . . . . . . . . . . . . 31.3 Topological and approach spaces as monoids and lax algebras . . . . . . 41.4 The case for convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Filter convergence and Kleisli monoids . . . . . . . . . . . . . . . . . . . 8

2 Spaces as categories, and categories of spaces . . . . . . . . . . . . . . . . . . 102.1 Ordinary small categories . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Considering a space as a category . . . . . . . . . . . . . . . . . . . . . 112.3 Moving to the large category of all spaces . . . . . . . . . . . . . . . . . 12

3 Chapter highlights and dependencies . . . . . . . . . . . . . . . . . . . . . . 14

II Monoidal Structures 191 Ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.1 The cartesian structure of sets and its monoids . . . . . . . . . . . . . . 191.2 The compositional structure of relations . . . . . . . . . . . . . . . . . . 201.3 Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.4 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.5 Adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.6 Closure operations and closure spaces . . . . . . . . . . . . . . . . . . . 241.7 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.8 Adjointness criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.9 Semilattices, lattices, frames, and topological spaces . . . . . . . . . . . 291.10 Quantales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.11 Complete distributivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.12 Directed sets, filters, and ideals . . . . . . . . . . . . . . . . . . . . . . . 341.13 Ultrafilters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361.14 Natural and ordinal numbers . . . . . . . . . . . . . . . . . . . . . . . . 38Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2 Categories and adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.1 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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CONTENTS xi

2.2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.3 Natural transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.4 The Yoneda embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.5 Adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.6 Reflective subcategories, equivalence of categories . . . . . . . . . . . . 532.7 Initial and terminal objects, comma categories . . . . . . . . . . . . . . 552.8 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.9 Colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.10 Construction of limits and colimits . . . . . . . . . . . . . . . . . . . . . 612.11 Preservation and reflection of limits and colimits . . . . . . . . . . . . . 632.12 Adjoint Functor Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 652.13 Kan Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672.14 Dense functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3 Monads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.1 Monads and adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 743.2 The Eilenberg–Moore category . . . . . . . . . . . . . . . . . . . . . . . 763.3 Limits in the Eilenberg–Moore category . . . . . . . . . . . . . . . . . . 783.4 Beck’s monadicity criterion . . . . . . . . . . . . . . . . . . . . . . . . . 793.5 Duskin’s monadicity criterion . . . . . . . . . . . . . . . . . . . . . . . . 823.6 The Kleisli category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.7 Kleisli triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.8 Distributive laws, liftings, and composite monads . . . . . . . . . . . . . 863.9 Distributive laws and extensions . . . . . . . . . . . . . . . . . . . . . . 90Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4 Monoidal and ordered categories . . . . . . . . . . . . . . . . . . . . . . . . . 974.1 Monoidal categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.2 Monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.3 Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.4 Monoidal closed categories . . . . . . . . . . . . . . . . . . . . . . . . . 1014.5 Ordered categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.6 Lax functors, pseudo-functors, 2-functors, and their transformations . . 1064.7 Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.8 Quantaloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.9 Kock–Zoberlein monads . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.10 Enriched categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5 Factorizations, fibrations and topological functors . . . . . . . . . . . . . . . 1165.1 Factorization systems for morphisms . . . . . . . . . . . . . . . . . . . . 1165.2 Subobjects, images and inverse images . . . . . . . . . . . . . . . . . . . 1195.3 Factorization systems for sinks and sources . . . . . . . . . . . . . . . . 1205.4 Closure operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

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5.5 Generators and cogenerators . . . . . . . . . . . . . . . . . . . . . . . . 1275.6 U -initial morphisms and sources . . . . . . . . . . . . . . . . . . . . . . 1285.7 Fibrations and cofibrations . . . . . . . . . . . . . . . . . . . . . . . . . 1305.8 Topological functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.9 Selfdual characterization of topological functors . . . . . . . . . . . . . . 1335.10 Epireflective subcategories . . . . . . . . . . . . . . . . . . . . . . . . . 1355.11 The Taut Lift Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 137Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Notes on Chapter II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

III Lax Algebras 1451 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

1.1 V-relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1451.2 Maps in V-Rel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1471.3 V-categories, V-functors and V-modules . . . . . . . . . . . . . . . . . . 1501.4 Lax extensions of functors . . . . . . . . . . . . . . . . . . . . . . . . . 1541.5 Lax extensions of monads . . . . . . . . . . . . . . . . . . . . . . . . . . 1561.6 (T,V)-categories and (T,V)-functors . . . . . . . . . . . . . . . . . . . . 1581.7 Kleisli convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1611.8 Unitary (T,V)-relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1631.9 Associativity of unitary (T,V)-relations . . . . . . . . . . . . . . . . . . 1641.10 The Barr extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1691.11 The Beck–Chevalley condition . . . . . . . . . . . . . . . . . . . . . . . 1721.12 The Barr extension of a monad . . . . . . . . . . . . . . . . . . . . . . . 1751.13 A double-categorical presentation of lax extensions . . . . . . . . . . . . 178Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

2 Fundamental examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1862.1 Ordered sets, metric spaces and probabilistic metric spaces . . . . . . . 1862.2 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1872.3 Compact Hausdorff spaces . . . . . . . . . . . . . . . . . . . . . . . . . 1912.4 Approach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1922.5 Closure spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

3 Categories of lax algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2023.1 Initial structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2023.2 Discrete and indiscrete lax algebras . . . . . . . . . . . . . . . . . . . . 2033.3 Induced orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2043.4 Algebraic functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2073.5 Change-of-base functors . . . . . . . . . . . . . . . . . . . . . . . . . . . 2103.6 Fundamental adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 212Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

4 Embedding lax algebras into a quasitopos . . . . . . . . . . . . . . . . . . . 2174.1 (T,V)-graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

CONTENTS xiii

4.2 Reflecting (T,V)-RGph into (T,V)-Cat . . . . . . . . . . . . . . . . . . . 2224.3 Coproducts of (T,V)-categories . . . . . . . . . . . . . . . . . . . . . . . 2234.4 Interlude on partial products and local cartesian closedness . . . . . . . 2274.5 Local cartesian closedness of (T,V)-Gph . . . . . . . . . . . . . . . . . . 2304.6 Local cartesian closedness of subcategories of (T,V)-Gph . . . . . . . . . 2344.7 Interlude on subobject classifiers and partial-map classifiers . . . . . . . 2374.8 The quasitopos (T,V)-Gph . . . . . . . . . . . . . . . . . . . . . . . . . 2444.9 Final density of (T,V)-Cat in (T,V)-Gph . . . . . . . . . . . . . . . . . . 246Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

5 Representable lax algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2525.1 The monad T on V-Cat . . . . . . . . . . . . . . . . . . . . . . . . . . . 2525.2 T-algebras in V-Cat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2545.3 Comparison with lax algebras . . . . . . . . . . . . . . . . . . . . . . . 2565.4 The monad T on (T,V)-Cat . . . . . . . . . . . . . . . . . . . . . . . . . 2605.5 Dualizing (T,V)-categories . . . . . . . . . . . . . . . . . . . . . . . . . 2635.6 The ultrafilter monad on Top . . . . . . . . . . . . . . . . . . . . . . . . 2655.7 Representable topological spaces . . . . . . . . . . . . . . . . . . . . . . 2685.8 Exponentiable topological spaces . . . . . . . . . . . . . . . . . . . . . . 2715.9 Representable approach spaces . . . . . . . . . . . . . . . . . . . . . . . 274Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

Notes on Chapter III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

IV Kleisli Monoids 2811 Kleisli monoids and lax algebras . . . . . . . . . . . . . . . . . . . . . . . . . 281

1.1 Topological spaces via neighborhood filters . . . . . . . . . . . . . . . . 2821.2 Power-enriched monads . . . . . . . . . . . . . . . . . . . . . . . . . . . 2831.3 T-monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2861.4 The Kleisli extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2881.5 Topological spaces via filter convergence . . . . . . . . . . . . . . . . . . 291Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

2 Lax extensions of monads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2982.1 Initial extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2982.2 Sup-dense and interpolating monad morphisms . . . . . . . . . . . . . . 3012.3 (S, 2)-categories as Kleisli monoids . . . . . . . . . . . . . . . . . . . . . 3032.4 Strata extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3072.5 (S,V)-categories as Kleisli towers . . . . . . . . . . . . . . . . . . . . . . 310Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

3 Lax algebras as Kleisli monoids . . . . . . . . . . . . . . . . . . . . . . . . . 3173.1 The ordered category (T,V)-URel . . . . . . . . . . . . . . . . . . . . . . 3173.2 The discrete presheaf monad . . . . . . . . . . . . . . . . . . . . . . . . 3203.3 Approach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3253.4 Revisiting change-of-base . . . . . . . . . . . . . . . . . . . . . . . . . . 328Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330

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4 Injective lax algebras as Eilenberg–Moore algebras . . . . . . . . . . . . . . . 3324.1 Eilenberg–Moore algebras as Kleisli monoids . . . . . . . . . . . . . . . 3324.2 Monads on categories of Kleisli monoids . . . . . . . . . . . . . . . . . . 3334.3 Eilenberg–Moore algebras over S-Mon . . . . . . . . . . . . . . . . . . . 3364.4 Continuous lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3404.5 Kock–Zoberlein monads on T-Mon . . . . . . . . . . . . . . . . . . . . . 3424.6 Eilenberg–Moore algebras and injective Kleisli monoids . . . . . . . . . 344Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348

5 Domains as lax algebras and Kleisli monoids . . . . . . . . . . . . . . . . . . 3505.1 Modules and adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 3505.2 Cocontinuous ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . 3515.3 Observable realization spaces . . . . . . . . . . . . . . . . . . . . . . . . 3535.4 Observable realization spaces as lax algebras . . . . . . . . . . . . . . . 3555.5 Observable specialization systems . . . . . . . . . . . . . . . . . . . . . 3575.6 Ordered abstract bases and round filters . . . . . . . . . . . . . . . . . . 3585.7 Domains as Kleisli monoids of the ordered-filter monad . . . . . . . . . 3595.8 Continuous dcpos as sober domains . . . . . . . . . . . . . . . . . . . . 3615.9 Cocontinuous lattices among lax algebras . . . . . . . . . . . . . . . . . 364Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

Notes on Chapter IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

V Lax Algebras as Spaces 3691 Hausdorff separation and compactness . . . . . . . . . . . . . . . . . . . . . 369

1.1 Basic definitions and properties . . . . . . . . . . . . . . . . . . . . . . 3691.2 Tychonoff’s Theorem, Cech–Stone compactification . . . . . . . . . . . . 3731.3 Compactness for Kleisli-extended monads . . . . . . . . . . . . . . . . . 3741.4 Examples involving monoids . . . . . . . . . . . . . . . . . . . . . . . . 378Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

2 Low separation, regularity and normality . . . . . . . . . . . . . . . . . . . . 3832.1 Order separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3832.2 Between order separation and Hausdorff separation . . . . . . . . . . . . 3852.3 Regular spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3872.4 Normal and extremally disconnected spaces . . . . . . . . . . . . . . . . 3912.5 Normal approach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 396Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

3 Proper and open maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4013.1 Finitary stability properties . . . . . . . . . . . . . . . . . . . . . . . . . 4013.2 First characterization theorems . . . . . . . . . . . . . . . . . . . . . . . 4053.3 Notions of closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4103.4 The Kuratowski–Mrowka Theorem . . . . . . . . . . . . . . . . . . . . . 4143.5 Products of proper maps . . . . . . . . . . . . . . . . . . . . . . . . . . 4193.6 Coproducts of open maps . . . . . . . . . . . . . . . . . . . . . . . . . . 4223.7 Preservation of space properties . . . . . . . . . . . . . . . . . . . . . . 424

CONTENTS xv

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4264 Topologies on a category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

4.1 Topology, fibrewise topology, derived topology . . . . . . . . . . . . . . 4294.2 P-compactness, P-Hausdorffness . . . . . . . . . . . . . . . . . . . . . . 4334.3 A categorical characterization theorem . . . . . . . . . . . . . . . . . . . 4354.4 P-dense maps, P-open maps . . . . . . . . . . . . . . . . . . . . . . . . 4384.5 P-Tychonoff and locally P-compact Hausdorff objects . . . . . . . . . . 443Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

5 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4495.1 Extensive categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4495.2 Connected objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4535.3 Topological connectedness governs . . . . . . . . . . . . . . . . . . . . . 4555.4 Products of connected spaces . . . . . . . . . . . . . . . . . . . . . . . . 457Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

Notes on Chapter V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

Bibliography 463

Index 478

xvi CONTENTS

I IntroductionRobert Lowen, Walter Tholen

In this introductory chapter we explain, in largely non-technical terms, how monoids andtheir actions occur not only everywhere in algebra, but how they also provide a commonframework for the ordered, metric, topological, or similar structures targeted in this book.This framework is categorical, both at a micro level, since individual spaces may be viewedas generalized small categories, and at a macro level, as we are providing a common settingand theory for the categories of all ordered sets, all metric spaces and all topologicalspaces—and many other categories.While this Introduction uses some basic categorical terms, we actually provide all neededcategorical language and theory in the next chapter, along with the basic terms about order,metric and topology, before we embark on presenting the common setting for our targetcategories. Many readers may therefore want to jump directly to Chapter III, using theIntroduction just for motivation and Chapter II as a reference for terminology and notation.

1 The ubiquity of monoids and their actions

Nothing seems to be more benign in algebra than the notion of monoid, that is, of a setM that comes with an associative binary operation m : M ×M −→ M and a neutralelement, written as a nullary operation e : 1 −→ M . If mentioned at all, normally thenotion finds its way into an algebra course only as a brief precursor to the segment ongroup theory. However, with the advent of monoidal categories, as first studied by Benabou[1963], Eilenberg and Kelly [1966], Mac Lane [1963], and others, came the realization thatmonoids and their actions occur everywhere in algebra, as the fundamental building blocksof more sophisticated structures. This book is about the extension of this realization fromalgebra to topology.

1.1 Monoids and their actions in algebra. Every algebraist of the past hundred yearswould subscribe to the claim that free algebras amongst all algebras of a prescribed typecontain all the information needed to study these algebras in general. However, what“contain” means was made precise only during the second half of this period. First, there wasthe observation of the late 1950s ([Godement, 1958], [Huber, 1961]) that the endofunctorT = GF induced by a pair F a G : A −→ X of adjoint functors comes equipped with natural

1

2 CHAPTER I. INTRODUCTION

transformationsm : TT −→ T and e : 1X −→ T

which, when we trade the cartesian product of sets and the singleton set 1 for functorcomposition and the identity functor on X, respectively, are associative and neutral inan easily described diagrammatic sense. Hence, they make T a monoid in the monoidalcategory of all endofunctors on X, that is, a monad on X ([Mac Lane, 1971]). If G isthe underlying-set functor of an algebraic category, like the variety of groups, rings, or aparticular type of algebras, the free structure TX on X-many generators is just a componentof that monad.On the question of how to recoup the other objects of the algebraic category from the monadthey have induced, let us look at the easy example of actions of a fixed monoid M in Set.Hence, our algebraic objects are simply sets X equipped with an action a : M ×X −→ Xmaking the diagrams

M ×M ×X 1M×a //

m×1X

M ×Xa

M ×X a // X

X〈e,1X〉

//

1X%%

M ×Xa

X

commutative. Realizing that TX = M ×X is in fact the carrier of the free structure overX, we may now rewrite these diagrams as

TTX Ta //

mX

TX

a

TXa // X

XeX //

1X !!

TX

a

X .

(1.1.i)

Using a similar presentation of the relevant morphisms, i.e., of the action-preserving orequivariant maps, Eilenberg and Moore [1965] realized that with every monad T = (T,m, e)on a category X (in lieu of Set) one may associate the category XT whose objects areX-objects X equipped with a morphism a : TX −→ X making the two diagrams (1.1.i)commutative. Furthermore, there is an adjunction F T a GT : XT −→ X inducing T, suchthat when T is induced by any adjunction F a G : A −→ X, there is a “comparison functor”K : A −→ XT which, at least for X = Set, measures the “degree of algebraicity” of A overX. In fact, for any variety of general algebras (with “arities” of operations allowed to bearbitrarily large, as long as the existence of free algebras is guaranteed), K is an equivalenceof categories and therefore faithfully recoups the algebras from their monad. By contrast,an application of this procedure to the underlying set functors of categories of ordered setsor topological spaces in lieu of general algebras would just render the identity monad on Setwhose Eilenberg–Moore category is Set itself, i.e., all structural information would be lost.While all categories of general algebras allowing for free structures may be seen as categoriesof generalized monoid actions as just described, this fact by no means describes the full

1. THE UBIQUITY OF MONOIDS AND THEIR ACTIONS 3

extent of the ubiquity of monoids and their actions in algebra. For example, a unital ringR is nothing but an abelian group R equipped with homomorphisms

m : R⊗R −→ R and e : Z −→ R ,

which are associative and neutral in a quite obvious diagrammatic sense. Hence, when onetrades the cartesian category (Set,×, 1) for the monoidal category (AbGrp,⊗,Z), monoids Rare simply rings, and their actions are precisely the left R-modules. This example, however,is just the tip of an iceberg which places the systematic use of monoidal structures, monoids,and their actions, at the core of post-modern algebra.

1.2 Orders and metrics as monoids and lax algebras. While trying to describeordered sets via the monad induced by the forgetful functor to Set is hopeless, since itinduces just the identity monad on Set, a “monoidal perspective” on structures is neverthelessbeneficial. First, departing from the notion of a monad, but trading endofunctors T on acategory X for relations a on a set X, one can express transitivity and reflexivity of a by

a · a ≤ a and 1X ≤ a , (1.2.i)

with ≤ to be read as set-theoretical inclusion if a is presented as a ⊆ X ×X. Hence, withthe morphisms m : a · a −→ a and e : 1X −→ a simply given by ≤, what we regard as the twoindispensable requirements of an order a on X, transitivity and reflexivity, are expressed bya carrying the structure of a monoid in the monoidal category of endorelations of X1. (Thefact that such a relation actually satisfies the equation a · a = a is of no particular concernat this point.) But it is also possible to consider an order a on X in its role as a structureon X in the spirit of 1.1, as follows. Replacing Set by the category Rel of sets with relationsas morphisms and choosing for T the identity monad on Rel, we see that the inequalities(1.2.i) are instances of lax versions of the Eilenberg–Moore requirements (1.1.i). Indeed,when formally replacing strict (“=”) by lax (“≤”) commutativity in (1.1.i), we obtain

TTX Ta //

mX

≥

TX

a

TX a // X

XeX //

1X !!

TX

a

≤

X .

(1.2.ii)

In doing so, we suppose that the ambient category X (which is Rel in the case at hand) isordered, so that its hom-sets are ordered, compatibly with composition. Briefly: orderedsets are precisely the lax Eilenberg–Moore algebras of the identity monad on the orderedcategory Rel.

1In this book, in order to avoid the proliferation of meaningless prefixes, we refer to what is usually calleda preorder as an order, considering the much less-used antisymmetry axiom as an add-on separation conditionwhenever needed. In fact, with respect to the induced order topology, antisymmetry amounts to the T0-separationrequirement.


Next, presenting relations a on X as functions a : X ×X −→ 2 = ⊥ < > with at mosttwo truth values, let us rewrite the transitivity and reflexivity requirements as

a(x, y) ∧ a(y, z) ≤ a(x, z) and > ≤ a(x, x)

for all x, y, z ∈ X. In this way, there appears a striking formal similarity with what weregard as the two principal requirements of a metric a : X ×X −→ [0,∞] on X, the triangleinequality and the 0-distance requirement for a point to itself2:

a(x, y) + a(y, z) ≥ a(x, z) and 0 ≥ a(x, x) .

Hence, the set 2 with its natural order ≤ and its inherent structure ∧ and > has beenformally replaced by the extended real half-line [0,∞], ordered by the natural ≥ (!), andstructured by + and 0. Just as for orders, one can now interpret metrics as both monoidsand lax Eilenberg–Moore algebras with respect to the identity monad, after extending therelational composition

(b · a)(x, z) =∨

y∈Y(a(x, y) ∧ b(y, z))

for a : X × Y −→ 2, b : Y × Z −→ 2 and all x ∈ X, y ∈ Y , by3

(b · a)(x, z) = infy∈Y (a(x, y) + b(y, z))

for a : X × Y −→ [0,∞], b : Y × Z −→ [0,∞] and all x ∈ X, y ∈ Y .The generalized framework encompassing both structures that we will use in this book isprovided by a unital quantale V in lieu of 2 or [0,∞], that is: of a complete lattice equippedwith a binary operation ⊗ (in lieu of ∧ or +) respecting arbitrary joins in each variable,and a ⊗-neutral element k (in lieu of > or 0). The role of the monad T that appears to berather artificial in the presentation of ordered sets and metric spaces will become muchmore pronounced in the presentation of the structures discussed next.

1.3 Topological and approach spaces as monoids and lax algebras. In 1.2 wedescribed ordered sets and metric spaces as lax algebras with respect to the identity monadon the category of relations and “numerical” relations, respectively. Taking a historicalperspective, we can now indicate how topological spaces fit into this setting once we allowthe identity monad to be traded for an arbitrary “lax monad”, and how the less-knownapproach spaces [Lowen, 1997] emerge as the natural hybrid of metric and topology in thiscontext.

2Similarly to the use of the term ordered set, in this book we refer to a distance function a satisfying these twobasic axioms as a metric, using additional attributes for the other commonly used requirements when needed, likefiniteness, symmetry, and separation.

3While we use∧

,∨

to refer to infima and suprema in general, in order to avoid ambiguity arising from the“inversion of order” in [0, ∞], we use sup and inf when denoting suprema and infima with respect to the naturalorder.


Although the axiomatization of topologies in terms of convergence, via filters or nets,has been pursued early on in the development of these structures since Hausdorff [1914],notably by Frechet [1921] and others, the geometric intuition provided by the open-set andneighborhood perspective clearly dominates the way in which mathematicians perceivetopological spaces. Nevertheless, the proof by Manes [1969] that compact Hausdorff spacesare precisely the Eilenberg–Moore algebras of the ultrafilter monad = (β,m, e) on Setcould not be ignored, as it gives the ultimate explanation for why the category CompHausbehaves in many ways just like algebraic categories do. (For example, just as in algebra,but unlike in the case of arbitrary topological spaces, the set-theoretic inverse of a bijectivemorphism in CompHaus is automatically a morphism again.) In this description, a compactHausdorff space is a set X equipped with a map a : βX −→ X assigning to every ultrafilterx on X (what turns our to be) its point of convergence in X, requiring the two basic axiomsof an Eilenberg–Moore algebra:

a(βa(X )) = a(mX(X )) and a(eX(x)) = x (1.3.i)

for all X ∈ ββX and x ∈ X; here the following ultrafilters on X are used:

eX(x) = x = A ⊆ X | x ∈ A

is the principal filter on x,

mX(X ) = ∑X =

A ⊆ X | x ∈ βX | A ∈ x ∈ X

is the Kowalsky sum of X , and

βa(X ) = a[X ] =A ⊆ X | x ∈ βX | a(x ) ∈ A ∈ X

is simply the image filter of X under the map a. Writing x −→ y instead of a(x ) = y andX −→ y instead of a[X ] = y , the conditions (1.3.i) take the more intuitive form

∃y ∈ βX (X −→ y & y −→ z) ⇐⇒ ∑X −→ z and x −→ x

for all X ∈ ββX and x ∈ X. In fact, in the presence of the implication “ =⇒ ” in thedisplayed equivalence, the implication “ ⇐= ” comes for free (as y = a[X ] necessarilysatisfies y −→ z when ∑X −→ z), and conditions (1.3.i) take the form

X −→ y & y −→ z =⇒ ∑X −→ z and x −→ x (1.3.ii)

for all X ∈ ββX, y ∈ βX, x, z ∈ X.As Barr [1970] observed, if one allows a to be an arbitrary relation between ultrafilterson X and points of X, rather than a map, so that we are no longer assured that everyultrafilter has a point of convergence (compactness) and that there is at most one suchpoint (Hausdorffness), then the relations −→ satisfying (1.3.ii) describe arbitrary topologieson X, with continuous maps characterized as convergence-preserving maps. Furthermore,


given the striking similarity of (1.3.ii) with the transitivity and reflexivity conditions ofan ordered set, it is not surprising that (1.3.ii) gives rise to the presentation of topologicalspaces as both monoids and lax algebras of the ultrafilter monad.In this statement, however, we glossed over an important point: having the Set-functor β,one knows what βa is when a is a map, but not necessarily when a is just a relation. Whilethere is a fairly straightforward answer in the case at hand, in general we are confrontedwith the problem of having to extend a monad T = (T,m, e) on Set to Rel or, even moregenerally, to V-Rel, the category of sets and V-relations r : X −→7 Y , given by functionsr : X × Y −→ V . Although for our purposes it suffices that this extension be lax, i.e., quitefar from being a genuine monad on V-Rel, the study of the various needed methods of justlaxly extending monads on Set to V-Rel can be cumbersome and takes up significant spacein this book.The general framework that emerges as a common setting is therefore given by a unital(but not necessarily commutative) quantale (V ,⊗, k) and a monad T = (T,m, e) on Setlaxly extended to V-Rel, with the lax extension usually denoted by T : V-Rel −→ V-Rel(although a given T may have several lax extensions). The lax algebras considered are setsX equipped with a V-relation a : TX −→7 X satisfying the two basic axioms

T a(X , y )⊗ a(y , z) ≤ a(mX(X ), z) and k ≤ a(eX(x), x) (1.3.iii)

for all X ∈ TTX, y ∈ TX, z ∈ Z. The lax algebras are to be considered generalizedcategories enriched in V , with the domain x of the hom-object a(x , y) not lying in X butin TX. Furthermore, relational composition can be generalized to Kleisli convolution forV-relations r : TX × Y −→ V , s : TY × Z −→ V via

(s r)(x , z) =∨

X∈TTXmX(X )=x

∨y∈TY

T r(X , y )⊗ s(y , z)

for all x ∈ TX, z ∈ Z. The lax algebra axioms for (X, a) are then represented via themonoidal structures

a a ≤ a and 1]X ≤ a ,

where 1]X is neutral with respect to the Kleisli convolution.In this general framework we have so far encountered the objects in the following table,displayed with the corresponding monad T and quantale V (here, P+ = (([0,∞],≥),+, 0) isthe extended non-negative real half-line):

T V 2 P+

identity monad ordered sets metric spacesultrafilter monad topological spaces ?

Fortunately, the field left blank is filled with a well-studied, but much less familiar structure,called approach space. It is perhaps easiest described in metric terms: an approach structure


on a set X can be given by a point-set distance function δ : X × PX −→ [0,∞] satisfyingsuitable conditions. A metric space (X, d) becomes an approach space via

δ(x,B) = infy∈B d(y, x)

for all x ∈ X, B ⊆ X. When an approach space is presented as a lax algebra (X, a) witha : βX × X −→ [0,∞], one can think of the value a(x , y) as the distance that the pointy is away from being a limit point of x . Indeed, a topological space X has its approachstructure given by

a(x , y) =

0 if x −→ y,

∞ otherwise.

As for topological spaces, the more categorical view of approach spaces in terms of conver-gence proves useful.

1.4 The case for convergence. A topology (of open sets) on a set X is most elegantlyintroduced as a subframe of the powerset X, i.e., a collection of subsets of X closed underfinite intersection and arbitrary union. Via complementation, a topology (of closed sets) isequivalently described as a collection closed under finite union and arbitrary intersection,and this simple tool of Boolean duality (switching between open and closed sets) proves tobe very useful. There is, however, an unfortunate breakdown of this duality when it comesto morphisms. Although continuous maps are equivalently described by their inverse-imagefunction preserving openness or closedness of subsets, the seemingly most important andnatural subclasses of morphisms, namely those continuous maps whose image functionpreserve openness or closedness (open or closed continuous maps) behave very differently:while open maps are stable under pullback, closed maps are not; not even the subspacerestriction f−1B −→ B of a closed maps f : X −→ Y with B ⊆ Y will generally remainclosed. Hence, as recognized by Bourbaki [1989], more important than the closed maps arethe proper maps, i.e., the morphisms f that are stably closed, so that every pullback of themap f is closed again, also characterized as the closed maps f with compact fibres.While under the open- or closed-set perspective no immediate “symmetry” between openand proper maps becomes visible, their characterization in terms of ultrafilter convergencereveals a remarkable duality: a continuous map f : X −→ Y is

• open if y −→ f(x) (with x ∈ X and y ∈ βY ) implies y = f [x ]with x −→ x for some x ∈ βX,

x // x

y // f(x)• proper if f [x ] −→ y (with x ∈ βX and y ∈ Y ) implies y = f(x)

with x −→ x for some x ∈ X.x // x

f [x ] // y

In fact, once presented as lax homomorphisms between lax Eilenberg–Moore algebras withrespect to the ultrafilter monad (laxly extended from Set to Rel), these two types of special


morphisms occur most naturally as the ones for which an inequality characterizing theircontinuity may be replaced by equality, i.e., by a strict homomorphism condition.Another indicator why convergence provides a most useful complementary view of topologicalspaces is the following. For a set X and maps fi : X −→ Yi into topological spaces Yi, i ∈ I,there is a “best” topology on X making all fi continuous, often called “weak”, but “initial”in this book. Its description in terms of open sets is a bit cumbersome, as it is generated bythe sets f−1(B), B ⊆ Yi open, i ∈ I, whereas the characterization in terms of ultrafilterconvergence is immediate: x −→ x in X precisely when fi[x ] −→ f(x) for all i ∈ I. Forexample, when X = ∏

i∈I Yi with projections fi, so that the topology on X just describedis the product topology, a proof of the Tychonoff Theorem (on the stability of compactnessunder products) becomes almost by necessity cumbersome when performed in the open-setenvironment, but is in fact a triviality in the convergence setting.However, we stress the fact that the roles of open sets versus convergence relations arereversed in the dual situation, when one wants to describe the “best” (or “final”) topologyon a set Y with respect to given maps fi : Xi −→ Y originating from topological spacesXi, i ∈ I. Its description in terms of open sets is immediate, as B ⊆ Y is declared openwhenever all f−1

i (B) are open, whereas a characterization in terms of convergence involvesa cumbersome generation process.In conclusion, we regard the two perspectives not at all as mutually exclusive but ratheras complementary to each other. Consequently, this book provides a number of resultson topological and approach spaces which arise naturally from the general convergenceperspective, but which are far from being obvious when expressed in the more classicalopen-set or point-set-distance language.

1.5 Filter convergence and Kleisli monoids. To what extent is it possible to tradeultrafilter convergence for filter convergence when presenting topological spaces as in 1.3or characterizing open and proper maps as in 1.4? In order to answer this question, it isuseful to axiomatize topologies on a set X in terms of maps ν : X −→ FX into the set FXof filters on X, to be thought of as assigning to each point its neighborhood filter. Orderingsuch maps pointwise by reverse inclusion and using the same notation as in 1.3, exceptthat now denotes the Kleisli composition rather than the Kleisli convolution, one obtainsanother (and, in fact, more elementary) monoidal characterization of topologies on a set X:

ν ν ≤ ν and eX ≤ ν ;

in pointwise terms, this reads as∑ν[ν(x)] ⊇ ν(x) and x ⊇ ν(x)

for all x ∈ X. We say that topological spaces are represented as Kleisli monoids (X, ν), orsimply as F-monoids, since the filter monad F = (F,m, e) may be traded for any monadT on Set such that the sets TX carry a complete-lattice order, suitably compatible withthe monad operations. As such a monad T may be characterized via a monad morphism


τ : P −→ F, with P the powerset monad, we call T power-enriched. The basic correspondencebetween filter convergence and neighborhood systems, given by

f −→ x ⇐⇒ f ⊇ ν(x) ,

may now be established at the level of a power-enriched monad T. With a suitable laxextension of T to Rel it yields a presentation of T-monoids as lax algebras. For T = F ittells us that, remarkably, the characterization (1.3.ii) of topological spaces remains valid ifwe trade ultrafilters for filters. This fact, although established by Pisani [1999] in slightlyweaker form, remained unobserved until proved by Seal [2005]. All previous axiomatizationsof the notions of topology in terms of filter convergence entailed redundancies.The answer to our initial question is therefore affirmative with respect to the convergencepresentation of topological space. Also the characterization of open maps given in 1.4survives the filters-for-ultrafilters exchange, but that of proper maps does not. Hence, wemust be cognizant of the fact that the notions introduced for lax algebras will in generaldepend on the parameters T and V , not just on the category of lax algebras described bythem, such as the category of topological spaces considered here.


2 Spaces as categories, and categories of spaces

It has been commonplace since the very beginning of category theory to regard individualordered sets as categories: they are precisely the categories whose hom-sets have at mostone element. By contrast, it was a very bold step for Lawvere [1973] to interpret thedistance a(x, y) in a metric space as hom(x, y). To understand this interpretation, wefirst recall how ordinary categories fare in the context of orders and metrics as describedin 1.2. We then indicate how the consideration of individual ordered sets, metric spaces,topological spaces, and similar objects as small generalized categories leads to new insightsand cross fertilization between different areas, as does the investigation of the properties ofthe category of all such small categories of a particular type.

2.1 Ordinary small categories. Replacing “truth values” (2-valued or [0,∞]-valued)by arbitrary sets, for a given set X of “objects” let us consider functions

a : X ×X −→ Set .

X is then the set of objects of a category with hom-sets a(x, y) if there are families of maps

mX,Y,Z : a(x, y)× a(y, z) −→ a(x, z) and eX : 1 −→ a(x, x)

satisfying the obvious associativity and neutrality conditions, expressible in terms ofcommutative diagrams. Hence, the notion of small category fits into the same structuralpattern already observed for orders and metrics, where now the composition of functionsa : X × Y −→ Set, b : Y × Z −→ Set is given by

(b · a)(x, z) =∐

y∈Y(a(x, y)× b(y, z))

for all x ∈ X, z ∈ Z.Briefly, if one allows the above-mentioned setting of a unital quantale (V ,⊗, k) to beextended to that of a monoidal closed category, ordinary small categories occur as monoidsor lax Eilenberg–Moore algebras of an identity monad when V is taken to be (Set,×, 1).We note in passing that the presentation of ordinary small categories just given becomesperhaps more familiar when one exhibits functions a : X×X −→ Set equivalently as directedgraphs

Edomain //

codomain// X

(with a fixed set X of vertices), where a and E determine each other via

E =∐

x,y∈Xa(x, y) and a(x, y) = f ∈ E | domain(f) = x, codomain(f) = y .

Hence, ordinary categories are monoids in the monoidal category of directed graphs, thetensor product (that is, composition) of which corresponds to the above composition ofSet-valued functions.

2. SPACES AS CATEGORIES, AND CATEGORIES OF SPACES 11

While we have made clear now that the setting of a monoidal closed category (V,⊗, k) andthe theory of categories enriched over V (so that their “hom-sets” and structural componentslive in V rather than Set, see [Kelly, 1982]) provide the right environment for studyingnot only orders and metrics, but also categories themselves, ordinary or additive (withV = AbGrp), and much more, in this book we restrict ourselves to considering the highlysimplified case of a quantale (V ,⊗, k). This is sufficient for reaching the intended targetcategories, and it makes the theory “technically” simpler since the triviality of 2-cells (givenby order in this case) makes all coherence issues disappear as all diagrams in V commute.Nevertheless, the categorical perspective of interpreting the entity a(x, y) as hom(x, y) turnsout to be very useful even in this simplified situation, as we indicate next.

2.2 Considering a space as a category. A key tool of category theory is the Yonedaembedding

y : X −→ SetXop, y 7−→ X(−, y)

which assigns to every object y of a category X its contravariant hom-functor X(−, y) =homX(−, y) : Xop −→ Set. It fully embeds X into a category with all (small-indexed) colimits;moreover, it is dense, in the sense that every object of its codomain is, in a natural way, acolimit of representable functors, i.e., of objects in the image of y. In other words, SetXop

serves as a cocompletion of X. Other types of cocompletions of X may be found inside SetXop

through suitable closure processes, and this statement remains valid even when one movesfrom ordinary to enriched category theory, trading Set for a monoidal closed category V.For ordered sets, so that V = 2 = 0 < 1 is the two-chain, monotone maps Xop −→ 2correspond to down-closed subsets of the ordered set X, so that

y : X −→ DnX , y 7−→ ↓ y = x ∈ X | x ≤ y

is the embedding of X into the sup-completion of X, i.e., the lattice with all suprema freelygenerated by X.For metric spaces, in the generality adopted in 1.2, it is natural to endow V = [0,∞] withits “internal hom” given by the non-symmetric distance function

µ(v, w) =

w − v if v ≤ w <∞,0 if w ≤ v,

∞ if v < w =∞,

the symmetrization of which gives the Euclidian metric suitably extended to∞. Dualizationof a metric space X = (X, a) is as trivial as for ordered sets: Xop = (X, a) with a(x, y) =a(y, x). Now y provides an isometric embedding into the space of all non-expansive mapsXop −→ [0,∞], provided with the sup-metric. This space inherits various completenessproperties from [0,∞], and inside of it one finds the Cauchy completion of X: for everyCauchy sequence (xn)n∈N in X one considers the non-expanding map

ψ : Xop −→ [0,∞] , x 7−→ limn−→∞ a(x, xn) ,


and the subspace formed by all such maps is the Cauchy completion of X. Amazingly,as first observed by Lawvere [1973], this construction may be performed for arbitraryV-categories, since the Cauchy property of (xn)n∈N is fully characterized by an adjointnessproperty of ψ when viewed as a module, that is, as a generalized compatible relation.It is now natural to ask whether such constructions may be performed for topologicalspaces, presented as lax algebras as in 1.3. The additional parameter given by the monadT = (T,m, e) does in fact introduce a serious obstacle, which starts with trying to determinewhat Xop should be: a simple switch of arguments of the structure a of X is no longerpossible! It turns out that by changing carrier sets from X to TX when forming the dual,it is possible to develop a comprehensive completion theory in the general (T,V)-context,with the Yoneda embedding providing the central tool also at this level of generality.For a topological space, among other constructions, the Yoneda embedding leads to itssobrification. While the core of this general completion theory, along with other advancedtopics, will be presented only in a forthcoming book by Clementino, Hofmann, and Tholen,many of the needed tools are in fact presented in this volume.

2.3 Moving to the large category of all spaces. The internally-defined propertyof completeness may be externally characterized within the category of all spaces of aparticular type: Banaschewski and Bruns [1967] and Isbell [1964] respectively characterizedcompleteness of ordered sets and metric spaces by injectivity. Remarkably this categoricalcharacterization can be established in the general (T,V)-context for a whole scheme ofcompleteness notions. While this characterization may be seen as depending only on thecategory of lax algebras of a particular type, hence as independent of the parameters Tand V presenting them, there is also an equational characterization of completeness, whichuses these parameters in a substantial way. Indeed, within the context of (suitably defined)separated lax algebras, the cocomplete objects are precisely the Eilenberg–Moore algebrasof a certain monad on the category of all lax algebras.The characterization of equationally-defined objects as the injectives in a category of laxalgebras or monoids is in fact a recurring theme in the book. For example, in Chapter IV wepresent general theorems that entail the identification of continuous lattices as the regular-injective objects in the category of T0-spaces. Furthermore, the general context of laxalgebras allows us to make precise the connection of the equationally-defined-versus-injectiveparadigm with the fundamental categorical notion of exponentiability. In the context oftopological spaces it facilitates the formation of function spaces, and Day and Kelly [1970]identified the exponentiable objects as the core-compact spaces. When topological spacesare described as lax algebras (X, a) by the inequalities (1.3.iii) which, with V-relationalcomposition, may be transcribed as

a · T a ≤ a ·mX and 1X ≤ a · eX , (2.3.i)

then the core-compact spaces are precisely those that make the first of these two inequalitiesan equality. If in (2.3.i) one lets T be the filter monad (with its Kleisli extension), rather

2. SPACES AS CATEGORIES, AND CATEGORIES OF SPACES 13

than the ultrafilter monad, those spaces which satisfy the multiplicative law (2.3.i) up toequality form again an important subclass of spaces, called observable realization spaces inthis book, for which we give alternate characterizations in Chapter IV.With these facts in mind, it is not surprising that the category of reflexive graphs, givenby all pairs (X, a) required only to satisfy the second of the two inequalities (2.3.i), forma quasitopos which contains the category of lax algebras as a reflexive subcategory, undermild conditions on the parameters T and V . In fact, for T = and V = 2, this extension isminimal, producing the category of pseudotopological spaces. In the general (T,V)-context,however, the quasitopos hull of the category of lax algebras will form a proper subcategoryof that of all reflexive graphs, which leads to the consideration of important intermediatecategories (Chapter III).


3 Chapter highlights and dependencies

Chapter II gives a rapid introduction into ordered sets and category theory, to the extentneeded in this book. It provides not only the notation, terminology and theory used in themain body of the book (starting with Chapter III), but emphasizes areas of importance inthe sequel that may play a less prominent role in other introductory texts, such as monadicand topological functors. For enriched and higher-order category theory, we get by with abrief exposition of monoidal and of ordered categories.While the presentation of topics is self-contained, the arguments provided are often quitecompact and pitched at a level that requires a degree of mathematical maturity that mayat times be challenging for a beginning graduate student. Some of the exercises at the endof each section should help to overcome these challenges. Others are complementary to themain body of the text and may be used later on.Chapter III, Sections 1–3, provide the first key notions, properties and examples of thetheory and applications of lax algebras. Introduced under the name (T,V)-category in 1.6, inorder to stress their status as individual small generalized categories, they are alternativelycalled (T,V)-algebras or (T,V)-spaces, depending on whether we want to emphasize theiralgebraic or geometric-topological roles. It is important that the reader does not skipthe preceding subsections 1.1–1.5 in which many of the syntactical tools pertaining to laxextensions of the monad T are being developed.Topologicity of the resulting large category (T,V)-Cat over Set is shown at the beginning ofSection 3, followed by a discussion of the impact of change in the parameters, arising frommorphisms S −→ T and V −→ W .As (T,V)-Cat fails to be cartesian closed, a presentation of quasitopoi containing (T,V)-Catfollows in Section 4, along with an introduction to the categorical tools on exponentiabilityof morphisms. For the role model Top ∼= (, 2)-Cat, the quasitopos extension (T,V)-Gph(“(T,V)-graphs”) of (T,V)-Cat leads to the category of pseudotopological spaces. Section 5gives a first demonstration of how the general theory feeds into applications and providesnew insights. There is a key adjunction

(T,V)-Cat //⊥ (V-Cat)Too

which compares (T,V)-Cat with the Eilenberg–Moore category of T extended from Set toV-Cat. In our role model, it relates Top with the category of ordered compact Hausdorffspaces and emphasizes the importance of the order

x ≤ y ⇐⇒ ∀A ⊆ X closed (A ∈ x =⇒ A ∈ y )⇐⇒ ∀B ⊆ X open (B ∈ y =⇒ B ∈ x )

on the set βX of ultrafilters on the set X, for every topological space X. This approachleads to the powerful notion of representable (T,V)-category which, in the role model,entails core-compactness, i.e., exponentiability in the category Top. While Sections 1–3 of

3. CHAPTER HIGHLIGHTS AND DEPENDENCIES 15

Chapter III are a necessary prerequisite for Chapters IV and V, the slightly more demandingSections 4 and 5 will be used only sporadically.Chapter IV provides powerful alternate descriptions of the category (T,V)-Cat, the moststriking of which arises from the fact that the quantale V and the monad T on Set laxlyextended to V-Rel allow for the construction of a new monad = (T,V) (read “Pi”) onSet laxly extended to Rel = 2-Rel such that

(T,V)-Cat = (, 2)-Cat ,

associativity of the Kleisli convolution granted. Consequently, (T,V)-categories may bepresented equivalently as relational lax algebras, with respect to a power-enriched (see 1.5)monad . The fact that all relevant information provided by T and V can be encoded by anew monad gives the parameter T some prominence over V. This result, presented inSection 3 along with applications, including the relational description of approach spaceswhich initiated this research, needs some preparations from Sections 1 and 2 that are ofindependent interest.Guided by the role model of the filter monad, for a power-enriched monad T we describe inSection 1 the isomorphism

(T, 2)-Cat ∼= T-Mon ,

presenting relational algebras as Kleisli monoids. In Section 2, taking the inclusion −→ Fof ultrafilters into filters as the role model, for a suitable morphism S −→ T of monads wepresent an isomorphism

T-Mon ∼= (S, 2)-Cat ,

which provides the general framework for the identical description of topological spaces interms of either filter or ultrafilter convergence. A V-level generalization (in lieu of V = 2)of this last isomorphism is also provided.In the context of a morphism S −→ T of power-enriched monads one can construct a monadT′ on S-Mon which has the same Eilenberg–Moore category as T:

SetT ∼= (S-Mon)T′ .

For S = T, the right-hand side category becomes isomorphic to a category of injectiveT-monoids, as we show in Section 4. For T the filter monad, so that T-Mon ∼= Top, oneobtains in particular the simultaneous description of continuous lattices as Eilenberg–Moorealgebras and as injectives in Top (with respect to the class of initial morphisms).Section 5 is devoted exclusively to the study of those topological spaces that, when presentedas lax algebras with the filter monad, satisfy the multiplicative law (2.3.i) up to equality,for which alternative descriptions are given, and the continuous lattices amongst them arefully characterized.Chapter V looks at (T,V)-categories as spaces and explores topological properties, such asseparation, regularity, normality and compactness in (T,V)-Cat (Sections 1–4). Emphasis


is given to those properties which arise “naturally” in the (T,V)-setting, such as thesymmetric descriptions of Hausdorff separation and compactness, or the symmetrically-described properties of properness and openness for morphisms, as already alluded to abovein 1.4. There is also a much more hidden symmetry between normality and extremaldisconnectedness.Closure of the relevant properties under direct products (for compact objects or propermorphisms) is a prominent theme (including the Tychonoff Theorem), and so is thegeneralization of the Kuratowski–Mrowka Theorem characterizing compact spaces in thegeneral (T,V)-context.Section 4 gives an axiomatic categorical framework for treating some of these key propertiesin a most economical fashion, and Section 5 explores the notion of connectedness in extensivecategories in general and in the (T,V)-context in particular.The range of example categories is expanded beyond the realm of metric and topology. Itincludes multi-ordered sets, to be thought of as a “thin” version of Lambek’s multicategories[Lambek, 1969] (also known as colored operads) that are gaining considerable attention inalgebraic topology.

Summary of chapter dependencies. The left column in the graph below indicates theprincipal stream of suggested reading, while the right column lists the order-theoretic andcategorical prerequisites, as well as special topics that may be omitted initially.

I II

III.1–3III.4

III.5

IV.1–3IV.4

IV.5

V

References to items occurring outside a current chapter are always preceded by the Romannumeral for the relevant chapter, but the chapter number is omitted for references withinthe same chapter.

3. CHAPTER HIGHLIGHTS AND DEPENDENCIES 17

A word about sets, classes, and choice. Without reference to any particular kind of set-theoretic foundations, in this book we distinguish between sets, classes and conglomerates,to be able to form the class of all sets and the conglomerate of all classes, leading usin particular to the category Set and the metacategory SET, respectively (as in II.2.1and II.2.2). Classes whose elements may be labelled by a set are also called small; othersare large or proper classes. We refer to the Notes on Chapter II for suggested furtherreading on this topic.We frequently use the Axiom of Choice (guaranteeing that surjective maps of sets areretractions). In fact, key results (such as the equivalence of the open-set and the ultrafilter-convergence presentations of topologies) rely on it. We alert the reader to each new use ofthe Axiom of Choice by putting the symbol

c©

in the margin. The symbol merely indicates our use of Choice at the instance in question,without any affirmation that the use is actually essential.

II Monoidal StructuresGavin J. Seal, Walter Tholen

This chapter provides a compactly written introduction into the order- and category-theoretic tools most commonly used throughout the remainder of the book. A newcomer tothe subject may at times need to consult a standard reference on category theory, while thechapter may be skipped by the more advanced reader who might use it just as a referencepoint for notation and terminology.It is hoped that all readers will appreciate the ubiquity of “monoidal structures” appearingin the text. We allude to them quite explicitly only in Sections 1, 3, and 4 but note that,after all, categories are generalized monoids.

1 Ordered sets

1.1 The cartesian structure of sets and its monoids. Any two sets A,B may be“multiplied” in terms of their cartesian product

A×B = (x, y) | x ∈ A, y ∈ B ,

and this multiplication extends to maps f : A −→ A′, g : B −→ B′ via

f × g : A×B −→ A′ ×B′ , (x, y) 7−→ (f(x), g(y)) .

The cartesian product respects identity maps, since

1A × 1B = 1A×B ,

as well as composition of maps, since for f ′ : A′ −→ A′′, g′ : B′ −→ B′′, one has themiddle-interchange law

(f ′ × g′) · (f × g) = (f ′ · f)× (g′ · g) .

The cartesian product is associative “up to isomorphism”, and any one-element set E = ?acts as a neutral element. More precisely, there are obvious natural bijections

αA,B,C : A× (B × C) −→ (A×B)× C , λA : E × A −→ A , ρA : A× E −→ A

19

20 CHAPTER II. MONOIDAL STRUCTURES

satisfying the so-called coherence conditions

λE = ρE , (ρA × 1B) · αA,E,B = 1A × λB ,

(αA,B,C × 1D) · αA,B×C,D · (1A × αB,C,D) = αA×B,C,D · αA,B,C×D ,

for all sets A,B,C,D. Moreover, the cartesian structure is symmetric, since there is anatural bijection

σA,B : A×B −→ B × A

with

σB,A · σA,B = 1A×B , ρA = λA · σA,E ,

αC,A,B · σA×B,C · αA,B,C = (σA,C × 1B) · αA,C,B · (1A × σB,C) .

A monoid M (with respect to the cartesian structure of sets) is a set M that comes with abinary and a nullary operation

m : M ×M −→M , e : E −→M

that are associative and make e = e(?) a neutral element of M ; equivalently, the diagrams

M × (M ×M) α //

1×m

(M ×M)×M m×1//M ×M

m

M ×M m //M

E ×M e×1//

λ&&

M ×Mm

M × E1×eoo

ρxx

M

commute. The monoid is commutative if m · σ = m. A homomorphism f : M −→ N ofmonoids preserves both operations: f ·mM = mN · (f × f) and f · eM = eN .

1.2 The compositional structure of relations. A relation r from a set X to a set Ydistinguishes those elements x ∈ X and y ∈ Y that are r-related; we write x r y if x isr-related to y. Hence, depending on whether we display r as a subset, a two-valued function,or a multi-valued function via

r ⊆ X × Y , r : X × Y −→ true, false , r : X −→ PY

respectively, x r y may be equivalently written as

(x, y) ∈ r , r(x, y) = true , y ∈ r(x) ,

where PY denotes the powerset of Y . Writing r : X −→7 Y when r is a relation from X toY , we can “multiply” r with s : Y −→7 Z via ordinary relational composition:

x (s · r) z ⇐⇒ ∃y ∈ Y (x r y & y s z) .

1. ORDERED SETS 21

Writing r ≤ r′ (with r′ : X −→7 Y ) when, equivalently,

r ⊆ r′ , ∀x ∈ X ∀y ∈ Y (r(x, y) |= r′(x, y)) , ∀x ∈ X (r(x) ⊆ r′(x)) ,

we see that the multiplication respects ≤, since

r ≤ r′, s ≤ s′ =⇒ s · r ≤ s′ · r′ . (1.2.i)

Moreover, relational composition is associative, so that

t · (s · r) = (t · s) · r

when t : Z −→7 W , and for the identity relation 1X (with x 1X x′ ⇐⇒ x = x′) one has

r · 1X = r = 1Y · r .

Hence, comparing with 1.1, we observe that sets A,B, . . . have been replaced by relationsr, s, . . . and the cartesian product by composition. While in 1.1 there is room for maps f, g,here we have only ≤ between relations, so that the middle-interchange law of 1.1 reducesto a mere property (1.2.i). The natural bijections α, λ, ρ of 1.1 have become identities, butthe multiplicative structure is no longer “symmetric”. However, for r : X −→7 Y one has theopposite (or dual) relation r : Y −→7 X with

y r x ⇐⇒ x r y

for all x ∈ X, y ∈ Y , which satisfies

(s · r) = r · s , (1X) = 1X , (r) = r , r ≤ r′ =⇒ r ≤ (r′) .

Note that when r is the graph of a map f : X −→ Y (so that x r y ⇐⇒ f(x) = y), thenr(y) = f−1(y) is simply the fibre of f over y ∈ Y . In what follows we make no notationaldistinction between a map and its graph.

1.3 Orders. An order on a set X is a relation a : X −→7 X that carries a monoid structurewith respect to the compositional structure of relations; that is, a satisfies

a · a ≤ a , 1X ≤ a .

Hence, a is simply a transitive and reflexive relation on X:

(x ≤ y & y ≤ z =⇒ x ≤ z) , x ≤ x

for all x, y, z ∈ X, when we write x ≤ y for x a y. The order is

(1) separated if a ∩ a = 1X (so that x ≤ y & y ≤ x =⇒ x = y);

(2) total if a ∪ a = X ×X (so that x ≤ y or y ≤ x, for all x, y ∈ X).


In the literature, orders on X are usually called preorders, and separated (that is, anti-symmetric) orders are often called partial orders on X. In this book, an ordered set X issimply a set X equipped with an order, and X is separated if the order is separated. Ifa is an order on X (respectively, a separated, or a total order), then so is a. A chain isa set with a separated total order. A map f : X −→ Y of ordered sets is monotone (ororder-preserving) if

f · a ≤ b · f ,where a, b denote the orders on X, Y respectively, and f is identified with its graph; hence,if we write ≤ for both a and b,

x ≤ y =⇒ f(x) ≤ f(y)

for all x, y ∈ X. If the implication “⇐= ” also holds, so that a = f · b · f , then f is fullyfaithful. For an ordered set X, we write Xop for the same set equipped with the oppositeorder; thus, when f : X −→ Y is monotone, so is f op : Xop −→ Y op (with f op(x) = f(x) forall x ∈ X).Every relation r : X −→7 X has an ordered hull r, which may be described as

r = ⋃n≥0 r

n ,

(where r0 := 1X , rn+1 = r · rn), and if r is separated, so is r. For any order a on X, a ∩ ais an equivalence relation on X which, when a is written as ≤, is denoted by ', so that

x ' y ⇐⇒ x ≤ y & y ≤ x .

There is a least order b on the quotient set X/' which makes the projection p : X −→ X/'monotone, namely b = p · a · p, also described by

p(x) ≤ p(y) ⇐⇒ x ≤ y .

The point of this construction is that b is separated; we call X/' the separated reflectionof X.

1.4 Modules. A relation r : X −→7 Y between ordered sets is a module if (≤Y )·r ·(≤X) ≤ r,that is, if

x′ ≤ x & x r y & y ≤ y′ =⇒ x′ r y′

for all x, x′ ∈ X, y, y′ ∈ Y . Hence, the relation r is a module if and only if the mapr : Xop × Y −→ true, false is monotone (where Xop × Y is ordered componentwise).Graphically, we indicate modularity of a relation r : X −→7 Y by

r : X −→ Y .

Every monotone map f : X −→ Y gives rise to the modules

f∗ = (≤Y ) · f : X −→ Y and f ∗ = f · (≤Y ) : Y −→ X ,

1. ORDERED SETS 23

that is,x f∗ y ⇐⇒ f(x) ≤ y and y f ∗ x ⇐⇒ y ≤ f(x)

for all x ∈ X, y ∈ Y . The following rules may be easily verified when g : Y −→ Z ismonotone:

(1) 1∗X = (1X)∗ = (≤X),

(2) (g · f)∗ = g∗ · f∗ and (g · f)∗ = f ∗ · g∗,

(3) 1∗X ≤ f ∗ · f∗ and f∗ · f ∗ ≤ 1∗Y .

Modularity is also closed under relational composition. Indeed, for modules r : X −→ Yand s : Y −→ Z, one has

(≤Z) · (s · r) · (≤X) ≤ (≤Z) · s · (≤Y ) · (≤Y ) · r · (≤X) ≤ s · r ,

so that s · r : X −→ Z is again a module.

1.5 Adjunctions. For ordered sets X, Y , the set

Ord(X, Y ) = f | f : X −→ Y monotone

is itself ordered pointwise by

f ≤ f ′ ⇐⇒ ∀x ∈ X (f(x) ≤ f ′(x)) .

This order is preserved by composition on either side: whenever h : W −→ X and k : Y −→ Zare monotone, then

f ≤ f ′ =⇒ k · f · h ≤ k · f ′ · h .

A monotone map g : Y −→ X is called

(1) right adjoint if there is a monotone map f : X −→ Y with 1X ≤ g · f , f · g ≤ 1Y ;

(2) an equivalence if there is a monotone map f : X −→ Y with 1X ' g · f , f · g ' 1Y ;

(3) an isomorphism if there is a monotone map f : X −→ Y with 1X = g · f , f · g = 1Y .

By definition, one has the implications

isomorphism =⇒ equivalence =⇒ right adjoint.

The map f occurring in the definition of right adjointness is, up to “'”, uniquely determinedby g: if 1X ≤ g · f ′ and f ′ · g ≤ 1Y , then

f ′ = f ′ · 1X ≤ f ′ · g · f ≤ 1Y · f = f ,


and dually f ≤ f ′. If g is right adjoint, the corresponding f is called left adjoint to g, andone writes

f a g .This terminology becomes more plausible when we consider the following fact:

1.5.1 Proposition. A map g : Y −→ X (not assumed to be monotone a priori) is rightadjoint if and only if there is a map f : X −→ Y such that

f(x) ≤ y ⇐⇒ x ≤ g(y)

for all x ∈ X, y ∈ Y .

Proof. The necessity of the condition is obvious since x ≤ g(y) implies f(x) ≤ f · g(y) ≤ y,and dually for “ =⇒ ”. For its sufficiency, observe that f(x) ≤ f(x) implies x ≤ g · f(x),and dually f · g(y) ≤ y. The monotonicity of f follows, since x ≤ x′ ≤ g · f(x′) yieldsf(x) ≤ f(x′), and likewise for g.

Calling a pair (f : X −→ Y, g : Y −→ X) of monotone maps an adjunction if f is left adjointto g, we see that (gop : Y op −→ Xop, f op : Xop −→ Y op) is also an adjunction. In other words,f is left adjoint (to g) if and only if f op is right adjoint (with left adjoint gop):

f a g : Y −→ X ⇐⇒ gop a f op : Xop −→ Y op .

An adjunction (f : Xop −→ Y, g : Y −→ Xop) is often called a Galois correspondence betweenX and Y .One sees easily that the following statement holds:

1.5.2 Corollary. A right adjoint map g (with left adjoint f) is fully faithful if and only iff · g ' 1Y .

The conjunction of this statement with its dual yields:

1.5.3 Corollary. Equivalences are given by those adjunctions for which both maps arefully faithful. In this case, each map serves as both a left and a right adjoint.

All properties for maps between ordered sets discussed so far (monotone, fully faithful, rightadjoint, left adjoint, equivalence, isomorphism) are closed under composition.

1.6 Closure operations and closure spaces. For any adjunction f a g : Y −→ X onehas

f · g · f ' f and g · f · g ' g ,

since 1X ≤ g ·f and f ·g ≤ 1Y imply f ≤ f ·g ·f = (f ·g) ·f ≤ f ; hence the first equivalenceholds, and the second follows by duality. Therefore, setting

c := g · f and d := f · g ,

1. ORDERED SETS 25

one obtains monotone maps c : X −→ X, d : Y −→ Y with

c · c ' c , 1X ≤ c and d ' d · d , d ≤ 1Y .

Any such map c is called a closure operation on X, and any such d an interior operation onX. Since 1X ≤ c and the monotonicity of c imply c ≤ c · c, it suffices to ask c to satisfy

c · c ≤ c , 1X ≤ c ,

i.e., a closure operation on X is nothing but an element of Ord(X,X) that carries a monoidstructure with respect to the compositional structure.With c and d induced by f a g as above, one easily sees that f and g can be restricted toan equivalence between the subsets

Fix(c) := x ∈ X | c(x) ' x and Fix(d) := y ∈ Y | d(y) ' y ,

of c-closed and d-open elements (or simply closed and open elements), also referred to asfixpoints of c and d, respectively. The following diagram summarizes this situation:

Xf

//⊥ Yg

oo

Fix(c)?

OO

f//

∼ Fix(d) .?

OO

goo

Any closure operation c on X is induced by an adjunction whose left adjoint is defined onX:

Xc //⊥ Fix(c) .? _oo

A closure space is a set X which comes with a closure operation on the powerset PX,ordered by inclusion. A map f : X −→ Y is continuous if

f(cX(A)) ⊆ cY (f(A))

for all A ⊆ X. Since for any map f : X −→ Y one has an adjunction

PXf

//⊥ PY

f−1(−)oo

given by image and preimage along f , the continuity condition is equivalently written as

cX(f−1(B)) ⊆ f−1(cY (B))

for all B ⊆ Y . Via the order isomorphism (−) : PXop −→ PX (which maps A ∈ PX to itscomplement A := X \ A in X), any closure operation c on PX corresponds to an interioroperation d on PX, and vice-versa:

c(A) = d(A) ,


for all A ⊆ X. Therefore, there is a concept of interior space, equivalent to that of a closurespace, and in this context, a map f : X −→ Y between interior spaces (X, dX) and (Y, dY )is continuous if

f−1(dY (B)) ⊆ dX(f−1(B))for all B ⊆ Y .

1.7 Completeness. For an element x in an ordered set X, let

↓X x = ↓x = y ∈ X | y ≤ x

be the down-set of x in X. The down-closure of A ⊆ X is

↓X A = ↓A = ⋃x∈A ↓x ,

and A is down-closed (or a down-set) if ↓A = A. There is a fully faithful map

↓ : X −→ DnX = Fix(↓X) = A ⊆ X | ↓A = A ,

where the set of down-sets in X is ordered by inclusion. The ordered set X is complete ifand only if this map is right adjoint; equivalently, if there is a map ∨X = ∨ : DnX −→ Xwhich for every A ∈ DnX satisfies

∀x ∈ X (∨A ≤ x ⇐⇒ A ⊆ ↓x) . (1.7.i)

Calling x an upper bound of A in X whenever A ⊆ ↓x, we may rephrase the characteristicproperty of the join (or supremum, or least upper bound) ∨A of A more familiarly by:

(1) ∨A is an upper bound of A in X (“ =⇒ ” of (1.7.i)), and

(2) if x is an upper bound of A in X, then ∨A ≤ x (“⇐= ” of (1.7.i)).

Of course, in general ∨A is uniquely determined by A only up to “'”. Note also thatour notion of completeness does not only give mere existence of ∨A, but comes with agiven choice of ∨A for every A ∈ DnX. Finally, existence of suprema for arbitrary subsetsB ⊆ X (not necessarily down-closed) follows from that of down-closed subsets: since Band ↓B have the same upper bounds, one can put∨

B = ∨ ↓B ;

phrased more sophisticatedly, the adjunctions

X↓//

> DnX∨oo

//> PX↓

oo

compose!

1. ORDERED SETS 27

Exploiting the adjunction ∨X a ↓X for Xop in lieu of X, we obtain

Xop↓Xop

//> Dn(Xop) ,∨Xop

oo

and dualization of this adjunction yields, with

Up X := (Dn(Xop))op , ↑X := (↓Xop)op ,∧X := (∨Xop)op ,

the adjunction

X↑X //⊥ Up X .∧X

oo

Note that, for ↑X to be monotone, Up X is (unlike DnX) ordered by reverse inclusion.The dual notions (like up-set, up-closure, up-closed, lower bound, meet, infimum, greatestlower bound) are all naturally describable in terms of this adjunction. For example, forA ∈ Up X, ∧A is characterized by

∀x ∈ X (x ≤ ∧A ⇐⇒ A ⊆ ↑x) .

Moreover, this adjunction exists (equivalently, Xop is complete) precisely when X iscomplete, since the meet of an (up-closed) set can be realized as the join of the set of itslower bounds; a more elegant argument is given in Corollary 1.8.4.

1.8 Adjointness criteria. A monotone map f : X −→ Y of ordered sets preserves thesupremum ∨

A of A ⊆ X if f(∨A) is a supremum of f(A) = f(x) | x ∈ A in Y . Moreover,f is a sup-map if it preserves every existing supremum in X:

f(∨A) ' ∨ f(A)

whenever ∨A exists. The dual notions are: preserves an infimum, inf-map. Sup-preservingmaps are useful for detecting left adjoints:

1.8.1 Proposition. Every left adjoint map f is a sup-map.

Proof. Indeed, if ∨A exists, then

f(A) ⊆ ↓ y ⇐⇒ ∀x ∈ A (f(x) ≤ y)⇐⇒ ∀x ∈ A (x ≤ g(y))⇐⇒ ∨

A ≤ g(y) ⇐⇒ f(∨A) ≤ y

for all y ∈ Y .


Dually, a right adjoint map is an inf-map. Being a sup-map (respectively, an inf-map) isnot only a necessary condition for being left adjoint (respectively, right adjoint), but alsosufficient, provided that the domain of the map is complete. More precisely:

1.8.2 Proposition. A monotone map f : X −→ Y is left adjoint if and only if there is amap g : Y −→ X such that for all y ∈ Y

g(y) '∨x ∈ X | f(x) ≤ y , (1.8.i)

and f preserves those suprema. Hence, when X is complete, the map g can be given as thecomposite map

Y↓

// DnY f−1(−)// DnX

∨// X .

Proof. The condition (1.8.i) is clearly necessary since f a g yields f−1(↓ y) = ↓ g(y), and∨ ↓ g(y) ' g(y) for all y ∈ Y . Conversely, existence of the join (1.8.i) gives x ≤ g(y)whenever f(x) ≤ y, and its preservation by f yields f · g(y) ≤ y, so that f(x) ≤ y wheneverx ≤ g(y).

1.8.3 Corollary. When X is a complete ordered set, a map f : X −→ Y is left adjoint ifand only if f is a sup-map.

As an application, let us prove the following result (see 1.7):

1.8.4 Corollary.

(1) Xop is complete when X is complete.

(2) When Y is complete, a map g : Y −→ X is right adjoint if and only if g is an inf-map.

Proof.

(1) It suffices to show that ↑X : X −→ Up X is a sup-map. But

↑∨A = ⋂a∈A ↑ a

for all A ⊆ X is just the defining property for suprema:∨A ≤ x ⇐⇒ ∀a ∈ A (a ≤ x) ⇐⇒ A ⊆ ↓x .

(2) By (1) one may apply Corollary 1.8.3 with gop in lieu of f .

1. ORDERED SETS 29

1.9 Semilattices, lattices, frames, and topological spaces. For a separated orderedset X, the map

↓X : X −→ DnXis an order-embedding, that is: the map is injective and fully faithful (separatedness is notessential, but is assumed for convenience). The set DnX is complete, with infima given byintersection. In particular, DnX with the binary operation ∩ and the nullary operation X(largest element in DnX) is a commutative monoid. When do these operations restrict toX along ↓X? That is, when do we have dotted maps making the diagrams

X ×X ↓×↓//

m

DnX ×DnX∩

X↓

// DnX

Ee

X

X↓// DnX

commute? Precisely when all finite infima exist in X, and then we must have

m(x, y) = x ∧ y =∧x, y

for all x, y ∈ X, and e must be the largest element of X: e = ∧∅.A meet-semilattice X is a separated ordered set with finite infima. A homomorphism ofmeet-semilattices preserves finite infima, that is, it preserves the binary ∧ and the largestelement. Trading infima for suprema (hence, ∧ for ∨ and largest for smallest), one obtainsthe notions of join-semilattice and homomorphism thereof.Both meet- and join-semilattices have a common algebraic description: (X,∧,>) and(X,∨,⊥), with > := ∧∅ and ⊥ := ∨∅ the top and bottom elements (or maximum andminimum) ofX, respectively, are simply commutative monoids in which, under mutliplicativenotation, every element is idempotent, so that x · x = x for all x ∈ X. One calls suchmonoids (X, ·, e) semilattices, since they may equivalently be considered as either a meet-or a join-semilattice, depending on whether one puts

(x ≤ y ⇐⇒ x · y = x) or (x ≤ y ⇐⇒ x · y = y) ,

in which case one obtains x · y = x ∧ y or x · y = x ∨ y (see Exercise 1.L). Homomorphismsof such monoids are equivalently described as homomorphisms of meet- or join-semilattices.A lattice is a separated ordered set X with finite infima and finite suprema. It may beequivalently described as a set X with binary operations ∧, ∨, and nullary operations >,⊥ such that both (X,∧,>), (X,∨,⊥) are commutative monoids such that

x ∧ x = x = x ∨ x , x ∧ (x ∨ y) = x = x ∨ (x ∧ y) .

A homomorphism of lattices is a map that preserves the operations ∧, ∨, >, ⊥.A frame is a complete meet-semilattice X such that, for all a ∈ X,

a ∧ (−) : X −→ X


is a sup-map. Hence, a frame is simply a complete lattice (that is, a complete separatedordered set) which satisfies the infinite distributive law

a ∧∨

i∈Ibi =

∨i∈I

a ∧ bi .

In particular, each complete chain is a frame (see Exercise 1.E). A homomorphism of framesmust be both a homomorphism of meet-semilattices, and a sup-map; i.e., it must preservefinite infima and arbitrary suprema.For example, if X is an ordered set, then DnX is a frame, and every monotone mapf : X −→ Y induces a homomorphism of frames f−1(−) : DnY −→ DnX (see Exercise 1.K).This applies in particular when X is discrete, that is, when its order is given by equality, sothat DnX = PX.A topology (of open sets) on X is nothing but a subframe of PX, that is, a subset of PXthat is closed under finite infima and arbitrary suprema, making X a topological space;the topology of X is usually denoted by OX. A map f : X −→ Y of topological spaces iscontinuous if f−1(−) : PY −→ PX restricts to a map f−1(−) : OY −→ OX. A base for OXis a subset B ⊆ PX with OX = ⋃ C | C ⊆ B. Every topological space is a closure space,and the notions of continuity given here and in 1.6 are equivalent (Exercise 1.F). Everytopological space can be endowed with its underlying (or induced) order

x ≤ y ⇐⇒ ∀U ∈ OX (y ∈ U =⇒ x ∈ U) .

This order is separated precisely when X is a T0-space. Every continuous map is monotonewith respect to the underlying orders. The dual of this order is called the specializationorder of a topological space.

1.10 Quantales. A quantale V (more precisely: a unital quantale) is a complete latticewhich carries a monoid structure with neutral element k (as in 1.1) such that, when thebinary operation is denoted as a tensor ⊗,

a⊗ (−) : V −→ V , (−)⊗ b : V −→ V

are sup-maps for all a, b ∈ V ; hence the tensor distributes over suprema:

a⊗∨i∈I

bi =∨i∈I

(a⊗ bi) ,∨

i∈Iai ⊗ b =

∨i∈I

(ai ⊗ b) .

A lax homomorphism of quantales f : V −→ W is a monotone map satisfying

f(a)⊗ f(b) ≤ f(a⊗ b) , l ≤ f(k)

for all a, b ∈ V , and l the neutral element of W ; monotonicity of f means equivalently laxpreservation of joins, that is ∨ f(A) ≤ f(∨A) for all A ⊆ V . For f to be a homomorphism,these three inequalities must be identities. A quantale is commutative if it is commutativeas a monoid. Every frame becomes a commutative quantale when we put ⊗ = ∧ and let k be

1. ORDERED SETS 31

the top element. In fact, frames are those commutative quantales V for which a⊗ a = a forall a ∈ V , and k is the top element (see Exercise 1.L). If V and W are frames (consideredas quantales), then a lax homomorphism of quantales f : V −→ W is a homomorphismprecisely when it is a sup-map. In a quantale V, for every a ∈ V, the sup-map a⊗ (−) isleft adjoint to a map a (−) : V −→ V which is uniquely determined by

a⊗ v ≤ b ⇐⇒ v ≤ a b

for all v, b ∈ V ; hencea b =

∨v ∈ V | a⊗ v ≤ b .

Likewise, for all a ∈ V , the sup-map (−)⊗ a is left adjoint to a map (−) a : V −→ V . Inthe case where V is commutative, a (−) and (−) a coincide, and either of the twonotations may be used.The following examples of commutative quantales are frequently used in this book. Forexamples of not necessarily commutative quantales, see Exercise 1.M.

1.10.1 Examples.

(1) The two-chain 2 = false |= true = ⊥,> with ⊗ = ∧, k = >. Here, a b isthe Boolean truth value of the implication a −→ b. More generally, we use this arrownotation for any frame considered as a quantale.

(2) The three-chain 3 = ⊥, k,> is a complete chain, and therefore a frame (see Ex-ercise 1.E). However, the quantale structure we will consider is given by choosingk to be the unit for the multiplication. In fact, the multiplication is now uniquelydetermined, and it is represented together with the right adjoints (−) a, for a ∈ 3,in the following tables:

⊗ ⊥ k >⊥ ⊥ ⊥ ⊥k ⊥ k >> ⊥ > >

⊥ k >⊥ > ⊥ ⊥k > k ⊥> > > >

(3) Allowing for an interval of truth values, we consider the extended real half-line [0,∞]which is a complete lattice with respect to its natural order ≤. But we reverse itsorder, so that 0 = > is the top and ∞ = ⊥ is the bottom element, and we consider ita quantale with ⊗ given by addition extended via

a+∞ =∞+ a =∞

for all a ∈ [0,∞], and necessarily k = 0 = >. Briefly, we will write

P+ = ([0,∞]op, + , 0) .


When working with this quantale, the relation ≤ always refers to the natural orderof [0,∞], and we use the symbols inf and sup when forming infima and suprema in[0,∞], while we use joins ∨ and meets ∧ when forming these in P+. Hence,

b a = b a := infv ∈ [0,∞] | b ≤ a+ v ,

so that b a = b − a if a ≤ b < ∞, and b a = 0 if b ≤ a, while b a = ∞ ifa < b =∞.

(4) In the previous example, addition may be replaced by multiplication extended via

a · ∞ =∞ · a =∞

for all a ∈ [0,∞] (this definition is necessary, since the tensor must preserve the emptyjoin, i.e., the bottom element ∞). Hence, we obtain the quantale

P× = ([0,∞]op, · , 1) .

Here,b a = b a := infv ∈ [0,∞] | b ≤ a · v ,

so that b a = ba

if 0 < a, b < ∞, and b 0 = ∞ = ∞ a if 0 < a, b, while0 a = 0 = b∞ for all a, b ∈ [0,∞].

(5) Since [0,∞]op is (like [0,∞]) a chain, it is a frame, and we may consider it a quantalewith its meet operation (which, according to our conventions, is the max with respectto the natural order of [0,∞]):

Pmax = ([0,∞]op,max, 0) .

Herea b = a −→ b := infv ∈ [0,∞] | b ≤ maxa, v ,

which turns out to be a −→ b = 0 if b ≤ a, and a −→ b = b if a < b.

There is a sup-map ι : 2 −→ [0,∞]op, sending > to 0, and ⊥ to ∞, which gives homomor-phisms of quantales

ι : 2 −→ P+ and ι : 2 −→ Pmax .

However,ι : 2 −→ P×

is only a lax homomorphism, since ι does not preserve the neutral elements of the respectivemonoid structures. This can be corrected if one replaces 2 by 3; then the sup-map

κ : 3 −→ P×

sending > to 0, k to 1, and ⊥ to ∞, is a homomorphism of quantales.

1. ORDERED SETS 33

1.11 Complete distributivity. As we saw in 1.7, completeness of the ordered set X ischaracterized by the existence of an adjunction∨ a ↓ : X −→ DnX .

We call a complete lattice completely distributive if the left adjoint ∨ has itself a left adjoint,i.e., if there is a map

⇓ : X −→ DnXwith

⇓ a ⊆ S ⇐⇒ a ≤ ∨Sfor all a ∈ X, S ∈ DnX. Necessarily,

⇓ a =⋂S ∈ DnX | a ≤ ∨S ,

so that when we write x a (read as: x is totally below a) instead of x ∈ ⇓ a, this relationis given by

x a ⇐⇒ ∀S ∈ DnX (a ≤ ∨S =⇒ x ∈ S) .Writing S = ↓A with A ⊆ X, an equivalent characterization is given by

x a ⇐⇒ ∀A ⊆ X (a ≤ ∨A =⇒ ∃y ∈ A (x ≤ y)) .

Since ⇓ is a monotone map with 1X ≤∨ · ⇓, one has for all a, b, x ∈ X:

(1) if x a ≤ b, then x b;

(2) a ≤ ∨x ∈ X | x a.

In (2), one actually has equality (so that a = ∨⇓ a): from a ≤ ∨ ↓ a follows ⇓ a ⊆ ↓ a byadjunction, so that ∨⇓ a ≤ a. We also note that by the very definition of , every elementin X is -atomic in the sense that

x ∨S =⇒ x ∈ S

for all S ∈ DnX, so that in (2) the join is taken “only” over -atomic elements. Keepingthis in mind, we see that the existence of the relation with properties (1) and (2) ischaracteristic of complete distributivity:

1.11.1 Proposition. If the complete lattice X allows for some relation ≺ satisfying

(1′) if x ≺ a ≤ b, then x ≺ b, and

(2′) a ≤ ∨x ∈ X | x is ≺-atomic and x ≺ a

for all a, b, x ∈ X, then X is completely distributive.


Proof. Indeed, assuming these conditions, and setting

Sa := ↓x ∈ X | x is ≺-atomic and x ≺ a ,

one has a ≤ ∨Sa by (2′), and whenever a ≤ ∨

S for some S ∈ DnX, then every x ∈ Sasatisfies x ≺ ∨S by (1′), and therefore lies in S. Consequently,

Sa =⋂S ∈ DnX | a ≤ ∨S = ⇓ a ,

as desired.

The complete lattices 2 and [0,∞] considered in Examples 1.10.1 are completely distributive,with the totally-below relation given by (x a ⇐⇒ a = true) and (x a ⇐⇒ x < a)respectively. For every set X, the ordered set PX is completely distributive; more generally,for every ordered set X, the ordered set DnX is completely distributive (see Exercise 1.O).We note that [0,∞]op is also completely distributive (with “totally-below” meaning “>”);in fact, one can prove that every chain is completely distributive and that Xop is completelydistributive whenever X has that property (see [Wood, 2004]).The term “completely distributive” still deserves some justification. By Corollary 1.8.3, themap ∨ : DnX −→ X is right adjoint if it is an inf-map:∨

(⋂

i∈ISi) =

∧i∈I

∨Si

for all families of down-sets Si in X, i ∈ I; equivalently, if∨(⋂i∈I↓Ai) =

∧i∈I

∨Ai (1.11.i)

for all families of subsets Ai in X, i ∈ I. Now, assuming the Axiom of Choicec© , the latteridentity may be written equivalently as∨

(ai)∈ΠIAi

∧i∈I

ai =∧

i∈I

∨Ai . (1.11.ii)

Indeed, denoting by s, t the left sides of (1.11.i), (1.11.ii) respectively, t ≤ s follows bynoticing that (ai)i∈I ∈

∏i∈I Ai yields ∧i∈I ai ∈ ⋂i∈I ↓Ai. Conversely, given x ∈ ⋂i∈I ↓Ai,

for every i ∈ I, one has ai ∈ Ai with x ≤ ai; by the Axiom of Choice, this defines anelement (ai)i∈I ∈

∏i∈I Ai with x ≤ ∧i∈I ai, and s ≤ t follows.

Most authors define complete distributivity via (1.11.ii), and reserve the name constructivelycompletely distributive for the Choice-free notion given by (1.11.i), that is, by the rightadjointness of ∨.

1.12 Directed sets, filters, and ideals. A subset A ⊆ Z of a separated ordered set Zis down-directed if every finite subset of A has a lower bound in A; this just means thatfor all x, y ∈ A, there is a z ∈ A with z ≤ x, z ≤ y, and that A 6= ∅. A subset A ⊆ Z is afilter in Z if A is down-directed and up-closed in Z; a filter A is proper if A 6= Z. When Z

1. ORDERED SETS 35

has finite infima, filters A in Z are up-closed meet-semilattices of Z; that is, the filters arethose A which satisfy

(1) x, y ∈ A =⇒ x ∧ y ∈ A,

(2) > ∈ A, and

(3) x ∈ A, x ≤ y =⇒ y ∈ A

for all x, y ∈ Z. If Z has a bottom element ⊥, properness then means

(4) ⊥ /∈ A.

Every down-directed set A in Z generates the filter ↑A in Z, in which case A is also calleda filter base for ↑A. In particular, for every element a ∈ Z, one has the principal filter↑ a in Z. The dual notions are those of up-directed set, ideal, proper ideal, ideal base andprincipal ideal.We use these notions predominantly for Z = PX (for some set X), ordered by inclusion.Hence, a is a filter on the set X if it is a filter in PX, that is, a must satisfy

(1′) A,B ∈ a =⇒ A ∩B ∈ a,

(2′) X ∈ a,

(3′) A ∈ a, A ⊆ B =⇒ B ∈ a,

and a is proper when

(4′) ∅ /∈ a.

Here are some frequently used filter-generation procedures:

(a) For every map f : X −→ Y and every filter a on X, one defines the image filter f [a] onY by

f [a] = ↑f(A) | A ∈ a = B ⊆ Y | f−1(B) ∈ a .For a filter b on Y , one defines the inverse image f−1[b ] by

f−1[b ] = ↑f−1(B) | B ∈ b = A ⊆ X | ∃B ∈ b (f−1(B) ⊆ A) ,

but this filter on X is proper only when f−1(B) 6= ∅ for all B ∈ b . If f is an inclusionmap X → Y , then

f−1[b ] = X ∩B | B ∈ b ⇐⇒ X ∈ b ;

when X ∈ b one calls b |X := f−1[b ] the restriction of b to X, and says that b is afilter on X. In fact, b is then determined by b |X , since ↑ f [b |X ] = b .


(b) For every A ⊆ X, one has the following principal filter on X:

A = ↑A = B ⊆ X | A ⊆ B .

Obviously, for f : X −→ Y , f [↑A] = ↑ f(A).

(c) For a set X, let us consider a filter A on the set

FX = a | a is a filter on X .

Then the filtered sum (or Kowalsky sum) ∑A, defined by

A ∈ ∑A ⇐⇒ AF ∈ A ,

for all A ⊆ X, where AF = a ∈ FX | A ∈ a, gives a filter on X. Hence, a set A ⊆ Xlies in ∑A precisely when the set of those filters on X that are actually filters on A,lies in A.

1.13 Ultrafilters. An ultrafilter x on a set X is a maximal element within the set ofproper filters on X, ordered by inclusion; that is, x is a proper filter on X such that if a isa proper filter on X with x ⊆ a, then x = a. A handier characterization is the following:

1.13.1 Lemma. For a proper filter x on X, the following statements are equivalent:

(i) x is an ultrafilter on X;

(ii) for all A,B ⊆ X, if A ∪B ∈ x then A ∈ x or B ∈ x ;

(iii) for every subset A ⊆ X, one has A ∈ x or A ∈ x (where A = X \ A denotes thecomplement of A in X).

Proof. (i) =⇒ (ii): if A∪B ∈ x but A /∈ x (for some A,B ⊆ X), then x ( ↑A∩C | C ∈ x ,so by maximality of x , the right-hand side filter cannot be proper; hence, A ∩ C = ∅ forsome C ∈ x , and (A∪B)∩C = B∩C ∈ x , so B∩C ⊆ B tells us that B ∈ x . (ii) =⇒ (iii):Immediate from X = A ∪ A ∈ x . (iii) =⇒ (i): x ( a implies that there is an A ∈ a withA /∈ x ; thus, A ∈ x ( a and A ∩ A = ∅ ∈ a which therefore is not proper.

The filter generation processes described in 1.12 may be specialized to ultrafilters; moreprecisely:

(a) For a map f : X −→ Y and an ultrafilter x on X, the image f [x ] is also an ultrafilteron Y . When f : X → Y is an inclusion map, and y an ultrafilter on Y with X ∈ y ,then y |X is an ultrafilter on X.

(b) For every x ∈ X, the principal filter x = ↑x is an ultrafilter on X.

1. ORDERED SETS 37

(c) If X is an ultrafilter on the set

βX = x | x is an ultrafilter on X ,

then ∑X is an ultrafilter on X.

For the creation of other ultrafilters, one resorts to the Axiom of Choice:

1.13.2 Proposition. c©Every proper filter a on X is contained in an ultrafilter x on X.

Proof. This is guaranteed by an easy application of Zorn’s Lemma, see Exercise 1.P.

In fact, this statement can be used to formulate a formally finer assertion:

1.13.3 Corollary. c©For a filter b and a proper filter a on X such that a ( b , there is anultrafilter x on X with a ⊆ x but b 6⊆ x .

Proof. Indeed, for some B ∈ b with B /∈ a, one considers the filter

a ′ = ↑B ∩ A | A ∈ a ,

which is proper since B ∩ A = ∅ would imply A ⊆ B ∈ a. So there is an ultrafilter xcontaining a ′, and therefore also a; as B ∈ x , we must have B /∈ x .

As an important consequence, we obtain:

1.13.4 Corollary. c©Every filter a on X is the intersection of all ultrafilters on X containinga.

Proof. We must show that every filter a ∈ FX may be obtained as

a =⋂x ∈ βX | a ⊆ x .

This equality holds trivially when ∅ ∈ a, and the inclusion “⊆” is also immediate. Moreover,when a is proper, so is the filter b obtained on the right-hand side; therefore, if a ( b , thereexists an ultrafilter x with a ⊆ x but b ( x , contradicting the definition of b .

One obtains alternatively the following result.

1.13.5 Corollary. c©For a proper filter a and an ideal j on X such that a ∩ j = ∅, there isan ultrafilter x with a ⊆ x and x ∩ j = ∅.

Proof. Since a is an up-set, the fact that a is disjoint from j translates as A * J forall A ∈ a and J ∈ j , or equivalently as A ∩ J 6= ∅ for all A ∈ a, J ∈ j . Thus,b := A∩J | A ∈ a, J ∈ j is a proper filter containing a, and Proposition 1.13.2 yields theexistence of an ultrafilter x with b ⊆ x , and consequently a ⊆ x . If there was J ∈ j ∩ x ,one would conclude J /∈ x , a contradiction.


1.14 Natural and ordinal numbers. We end this introductory section with somefoundational remarks. A natural numbers object for sets is a set N with a distinguishedelement 0 and a map s : N −→ N such that, for any set X equipped with a map t : X −→ Xand an element a ∈ X, there is a unique map f making the diagram

0_

a

N

f

s // N

f

Xt // X

(1.14.i)

commutative. Such a set N must necessarily have the form

N = 0 ∪ s(N) ,

and N = (N, s, 0) is uniquely determined up to a unique compatible bijection. Briefly, N ischaracterized by the requirement to allow for inductive definitions, via

f(0) = a , f(s(n)) = t(f(n)) (n ∈ N) .

Alternatively, in categorical language (as introduced in 2.7) N is initial amongst all generalalgebras with one nullary and one unary operation and no other requirements.Defining the sets Nn recursively by

N0 = 0 , Ns(n) = Nn ∪ s(n) (n ∈ N) ,

one may define the natural order on N by

n ≤ m ⇐⇒ Nn ⊆ Nm .

It is the only order that makes N a chain with n ≤ s(n) for all n ∈ N. Let us now assumethat X in (1.14.i) is ordered and t is pointed, in the sense that x ≤ t(x) for all x ∈ X. Thenthe recursively defined function f is monotone. Consequently, N is also initial amongstordered general algebras with a nullary and unary operation that is pointed. No such objectexists within the realm of finite sets or algebras. In other words, the inductive conditionmay also be seen as an infinity axiom.Missing a top element, N fails to be complete. It is therefore natural to ask: is therea separated complete ordered set O with a distinguished element 0 ∈ O and a pointedoperation s : O −→ O such that, for any separated complete ordered set X with adistinguished element a ∈ X and pointed operation t : O −→ O, there is a unique sup-mapf : O −→ X with f(0) = a and t(f(α)) = f(s(α)) for all α ∈ O? Naively, such set shouldhave the form

O = 0, s(0), s(s(0)), . . . , ω = supn<ω n, s(ω), s(s(ω)), . . . ,

with a considerably increased degree of uncertainty of how to “continue” the description ofits elements, in comparison to the description of those of N. In fact, just like the fact that a

1. ORDERED SETS 39

natural numbers object may not be found within the realm of finite sets, the desired objectO would be “too big” to be a set. But regardless of which foundational requirements onemay adopt to govern the use of “sets”, it is reasonable to require the existence of a class Owith a separated complete order (so that it has all set-indexed suprema), a distinguishedelement 0 and a pointed unary operation that is initial amongst all classes X structuredby a and t in the same way; in other words, such that maps f : O −→ X may be uniquelydefined by ordinal recursion, via

f(0) = a , f(s(α)) = t(f(α)) , f(supi∈I αi) = supi∈I f(αi)

for all α, αi ∈ O, i ∈ I, and set I.We will apply this ordinal recursion principle only twice in this book, for the proof of themain result of III.4.2 and for Proposition V.4.4.9, with rather limited impact on otherresults.

Exercises

1.A Universal property of the separated reflection. Every monotone map f : X −→ Y ofordered sets, with Y separated, factors through the separated reflection p : X −→ X/' by auniquely determined monotone map g : X/' −→ Y , so that f = g · p.

1.B Adjunctions and fully faithful maps. For an adjunction f a g : Y −→ X, the map g isfully faithful if and only if f · g ' 1Y . If h a f a g, then g is fully faithful if and only if h isfully faithful.

1.C Adjunctions for free. For a module r : X −→7 Y and an ordered set Z, denote by rZ themap

Mod(Z,X) −→ Mod(Z, Y ) , t 7−→ r · t ,where Mod(Z,X) denotes the ordered set of modules Z −→7 X. With a module s : Y −→7 X,show

∀Z (rZ a sZ) ⇐⇒ 1∗X ≤ s · r & r · s ≤ 1∗Y .

Conclude that for every monotone map f : X −→ Y and every ordered set Z, one has anadjunction (f∗)Z a (f ∗)Z .

1.D Closure operations on complete ordered sets. For every closure operation c on anordered set X, there is an adjunction e a j : Fix(c) → X. If X is complete, so is Fix(c),with infima in Fix(c) formed as in X while the supremum of A ⊆ Fix(c) in Fix(c) is givenby c(∨A).

1.E Complete chains are frames. For elements a, b in a chain X, put

a −→ b :=

> if a ≤ b, andb otherwise.


Thenx ∧ a ≤ b ⇐⇒ x ≤ (a −→ b) .

Hence, every complete chain is a frame.

1.F Topological spaces as closure spaces. Every topological space X becomes a closurespace via its Kuratowski closure operation:

A =⋂B ⊆ X | A ⊆ B,B ∈ OX = x ∈ X | ∀U ∈ OX (x ∈ U =⇒ A ∩ U 6= ∅)

for all A ⊆ X. The map (−) : PX −→ PX is not only a closure operation, but is alsofinitely additive:

A ∪B = A ∪B , ∅ = ∅

for all A,B ⊆ X. Hence, the map (−) is a monoid homomorphism of the join-semilatticePX. A map f : X −→ Y between topological spaces is continuous if and only if it iscontinuous as a map of closure spaces. The underlying order may then be written as

x ≤ y ⇐⇒ y ∈ x .

Of course, since closure spaces and interior spaces are equivalent concepts, every topologicalspace also becomes an interior space (whose interior operation preserves finite intersections).

1.G Closure spaces as topological spaces. For a set X, any finitely additive closure operationc on PX defines the topology

A ⊆ X | c(A) = A

on X, and this establishes a bijective correspondence between topologies on X and finitelyadditive closure operations on PX. The bijective correspondence between topologies on Xand interior operations on PX that preserve finite intersections is even more immediate, asit avoids the use of (−).

1.H A simple non-linear quantale. The diamond lattice 22 = ⊥, u, v,>, with u and v twoincomparable elements, is a frame, and as such is obviously isomorphic to the powerset of 2.Compute a b for all a, b ∈ 22.

1.I Lax homomorphisms from the extended real half-line. There are uniquely determinedmonotone maps o, p : [0,∞]op −→ 2 with

o a ι a p ,

where ι(⊥) =∞ and ι(>) = 0. For the “optimist’s map”, one has o(v) = > for all v <∞,while for the “pessimist’s map”, one has p(v) = > only when v = 0. When [0,∞]op isconsidered as a quantale P+, P×, or Pmax, which of these maps turn into lax homomorphisms,and among these, which are in fact homomorphisms?

1. ORDERED SETS 41

1.J Right and left adjoints to DnX → PX. For an ordered setX, the inclusion DnX → PXhas a left adjoint given by the down-closure ↓, and a right adjoint given by the down-interior↓, where

↓A = x ∈ X | ↓x ⊆ Afor A ⊆ X. In particular, DnX is a complete lattice with meets and joins formed as in PX.

1.K Right and left adjoints to f−1(−) : DnY −→ DnX. For a monotone map f : X −→ Y ofordered sets, the map f−1(−) : DnY −→ DnX has a left adjoint f! given by f!(A) = ↓ f(A),and a right adjoint f∗ given by f∗(A) = ↓∨B ⊆ Y | f−1(B) ⊆ A for all A ⊆ X. Inparticular, f−1(−) is a homomorphism of frames.

1.L Semilattices as monoids, frames as quantales. Let X be a separated ordered set whichis also a (multiplicatively written) commutative monoid in which the multiplication ismonotone in each variable. If every element in the monoid X is idempotent, and the neutralelement is the top element, then the ordered set X is a meet-semilattice: the order and themonoid structure determine each other via

(x ≤ y ⇐⇒ x · y = x) , and x · y = x ∧ y .

In particular, for a commutative quantale (V ,⊗, k) with k = > and v ⊗ v = v for all v ∈ V ,one has ⊗ = ∧.

1.M Quantales arising from monoids. For a monoid M = (M,m, e), the powerset PMbecomes a monoid with

B · A = y · x | x ∈ A, y ∈ B (A,B ⊆M)

and neutral element e. Ordered by inclusion, PM is a quantale which is commutativeprecisely when M is commutative. For a homomorphism f : M −→ N of monoids,f(−) : PM −→ PN is a homomorphism of quantales.

1.N Complete lattices as frames. A complete lattice X is a frame if and only if ∨ : DnX −→X is a meet-semilattice homomorphism. Every completely distributive lattice is a frame.

1.O Complete distributivity of DnX. For every ordered set X, DnX is completely distribu-tive. In particular, the powerset PX of a set X is completely distributive.Hint. The join map of DnX coincides with (↓X)−1(−) : DnDnX −→ DnX (where ↓X :X −→ DnX), and by Exercise 1.K it has the left adjoint (↓X)!.

1.P Zorn’s Lemma and ultrafilters. c©Zorn’s Lemma, which is in fact equivalent to the Axiomof Choice (for an elementary account, see for example [Davey and Priestley, 1990]), states:

If X is a separated ordered set and every chain A ⊆ X has an upper bound inX, then X has a maximal element, that is, an element x ∈ X such that if y ∈ Xsatisfies x ≤ y, then x = y.


As a consequence, every proper filter a on a set X is contained in an ultrafilter x on X.

1.Q Convergence of sequences. Given a sequence s : N −→ X in X (also denoted bys = (xn)n∈N), one can consider the sets Sn := xm ∈ X | n ≤ m, and the filter associatedto s

〈s〉 := ↑PXSn | n ∈ N .If X is a topological space (with topology OX), then the neighborhood filter at x ∈ X isthe filter

ν(x) := ↑PXU ∈ OX | x ∈ U .A neighborhood of x ∈ X is an element of ν(x). Show that the open sets of X are exactly thesubsets V that are neighborhoods of each of their points, and that a sequence s convergesto a point x, that is,

∀V ∈ ν(x)∃n ∈ N∀m ≥ n (xm ∈ V ) ,if and only if 〈s〉 ⊇ ν(x). Similarly, if s′ = (xni)i∈N is a subsequence of s (that is, 〈s′〉 ⊇ 〈s〉),and s converges to x, then s′ also converges to x. Finally, the constant sequence sx : N −→ Xat x (so that sx(N) = x) converges to x for any topology on X.

1.R Continuity and sequences. The image of a sequence s : N −→ X under a map f : X −→ Yis the sequence f · s : N −→ Y , or equivalently the sequence (f(xn))n∈N (where s = (xn)n∈N).The filter associated to f · s is then the image filter of 〈s〉:

f [〈s〉] = 〈f · s〉 .

If f : X −→ Y is a continuous map between topological spaces X and Y and a sequence sconverges to x ∈ X (in symbols 〈s〉 −→ x), then f · s converges to f(x):

〈s〉 −→ x =⇒ f [〈s〉] −→ f(x) ;

in other words, continuous maps preserve convergence of sequences.

1.S Sequences do not suffice. In general, topological concepts are not faithfully representedby convergence of sequences. For example, consider the real line R equipped with thetopology given by complements of countable subsets:

OR := U ⊆ R | U is countable .

In this topology, a sequence s = (xn)n∈N converges to a point x if and only if s is eventuallyconstant:

〈s〉 −→ x ⇐⇒ ∃n ∈ N∀m ≥ n (xm = x) .The identity map 1R : (R,OR) −→ (R, PR) preserves convergence of sequences, but is notcontinuous (the converse statement is true however, see Exercise 1.R).

2. CATEGORIES AND ADJUNCTIONS 43

2 Categories and adjunctions

2.1 Categories. A small category C is given by a set ob C of objects of C, a map homCwhich assigns to each pair of objects (A,B) a set homC(A,B), called their hom-set andmore briefly written as C(A,B), as well as composition and identity operations:

C(A,B)× C(B,C)−→ C(A,C) , ?−→ C(A,A)(f, g) 7−→ g · f ? 7−→ 1A

subject to the associativity and right and left identity laws

h · (g · f) = (h · g) · f , f · 1A = f = 1B · f

for all f ∈ C(A,B), g ∈ C(B,C), h ∈ C(C,D), A,B,C,D ∈ ob C. Given objects A, B, onewrites

f : A −→ B

instead of f ∈ C(A,B), and calls A = dom f the domain, and B = cod f the codomainof the morphism (or arrow) f in C, with the understanding that two morphisms f, g cancoincide only if

dom f = dom g and cod f = cod g .

The morphism f : A −→ B in C is an isomorphism of C if f ′ · f = 1A and f · f ′ = 1B for a(necessarily uniquely determined) morphism f ′ : B −→ A in C. One has the self-explanatoryrules

(g · f)−1 = f−1 · g−1 , 1−1A = 1A , (f−1)−1 = f .

An equivalence relation on ob C is defined by writing A ∼= B if there is an isomorphismf : A −→ B in C. An order on ob C is obtained by setting A ≤ B if there exists a morphismf : A −→ B. In this case, A ∼= B implies A ' B (meaning A ≤ B and B ≤ A, as in 1.3),but the converse is not true in general.In Section 1, we encountered two important types of small categories. First, every monoidM becomes a one-object category C when one puts C(?, ?) = M (with ? denoting theonly object of C) and interprets monoid operation as composition. In fact, monoids canbe thought of as precisely the one-object small categories, and small categories are simply“multi-object monoids”. Second, every ordered set X becomes a small category C withob C = X and C(x, y) = ? if x ≤ y in X, and C(x, y) = ∅ otherwise; the compositionand identity operations appear as the transitivity and reflexivity properties in X. Notethat in this case, the order on C defined above coincides with that of X. This shows thatordered sets can be thought of as precisely those small categories for which the hom map is∅, ?-valued, and small categories are simply sets with “multi-valued structured orders”.Arbitrary categories are defined just like small categories, but one allows ob C and C(A,B)to be classes rather than just sets (for all A,B ∈ ob C). The understanding here is that a


set is a particular class, and that we are allowed to form the class of all sets (see I.3). Forevery category C, the opposite or dual category Cop of C is given by

ob Cop = ob C , Cop(A,B) = C(B,A) , and g ·Cop f = f ·C g .

Here are some examples of categories that we encountered (implicitly) in Section 1:

Set: the category whose objects are sets, and whose morphisms are maps with ordinarycomposition of maps.

Rel: objects are sets, but morphisms are relations with ordinary relational composition.

Ord: ordered sets, with monotone maps and ordinary map composition.

Mod: ordered sets, with modules and ordinary relational composition.

Ordsep: separated ordered sets, with monotone maps and, as in all the following examples,ordinary map composition. A more common name for this category is PoSet, whichstands for “partially ordered sets”.

Sup: complete lattices, with sup-maps.

Inf: complete lattices, with inf-maps.

Mon: monoids, with monoid homomorphisms.

SLat: semilattices, with semilattice homomorphisms.

Lat: lattices, with lattice homomorphisms.

Frm: frames, with frame homomorphisms.

Qnt: quantales, with quantale homomorphisms.

Top: topological spaces, with continuous maps.

Cls: closure spaces, with continuous maps.

Int: interior spaces, with continuous maps.

2.2 Functors. A functor F : C −→ D from a category C to a category D is given byfunctions

F : ob C −→ ob D , and FA,B : C(A,B) −→ D(FA, FB)for all A,B ∈ ob C, with (when writing Ff = FA,B(f))

F (g · f) = Fg · Ff , F1A = 1FA

for all f : A −→ B, g : B −→ C in C. There is an obvious identity functor 1C : C −→ C,and the composition GF of two functors F : C −→ D and G : D −→ E is defined by using


composition of functions. The functor F is faithful if all functions FA,B are injective, and itis full if these functions are surjective. Fully faithful functors (that is, functors that areboth full and faithful) typically arise when forming a full subcategory D of a category C,so that ob D ⊆ ob C, D(A,B) = C(A,B) for all A,B ∈ ob D, and composition in D is as inC; then the inclusion functor F : D → C is full and faithful. Here, the functor F has theadditional property that its object function is injective, which makes F a full embedding.Small categories and functors form a category Cat. With suitable foundational provisions,we may also form the metacategory CAT of all categories and functors.Functors between monoids (considered as categories) are precisely monoid homomorphisms,and functors between ordered sets are precisely monotone maps; in this last case, the“fully faithful” terminology coincides with the one introduced in 1.3 for ordered sets. Manyfunctors also arise as forgetful functors, forgetting properties or structures of objects andmorphisms, such as in the sequence

Qnt −→ Sup −→ Ord −→ Set .

Some other functors that we implicitly encountered in Section 1 include:

· The covariant powerset functor P : Set −→ Set, with Pf(A) = f(A) for f : X −→ Y ,A ⊆ X.

· The contravariant powerset functor P • : Setop −→ Set, with P •f(B) = f−1(B) forf : X −→ Y , B ⊆ Y .

· The down-set functor Dn : Ord −→ Ord, with Dnf(A) = ↓ f(A) for f : X −→ Y , A ⊆ X(see 1.7).

· The open-set functor O : Topop −→ Frm (see 1.9).

· The underlying-order functor Top −→ Ord (see 1.9).

· The Kuratowski-closure functor Top −→ Cls (see Exercise 1.F).

· The functor Cls −→ Int arising from two-fold complementation (see 1.6), which is anisomorphism in CAT.

· The dualization functor (−)op : Ord −→ Ord (see 1.3). Since (Xop)op = X, this functor isan isomorphism (in CAT), and extends to an isomorphism (−)op : CAT −→ CAT (see 2.1).It maps a functor F : C −→ D to F op : Cop −→ Dop, where F opf = Ff . A restriction of thedualization functor yields an isomorphism Inf −→ Sup in CAT.

· The adjunction functor Supop −→ Inf which maps objects identically and assigns to asup-map its right adjoint (see Corollary 1.8.3); again this is an isomorphism in CAT.Composition with the isomorphism Inf −→ Sup shows that Sup is self-dual: Supop ∼= Sup;likewise for Inf.


· The module functors (−)∗ : Ord −→ Mod and (−)∗ : Ordop −→ Mod which map objectsidentically and assign to a monotone map f the induced modules f∗ and f ∗, respectively.

For every object A in a category C, one has the covariant hom-functor of A

C(A,−) : C −→ SET

(where the objects of the metacategory SET are classes) with

C(A,−)(B) := C(A,B) ,C(A,−)(g) := C(A, g) : C(A,B) −→ C(A,C) , f 7−→ g · f

for all f : A −→ B, g : B −→ C in C. Similarly, one has the contravariant hom-functor of A

C(−, A) = Cop(A,−) : Cop −→ SET .

These functors take value in Set if C is locally small, i.e., if all C(A,B) are sets. Note thatwhen C is a monoid M seen as a one-object category, then the hom-functor M(?,−) : M −→Set has as values those maps M −→M which are the “left translations by a” (a ∈M); witha suitable codomain restriction, M(?,−) is in most algebra books referred to as the Cayleyrepresentation of M , although M is normally assumed to be a group (that is, a monoid inwhich every element is invertible). When C is an ordered set X and x ∈ X, then X(x,−)is two-valued and therefore the characteristic function of a subset of X, namely ↑X x.

2.3 Natural transformations. Let F,G : C −→ D be functors. A natural transformationα : F −→ G, also depicted by

CF

))⇓αG

44 D ,

is given by a family of morphisms αA : FA −→ GA in D (with A running through ob C),making the naturality diagrams

FA

αA

Ff// FB

αB

GAGf// GB

commute for all f : A −→ B in C; hence, writing αf : FA −→ GB for the diagonal morphism,we have

αf = Gf · αA = αB · Ff .

For example, forming singleton sets is a natural transformation 1Set −→ P (where P is thepowerset functor), and forming unions is a natural transformation PP −→ P . The down-setfunction defines a natural transformation ↓ : 1Ord −→ Dn, and when we restrict Dn to afunctor Dn : Sup −→ Sup, there is a natural transformation ∨ : Dn −→ 1Sup.


Vertical pasting of naturality diagrams defines the vertical composition

β · α : F −→ H with (β · α)A = βA · αA (A ∈ ob C) ,

for α : F −→ G, β : G −→ H, and F,G,H : C −→ D; one also has the identity transformation

1F : F −→ F with (1F )A = 1FA (A ∈ ob C) .

The vertical composition is trivially associative, and makes the functors from C to D theobjects of the functor category DC whose morphisms are natural transformations. (Wenote that unless C is small, DC is actually a metacategory.) For functors S : B −→ C andT : D −→ E, one has the whiskering functors

(−)S : DC−→ DB T (−) : DC−→ EC

(α : F −→ G) 7−→ (αS : FS −→ GS) and (α : F −→ G) 7−→ (Tα : TF −→ TG)

which map natural transformations via

(αS)B = αSB , (Tα)A = TαA (B ∈ ob B, A ∈ ob C) ,

as suggested by

B S // CF

))⇓α

G

55 D T // E .

These whiskering functors are of course induced by either CAT(−,D) or CAT(C,−) appliedto S : B −→ C or T : D −→ E, respectively. In fact, whiskering by a functor (from the left orthe right) is just an instance of the horizontal composition for natural transformations

CF

))⇓α

G

55 DK

((⇓γ

L

66 E

defined byγ α : KF −→ LG , (γ α)A = γαA (A ∈ ob C) .

For a morphism f in C, one obtains

(γ α)f = γαf .

Moreover, since

(γ α)A = LαA · γFA = γGA ·KαA= (Lα)A · (γF )A = (γG)A · (Kα)A ,

the natural transformation γ α is completely determined by the values of the whiskeringfunctors on α and γ:

γ α = Lα · γF = γG ·Kα .


Equivalently, since

γ α = (1L α) · (γ 1F ) = (γ 1G) · (1K α) , (2.3.i)

the horizontal composition is determined by the special instances of composition withidentity transformations. The equality (2.3.i) is itself a special instance of the middle-interchange law; that is, given

CF##α⇓

G //

β⇓H

;;DK

""γ⇓L //

δ⇓M

<< E

one has(δ · γ) (β · α) = (δ β) · (γ α) .

The horizontal composition is associative, and α 11C = α = 11D α. With α : F −→ G,we also have a natural transformation αop : Gop −→ F op, where (αop)A = αA is an arrow inDop, for all A ∈ ob C.Finally, α is a natural isomorphism (that is, an isomorphism in DC) if and only if everymorphism αA is an isomorphism in D, and then one has (α−1)A = (αA)−1, briefly writtenas α−1

A .

2.4 The Yoneda embedding. For every morphism g : B −→ C in a category C, one hasa natural transformation

C(−, g) : C(−, B) −→ C(−, C)

of contravariant hom-functors. When C is locally small, there is therefore a functor

y : C −→ SetCop, g 7−→ y(g) = C(−, g) .

This functor is faithful, since C(−, g) = C(−, h) (for g, h : B −→ C) yields g = C(B, g)(1B) =C(B, h)(1B) = h. Moreover, on objects y is one-to-one: if yB = yC, then 1B ∈ C(B,B) =C(B,C) and B = cod 1B = C. It is less obvious that y is also full, so that

y : C −→ C = SetCop

is a full embedding, called the Yoneda embedding. As a preparation for that, we first provethe Yoneda Lemma:

2.4.1 Lemma. For every object A of a locally small category C, and every functor F :Cop −→ Set, the map

η : C(yA,F ) −→ FA , α 7−→ αA(1A)

is bijective.


Proof. Routine verification shows that the map

δ : FA −→ C(yA,F ) ,

with δ(a)B : C(B,A) −→ FB sending f to Ff(a), for all a ∈ FA, B ∈ ob C, is well-definedand inverse to η.

In the special case where F = yB for B ∈ ob C, the map

δ : C(A,B) −→ C(yA,yB)

is precisely the hom map yA,B of the functor y. Hence:

2.4.2 Corollary. The functor y : C −→ C is a full embedding.

Consequently, every locally small category may be considered as a full subcategory of afunctor category over Set.As we saw at the end of 2.2 in the dual situation, when C is an ordered set X and x ∈ X,the functor y(x) = X(−, x) may be identified with the down-set ↓X x. But the map↓X : X −→ DnX is injective only when X is separated, and in that case ↓X is just acodomain restriction of y.

2.5 Adjunctions. In generalization of the notion introduced in 1.5, one calls a functorG : A −→ X right adjoint if there is a functor F : X −→ A and natural transformationsη : 1X −→ GF and ε : FG −→ 1A satisfying the triangular identities

Gε · ηG = 1G and εF · Fη = 1F .

These data are, up to isomorphism, uniquely determined by G. Indeed, if F ′, η′, ε′ satisfythe corresponding conditions, then

α = (F Fη′// FGF ′ εF ′ // F ′) , β = (F ′ F ′η

// F ′GF ε′F // F )

are inverse to each other, as one shows by repeated applications of the middle-interchangelaw; for example:

β · α = ε′F · F ′η · εF ′ · Fη′= ε′F · εF ′GF · FGF ′η · Fη′ (middle-interchange)= εF · FGε′F · Fη′GF · Fη (middle-interchange, twice)= εF · 1FGF · Fη (triangular identity)= 1F (triangular identity).

Furthermore, η′ = Gα · η and ε′ = ε · βG. One calls F left adjoint to G, and η is the unit,while ε is the counit of the adjunction, denoted by

Fη

ε G : A // X .


A functor F is left adjoint to G : A −→ X if and only if Gop is left adjoint to F op : Xop −→ Aop;more precisely,

Fη

ε G ⇐⇒ Gop εop

ηop F op .

2.5.1 Examples.

(1) Adjunctions between ordered sets (considered as categories) are precisely those de-scribed in 1.5.

(2) The forgetful functor G : Mon −→ Set is right adjoint. Its left adjoint F is the free-monoid functor which, for a set X may be constructed as the set of “words over thealphabet” X

FX = X∗ = ⋃n≥0X

n ,

with concatenation as multiplication; the map ηX : X −→ X∗ (for a set X) considersan element of X as a one-letter word, and the homomorphism εA : A∗ −→ A (for amonoid A) sends words over A to the actual product of its letters in A.

(3) The forgetful functor Ord −→ Set has a left adjoint that provides each set with thediscrete order (given by the graph of 1X); similarly for Top −→ Set whose left adjointprovides a set X with the discrete topology. Both functors have also right adjointswhich provide a set X with the respective indiscrete (or chaotic) structure.

(4) For a fixed set A, the hom-functor Set(A,−) : Set −→ Set has a left adjoint, namely

(−)× A : Set −→ Set .

Writing BA = Set(A,B), the counit ε is given by the evaluation maps

εB : BA × A −→ B , (f, x) 7−→ f(x) ,

while the unit sends every element x of a set X to the section

ηX(x) : A −→ X × A , a 7−→ (x, a) .

(5) Considering the powerset PX of a set X as a complete lattice (ordered by inclusion)makes P : Set −→ Sup left adjoint to the forgetful functor U : Sup −→ Set. Theunit is described by the singleton maps ηX : X −→ PX that send x to x, and thecomponents of the counit εA : PA −→ A (for a complete lattice A) are simply givenby ∨.

(6) The contravariant powerset functor P • : Setop −→ Set is self-adjoint, that is, (P •)op :Set −→ Setop is its left adjoint, as shown by Proposition 2.5.2 below, and by the naturalbijection

Xf−→ PY

PXg←− Y


where f and g determine each other via (x ∈ g(y) ⇐⇒ y ∈ f(x)) for all x ∈ X,y ∈ Y . Of course, this bijection stems from the self-dual category Rel for whichRel(X, Y ) ∼= Rel(Y,X).

2.5.2 Proposition. A functor G : A −→ X is right adjoint if and only if there are a mapF : ob X −→ ob A and an ob X-indexed family of natural isomorphisms

φX : A(FX,−) −→ X(X,G−) = X(X,−)G .

This last condition yields bijectionsφX,A : A(FX,A) −→ X(X,GA)

for all X ∈ ob X, A ∈ ob A that are “natural in A”; casually, one writes

FXg−→ A

Xf−→ GA

with F appearing on the left, and G on the right. Note that one necessarily obtains thatthese bijections are also “natural in X”. Any pair of arrows (g, f) related to each other bythe bijection φX,A are called mates to each other under the adjunction.

Proof. If Fη

ε G , then φX,A is defined by

φX,A(g) = Gg · ηXfor all g : FX −→ A in A, with its inverse given by

(f : X −→ GA) 7−→ εA · Ff .

Conversely, having the bijections φX,A (natural in A), one obtains morphismsηX = φX,FX(1FX) , εA = φ−1

GA,A(1GA) .Then F , already defined on objects, can be made a functor via

(f : X −→ Y ) F−→ (φ−1X,FY (ηY · f) : FX −→ FY ) ,

such that η and ε become natural transformations satisfying the triangular identities.

2.5.3 Example. Considering 2 = ⊥,> successively as a set, an ordered set, a meet-semilattice, a lattice, or a frame, one obtains right adjoint functors

C(−, 2) : Cop −→ Set ,with C = Set, Ord, SLat, Lat, or Frm, respectively, with left adjoint X 7−→ PX. Ofcourse, for C = Set, C(−, 2) ∼= P • is isomorphic to the contravariant powerset functor ofExample 2.5.1(6), and the needed natural isomorphisms

Set(X,C(Y, 2)) ∼= C(Y, PX)(with X ∈ ob Set, Y ∈ ob C) are just restrictions of those described in Example 2.5.1(6).


Returning to Proposition 2.5.2, we note that naturality of φX,A in A gives, for g : FX −→ Ain A, the commutative diagram

A(FX,FX) A(FX,g)//

φX,FX

A(FX,A)φX,A

X(X,GFX) X(X,Gg)// X(X,GA)

so that with ηX = φX,FX(1FX), one has

φX,A(g) = φX,A · A(FX, g)(1FX) = X(X,Gg) · φX,FX(1FX) = Gg · ηX ;

in other words, the formula for φX,A in terms of the unit η is forced upon us by naturalityin A. The condition that φX,A is bijective now says:

for all f : X −→ GA in X (with A ∈ ob A), there is precisely one morphismg : FX −→ A in A with Gg · ηX = f .

XηX //

f $$

GFX

Gg

GA

FX

g

A .

(2.5.i)

For a functor G : A −→ X and an object X in X, a morphism u : X −→ GU in X with anobject U in A is called a G-universal arrow for X if it satisfies the universal property (2.5.i),with (U, u) in lieu of (FX, ηX). This property allows us to define φX,A as above wheneverwe have a chosen G-universal arrow for every X ∈ ob X. As a consequence, we get:

2.5.4 Theorem. A functor G : A −→ X is right adjoint if and only if there exist a mapF : ob X −→ ob A and an ob X-indexed family of G-universal arrows ηX : X −→ GFX. Thereis then a unique way of making F a functor such that η becomes a natural transformation.

Of course, F will be left adjoint to G with unit η. There is a dual way of describingadjunctions in terms of their counit: for any functor F : X −→ A and object A in A, amorphism εA : FGA −→ A in A together with an object GA is an F -couniversal arrow forA if, as an arrow in Aop, it is F op-universal, that is:

for all g : FX −→ A in A (with X ∈ ob X), there is precisely one morphismf : X −→ GA in X with εA · Ff = g.

GA

X

f

OO FGAεA // A .

FX

Ff

OO

g

:: (2.5.ii)


Usually, one also refers to (2.5.ii) as a universal property, although, strictly speaking, itis a “couniversal” property. In summary, one obtains from Theorem 2.5.4 the followingstatement by dualization:

2.5.4op Theorem. A functor F : X −→ A is left adjoint if and only if there exist a map G :ob A −→ ob X and an ob A-indexed family of F -couniversal arrows εX : FGX −→ X. Thereis then a unique way of making G a functor such that ε becomes a natural transformation.

Proof. Apply Theorem 2.5.4 with F op in lieu of G (recall from 2.5 that F a G ⇐⇒ Gop aF op).

In what follows, we rarely formulate explicitly such dual statements.Adjunctions compose; more precisely, direct calculation based on the definitions shows:

2.5.5 Proposition. If F η

ε G : A −→ X and H

γ

δ

J : C −→ A, then HFα

β

GJ : C −→X, with

α = (1Xη// GF

GγF// GJHF ) , β = (HFGJ HεJ // HJ δ // 1C) .

2.5.6 Corollary. If F a G : A −→ X, H a J : C −→ A and L a GJ , then L ∼= HF .

2.6 Reflective subcategories, equivalence of categories. Whether a right adjointfunctor is fully faithful is detected by the counit:

2.6.1 Proposition. For an adjunction Fη

ε G : A −→ X, the functor G is fully faithful

if and only if ε is a natural isomorphism. Dually, F is fully faithful if and only if η is anatural isomorphism.

Proof. Since, in the notations of 2.5, for all g : A −→ B in A,

φGA,B · A(εA, B)(g) = G(g · εA) · ηGA = Gg ,

the following diagram commutes for all A,B ∈ ob A:

A(FGA,B)φGA,B

&&

A(A,B)

A(εA,B)99

GA,B// X(GA,GB) .

Since φGA,B is bijective, one therefore has that for all A,B ∈ ob A the map GA,B is bijectiveprecisely when every εA is an isomorphism in A.


A full subcategory A of a category X is reflective in X if the inclusion functor J : A → X isright adjoint. A left adjoint R : X −→ A to J is called a reflector of X onto A, and the unitρ : 1X −→ JR gives the A-reflections ρX : X −→ RX that are characterized as J-universalarrows for X. By Theorem 2.5.4, A is reflective in X if and only if for every object in X,there is a chosen A-reflection. By Proposition 2.6.1, the counit of the adjunction is anisomorphism.An example of a reflective subcategory encountered in Section 1 is Ordsep in Ord, with theseparated reflections of 1.3 forming the unit. For a reflective subcategory A of X with unitρ, an object X in X is isomorphic to an object in A if and only if ρX is an isomorphism.Hence, when A is replete in X, that is, when X ∼= A ∈ ob A implies X ∈ ob A, an object ofX lies in A if and only if its A-reflection is an isomorphism.A full subcategory A is coreflective in X if Aop is reflective in Xop; equivalently, if theinclusion functor has a right adjoint, which is the coreflector of X onto A. The counit ofthe adjunction gives the A-coreflections.A functor G : A −→ X is an equivalence of categories if there is a functor F : X −→ Awith 1X ∼= GF and FG ∼= 1A. A category A is equivalent to a category X if there is anequivalence A −→ X. If G is an equivalence, then it is fully faithful (see the proof of thenext proposition), and therefore reflects isomorphisms, that is, if Gf is an isomorphismthen so is f (Exercise 2.A). In fact, these last two notions may be used to verify that G isan equivalence:2.6.2 Proposition. The following are equivalent for a functor G : A −→ X:

(i) G is an equivalence;

(ii) G reflects isomorphisms and has a fully faithful left adjoint;

(iii) G is fully faithful and there are a map F : ob X −→ ob A and an ob X-indexed familyof isomorphisms ηX : X −→ GFX (X ∈ ob X).

Proof. (i) =⇒ (iii): The hypothesis yields faithfulness of F and G. For fullness of G, considerf : GA −→ GB in X (with A,B ∈ ob A). In order to show Gg = f for g := εB · Ff · ε−1

A

(with ε : FG −→ 1A some isomorphism), it suffices to verify FGg = Ff . But this followsfrom

εB · FGg · ε−1A = g · εA · ε−1

A = εB · Ff · ε−1A .

(iii) =⇒ (ii): As a fully faithful functor, G reflects isomorphisms, and the isomorphism ηXis easily seen to be G-universal arrow for X ∈ ob X. Hence (b) follows from Theorem 2.5.4and Proposition 2.6.1.(ii) =⇒ (i): By Proposition 2.6.1, one has F a G with the unit η an isomorphism. SinceGε · ηG = 1G, the natural transformation Gε is an isomorphism, and so is ε whenever Greflects isomorphisms.

2.6.3 Corollary. If G : A −→ X is fully faithful and right adjoint, then A is equivalent to areplete reflective subcategory A of X.


Proof. Take A to be the full subcategory of objects X in X for which ηX is an isomorphism,with η the unit of an adjunction F a G.

2.7 Initial and terminal objects, comma categories. Let 1 = ? be the one-objectcategory whose only morphism is 1?. For every category C there is then a unique functor! : C −→ 1. A right adjoint to this functor is given by an object T in C such that

for all objects X in C, there is precisely one morphism X −→ T in C.

Any such object T is called terminal in C. As the value of a right adjoint, it is unique upto isomorphism (a fact that is easily established directly): if S, T are both terminal in C,then S ∼= T ; moreover, for any isomorphic objects S, T in C, one is terminal if the other is.A terminal object in a category is often denoted by 1.An object I is initial in C if it is terminal in Cop, so that

for all objects X in C, there is precisely one morphism I −→ X in C.

Equivalently, I is initial if the functor 1 −→ C with ? 7−→ I is left adjoint to the functor !.An initial object in a category is often denoted by 0.The empty set is initial in Set, and every one-element set is terminal; the same holds forOrd, Top, and Cat, for example. In Mon, the trivial (one-element) monoids are both initialand terminal; such objects are called zero-objects. An individual monoid M , consideredas a one-object category, can have an initial or terminal object only if it is trivial. In anordered set X considered as a category, a terminal object is given by a top element, and aninitial object by a bottom element.The process of defining initial and terminal objects via adjunctions may be reversed, asfollows. For a functor G : A −→ X and an object X in X, define the comma category(X ↓G) to have as objects pairs (A, f) with A ∈ ob A, and f : X −→ GA in X; a morphismh : (A, f) −→ (B, g) in (X ↓G) is given by a morphism h : A −→ B in A that makes thediagram

Xf

g

GAGh // GB

commute. Then

a G-universal arrow for X is simply an initial object in (X ↓G),

and Theorem 2.5.4 states that G is right adjoint if and only if there are chosen initialobjects in every comma category (X ↓G) (with X running through ob X).For a functor F : X −→ A, and A ∈ ob A, one also has the comma category

(F ↓A) := (A↓F op)op


whose morphisms h : (X, f) −→ (Y, g) are depicted by

FX

f

Fh // FY

g

A .

Its terminal objects are precisely the F -couniversal arrows for A. In the special case whereF = 1X, one calls

X/A := (1X ↓A)

the slice of X over A. Note that X/A always has a terminal object given by 1A:

X

f

f// A

1A

A .

We also denote the slice (A↓1X) of X under A by A/X.

2.8 Limits. A functor D : J −→ C is also called a diagram (of shape J) in C. If J is discrete,so that for all objects i, j in J,

J(i, j) :=

1i if i = j

∅ otherwise,

then D is simply a family (Ai)i∈ob J of objects in C. If J has precisely two objects, and twonon-identical arrows, as in

ax //

y// b ,

then a diagram of shape J is given by a pair of (not necessarily distinct) parallel arrows inC.For every object A in C and any J, one has the constant diagram of shape J in C given by

∆A : J −→ C , i 7−→ A , (δ : i −→ j) 7−→ 1A .

For a diagram D : J −→ C, any natural transformation α : ∆A −→ D is called a cone over Din C with vertex A. Naturality of α amounts to the property that for all δ : i −→ j in J, thediagram

Aαi

αj

DiDδ // Dj


commutes in C. Hence, a cone over D with vertex A is simply an object (A,α) in thecomma category (∆↓D), where D is considered as an object of CJ, and where ∆ has beenmade into a functor

∆ : C −→ CJ , (f : A −→ B) 7−→ (∆f : ∆A −→ ∆B) ,

with the natural transformation ∆f defined by (∆f)i = f for all i ∈ ob J .A limit of D : J −→ C is a terminal cone over D in C, or equivalently, a ∆-couniversal arrowfor D ∈ ob CJ. Hence, a limit of D is a cone λ : ∆L −→ D with L ∈ ob C such that

for every cone α : ∆A −→ D with A ∈ ob C, there is a unique morphismf : A −→ L in C with λ ·∆f = α, that is: λi · f = αi for all i ∈ ob J.

L

A

f

OO ∆L λ // D

∆A∆f

OO

α

;; Lλi // Di .

A

f

OO

αi

<<

Limits are uniquely determined by D, up to isomorphism, so that when (A,α) is also alimit of D, the unique morphism f : A −→ L is an isomorphism in C. We write L ∼= limDfor any object that serves as the vertex of a limit for D in C.Important instances of limits include: products, equalizers, and pullbacks (as well as certainspecial pullbacks), which we now proceed to describe.

Products. A product of a family of objects (Ai)i∈I in C is a limit of the diagram J −→ C,i 7−→ Ai, with J discrete and ob J = I. Hence, a product is an object P in C together with afamily of morphisms pi : P −→ Ai (i ∈ I) such that

for every family fi : B −→ Ai (i ∈ I) of morphisms in C, there is a uniquemorphism f : B −→ P with pi · f = fi (i ∈ I); this morphism is usually written

f = 〈fi〉i∈I .

Any object P that serves as a product for (Ai)i∈I is denoted by ∏i∈I Ai, and comes withprojections pi (i ∈ I). Note that I = ∅ is permitted, in which case P is simply a terminalobject of C. The case I = ? is trivial: a limit of the one-object family A can be given byA, with projection 1A. The product of a pair (A1, A2) of objects is denoted by A1 × A2,with projections p1, p2 whose universal property may be depicted by:

Bf1

zz

f

f2

$$

A1 A1 × A2p1oo

p2// A2 .


More casually, we may write:B −→ A1, B −→ A2

B −→ A1 × A2 .In Set, small-indexed products can be constructed as cartesian products. In many concretecategories (categories admitting a faithful functor to Set) like Ord and Mon, the cartesianproduct is provided with an appropriate structure that makes the projections morphisms ofthe category; see Propositions 5.8.3 and 3.3.1 respectively for specifications of this statementin two general contexts. In an ordered set X regarded as a category, a product of a family(ai)i∈I of elements in X is simply an infimum of ai | i ∈ I in X.

Equalizers. An equalizer of a pair (f, g : A −→ B) of “parallel” morphisms in C is a limit(E, λ) of the corresponding diagram D of shape a x //

y // b in C, with Dx = f , Dy = g.Since λb = Dx · λa = Dy · λa, an equalizer of (f, g) is given by a morphism u : E −→ A withf · u = g · u (namely, u = λa) such that

for every morphism h : C −→ A with f · h = g · h, there is a unique morphismt : C −→ E with u · t = h.

E u // Af//

g// B .

C

t

OO

h

??

We sometimes write u ∼= equ(f, g) in this case. If f = g, then u can be taken to be 1A;conversely, if equ(f, g) exists and is an isomorphism, then f = g.In Set, an equalizer equ(f, g) can be realized as the inclusion map

E = x ∈ A | f(x) = g(x) // A ,

and in concrete categories like Ord and Mon, one can endow E with an obvious structureinherited from A. An ordered set X (regarded as a category) trivially has equalizers, sinceany two parallel arrows are equal. In a group regarded as a one-object category, an equalizerof two elements x, y can exist only if x = y.

Pullbacks. A pair of morphisms f : A −→ C, g : B −→ C with common codomain can beviewed as a diagram D of shape J, where J is a three-object category with two non-identicalarrows, as in

a x // c b .yoo

A pullback of (f, g) in C is a limit (P, λ) of D in C. Since λc = Dx ·λa = Dy ·λb, a pullbackof (f, g) is given by a pair (p1 : P −→ A, p2 : P −→ B) with f · p1 = g · p2 (namely, p1 = λa,p2 = λb), such that

for all morphisms h1 : X −→ A, h2 : X −→ B with f ·h1 = g ·h2, there is a uniquet : X −→ P with p1 · t = h1, p2 · t = h2. The morphism t is often denoted by


〈h1, h2〉, and the square in the following diagram is called a pullback diagram:

Xh2

##t&&

h1

!!

Pp2//

p1

Bg

Af// C .

One often writes (somewhat casually) P ∼= A×C B and calls P a fibred product of A and Bover C, thus unduly neglecting the morphisms f and g. But pullbacks are in fact products,since a pullback of (f, g) in C is nothing but the product of the pair of objects (A, f), (B, g)in the comma category C/C. In case C = 1 is terminal in C, one has A×C B ∼= A×B, i.e.,the pullback is a product in C. Multiple pullbacks are simply products in C/C of families ofobjects in C (that is, of families of morphisms in C with common codomain C).In Set, a pullback of (f : A −→ C, g : B −→ C) can be constructed as

P = (x, y) ∈ A×B | f(x) = g(y) ,

with p1, p2 the restrictions of the product projections π1, π2. Hence, P is obtained as theequalizer of f · π1, g · π2 : A×B −→ C. This construction works in general categories, seeCorollary 2.10.2 below.

Special pullbacks. A kernel pair of a morphism f : A −→ B in C is a pullback of (f, f);hence, it is a pair (p1, p2 : K −→ A) of morphisms in C with f · p1 = f · p2 satisfying theuniversal property of a pullback diagram. A morphism f : A −→ B in C is a monomorphism(or is monic) if and only if f · x = f · y implies x = y whenever x, y : X −→ A; equivalently,if (1A, 1A) serves as a kernel pair, or more generally, if f has a kernel pair (p1, p2) withp1 = p2. For a monomorphism f , any commutative diagram

Ch //

1C

A

f

C g// B

is a pullback diagram. Composites of monomorphisms are monic, and monomorphisms arestable under pullback in the following sense: for every pullback diagram

Pg′//

f ′

A

f

C g// B


if f is monic, so is f ′. One calls f ′ a pullback of f along g, alluding to the fact that in Set,when f : A → B is a subset inclusion, f ′ can be taken to be g−1(A) → C.A frequently used property of pullback diagrams is:2.8.1 Proposition. If in a commutative diagram in C of the form

· //

· //

·

· // · // ·

the outer rectangle and the right square are pullback diagrams, then so is the left square.Proof. The statement follows by routine diagram chasing.

2.9 Colimits. A pair (K, γ) is a colimit of the diagram D : J −→ C in a category C if(K, γop) is a limit of Dop : Jop −→ Cop in Cop. Hence, γ : D −→ ∆K is a cocone over D withvertex K ∈ ob C such that

for every cocone α : D −→ ∆A with A ∈ ob C, there is a unique morphismf : K −→ A in C with ∆f · γ = α, that is: f · γi = αi for all i ∈ ob J.

Dγ//

α##

∆K∆f

∆A

K

f

A

Diγi //

αi""

K

f

A .

One writes K ∼= colimD in this case. Dually to products, equalizers, pullbacks, kernel pairs,and monomorphisms (or morphisms that are monic), one obtains coproducts, coequalizers,pushouts, cokernel pairs, and epimorphisms (or morphisms that are epic), respectively. Acoproduct (Ai)i∈I in C is denoted by ∐i∈I Ai, together with injections ki (i ∈ I). For I = ∅,the empty coproduct ∐i∈I Ai

∼= 0 is an initial object of C. In the binary case I = 1, 2,the coproduct is normally denoted by A1 + A2, and its universal property visualized by:

B

A1 k1//

f1

::

A1 + A2

f

OO

A2k2oo

f2

dd

or A1 −→ B,A2 −→ B

A1 + A2 −→ B .

As for products, we often write f = 〈f1, f2〉 for the arrow given by f1, f2. The universalproperties of coequalizers and pushouts may be displayed as follows:

Af//

g// B

q//

h

Q

t

C

andC

g//

f

B

q2

h2

A q1//

h100

Qt

%%X .


In Set, a coproduct of (Ai)i∈I may be constructed as its disjoint union ⋃i∈I Ai × i.

A coequalizer of f, g : A −→ B may be constructed as the projection onto Q = B/∼,where ∼ is the least equivalence relation on B with f(x) ∼ g(x) for all x ∈ A. For thepushout of (f : C −→ A, g : C −→ B), form the coproduct A+B with injections k1, k2 andthen the coequalizer of k1 · f, k2 · g. In Ord, and Top for example, one may “lift” theseconstructions from Set by putting appropriate structures on them (see the dual statementof Proposition 5.8.3 further on). In an ordered set X considered as a category, coproductsare given by suprema, and coequalizers are, like equalizers, trivial.

2.10 Construction of limits and colimits. For categories C and J, one says that C isJ-complete if the functor ∆ : C −→ CJ has a right adjoint, that is, if all diagrams of shape Jhave a chosen limit in C; the right adjoint of ∆ is usually denoted by lim. A category Cis finitely complete if C is J-complete for every finite category J (categories for which ob Jand all hom-sets of J are finite), and C is small-complete if C is J-complete for every smallcategory J. A category C has products if C is J-complete for every small discrete categoryJ, and C has finite products if this holds for every finite discrete category J. Similarly, · //

// · -completeness and · // · ·oo -completeness are phrased as “has equalizers”,“has pullbacks”, and so on. The dual notions are those of J-cocompleteness (with colimdenoting the left adjoint of ∆), finite cocompleteness, small-cocompleteness, has coproducts,has finite coproducts, has coequalizers, has pushouts, etc.General limits can be constructed in terms of products and equalizers, as follows. GivenD : J −→ C, one forms the product P = ∏

i∈ob J Di with projections pi. Since in generalthe pi do not form a cone (so that generally Dδ · pj 6= pk for δ : j −→ k in J), its existencegranted one also forms the product

Q = ∏δ∈mor J D(cod δ) with projections qδ ,

where mor J = ⋃j,k∈ob J J(j, k) (with the hom-sets of J assumed to be disjoint, without loss

of generality). With

f = 〈pcod δ〉δ∈mor J , g = 〈Dδ · pdom δ〉δ∈mor J ,

one obtains for every δ : j −→ k in J the diagram

Pf

//

g//

pj

pk

""

Q

qδ

DjDδ

// Dk

with qδ · f = pk, qδ · g = Dδ · pj (but generally Dδ · pj 6= pk). Now, for all morphismsh : A −→ P in C,

f · h = g · h ⇐⇒ ∀(δ : j −→ k) in J (Dδ · (pj · h) = pk · h) .


Hence, when h satisfies f · h = g · h, then αi = pi · h (i ∈ ob J) defines a cone α : ∆A −→ D;and conversely, for every cone α : ∆A −→ D, the morphism h = 〈αi〉i∈ob J satisfies f ·h = g ·h.Consequently, an equalizer h of (f, g) gives a limit cone over D, and a limit cone over Dgives an equalizer of (f, g). This sketches the proof of the following result:

2.10.1 Theorem. A category C is J-complete if it has equalizers as well as ob J- andmor J-indexed products. Hence, C is small-complete if and only if C has products andequalizers.

2.10.2 Corollary. The following conditions are equivalent for a category C:

(i) C is finitely complete;

(ii) C has finite products and equalizers;

(iii) C has pullbacks and a terminal object.

Proof. The equivalence (i) ⇐⇒ (ii) follows from the Theorem. (ii) =⇒ (iii): A terminalobject exists as a product of the empty family of objects in C. To obtain the pullback(P, p1, p2) of (f : A −→ C, g : B −→ C), let u ∼= equ(f · π1, g · π2), with π1, π2 the projectionsof A×B, and let p1 = π1 · u, p2 = π2 · u:

Pp2

//

p1

u$$

B

g

A×Bπ2

::

π1zzA

f//

C .

(iii) =⇒ (ii): Pullbacks over 1 give binary products; n-ary products can be constructed frombinary products via A1 × A2 × A3 ∼= (A1 × A2)× A3, etc. An equalizer u of f, g : A −→ Bcan be obtained as a pullback of δ = 〈1B, 1B〉 along 〈f, g〉:

E //

u

B

δ

A〈f,g〉

// B ×B .

2.10.3 Example. According to Theorem 2.10.1, a limit (L, λ) of D : J −→ Set with J smallis obtained as

L = (xi)i∈ob J ∈∏i∈ob J Di | ∀(δ : j −→ k) in J (Dδ(xj) = xk) ,

with λi : L −→ Di the restriction of the product projection.


For the colimit (K, γ) of D : J −→ Set with J small, one considers dually the least equivalencerelation ∼ on the disjoint union ∐i∈ob J Di = ⋃

i∈ob J Di× i such that for all δ : j −→ k inJ and x ∈ Dj,

(x, j) ∼ (Dδ(x), k) .

Then K = ∐i∈ob J Di/∼, and γi(x) is the equivalence class of (x, i).

There is an important special case when this equivalence relation has an easy description,namely when J is an up-directed ordered set I, considered as a category J, with ob J = I.Then the diagram D is simply a family of sets and maps fj,k : Xj −→ Xk (for j ≤ k in I),with

fk,l · fj,k = fj,l if j ≤ k ≤ l , fj,j = 1Xj .

Now ∼ is given by

(x, j) ∼ (y, k) ⇐⇒ ∃l ≥ j, k (fj,l(x) = fk,l(y)) .

Classically, in any category C, colimits of diagrams in C of shape J, where J is an up-directedordered set, are called direct limits, and limits of shape Jop are called inverse limits in C;we prefer to call them directed colimits and directed limits, respectively.

2.11 Preservation and reflection of limits and colimits. Let F : C −→ D be a functorand (L, λ) a limit of D : J −→ C in C. One says that F preserves this limit if (FL, Fλ) is alimit of FD in D. The functor F preserves J-limits (or is J-continuous) if F preserves allexisting limits in C of diagrams of shape J; if C is J-complete, this property may be casuallycommunicated by

F lim ∼= limF .

The functor F reflects J-limits if for every cone λ : ∆L −→ D (with D : J −→ C) such that(FL, Fλ) is a limit of FD in D, then (L, λ) is a limit of D in C. This language is appliedin various special contexts; for example, F preserves monomorphisms if Fm is monic inD whenever m is monic in C, and F reflects monomorphisms if m is monic whenever Fmis monic. This terminology is used similarly for colimits. For example, J-cocontinuity ispreservation of J-colimits.

2.11.1 Proposition.

(1) A right adjoint functor preserves all existing limits, and a left adjoint all existingcolimits.

(2) A fully faithful functor reflects all existing limits and colimits.

(3) A replete reflective subcategory of a J-complete category is J-complete. Also, a repletereflective subcategory of a J-cocomplete category is J-cocomplete.


Proof. For (1), if F a G : A −→ X, and L ∼= limD, the following natural correspondencessketch the proof of GL ∼= limGD:

∆X −→ GD

∆FX −→ D

FX −→ L

X −→ GL .

(F a G)(L ∼= limD)(F a G)

For (2), if G : A −→ X is full and faithful and GL ∼= limGD, one may sketch the proof forL ∼= limD as follows:

∆A−→ D

∆GA−→ GD

GA−→ GL

A−→ L .

(G fully faithful)(GL ∼= limGD)(G fully faithful)

For the completeness part of (3), we show the stronger statement that A is closed underlimits in X, that is: whenever (L, λ) is a limit of JD in X (with J : A → X), then L ∈ ob A.For this, it suffices to show that ρL is an isomorphism, with ρL : L −→ RL = JRL theA-reflection of L. But the inverse of ρL may be found as follows:

∆L λ−→ JD

∆RL−→ D

∆JRL−→ JD

JRLρ−1L−→ L .

(R a J)(J fully faithful)(L ∼= lim JD)

For the cocompleteness part of (3), if (K, γ) is a colimit of JD in X, then (RK,∆ρK · γ) isa colimit of D in A:

D−→ ∆AJD−→ ∆JAK −→ JA

RK −→ A .

(J fully faithful)(K ∼= colim JD)(R a J)

Hence, a left adjoint colimA to ∆A : A −→ AJ may be written as the composite functor

AJ J(−)// XJ colimX // X R // A .

In addition to (3), note that while the inclusion functor J preserves all limits, it generallyfails to preserve colimits. Nonetheless, existence in A of colimits of a specified shape isguaranteed by their existence in X.Limits and colimits in a functor category DC can be formed “pointwise” in D. More precisely,for every A ∈ ob C, one has the evaluation functor

EvA : DC −→ D , F 7−→ FA , α 7−→ αA .


A limit (L, λ) of D : J −→ DC can be formed by letting (LA, λA) be a limit of EvAD in D,if the latter exist for all A ∈ ob C; the same construction holds for colimits. This way oneproves:

2.11.2 Proposition. The functor category DC is J-complete if D is J-complete, and thenthe evaluation functors are J-continuous. Dually, DC is J-cocomplete if D is J-cocomplete,in which case the evaluation functors are J-cocontinuous.

2.11.3 Corollary. Every hom-functor C(A,−) : C −→ Set of a locally small category Cpreserves all existing limits, and so does the Yoneda embedding y : C −→ SetCop.

Proof. When (L, λ) is a limit of D in C, the isomorphism C(A,L) ∼= lim C(A,D(−)) followsfrom:

∆X α−→ C(A,D(−))∆A αx−→ D (x ∈ X)A

fx−→ L (x ∈ X)X

f−→ C(A,L) .

(L ∼= limD)

To show that yL ∼= lim yD in SetCop , it suffices to show that EvA yL ∼= lim EvA yD in Setfor all A ∈ ob C. But EvA y = C(A,−), which preserves the limit, as we just saw.

Note that since Set is small-complete and small-cocomplete, every functor category of Setis too. Hence, every locally small category is fully embedded into a small-complete andsmall-cocomplete category by its Yoneda embedding.

2.12 Adjoint Functor Theorem. Preservation of limits by a functor G : A −→ X is anecessary condition for its right adjointness (see Proposition 2.11.1). For ordered sets,completeness of A also makes this condition sufficient (see Corollary 1.8.3). For generalcategories however, small-completeness of A and preservation of limits by G do not generallysuffice to make G right adjoint (see Exercise 2.O). Here is a two-step procedure on how toconstruct a G-universal arrow for an object X in X.

Step 1. A G-solution set for X is a set L of objects in A with the property that for allf : X −→ GA (A ∈ ob A), there exist L ∈ L, e : X −→ GL in X, and h : L −→ A in A withGh · e = f . If X is locally small and A has products which are preserved by G, then thereexists a weakly G-universal arrow for X, consisting of an object V in A and a morphismv : X −→ GV in X such that for all f : X −→ GA (A ∈ ob A), there is a (not necessarilyunique) morphism g : V −→ A with Gg · v = f : indeed, simply take

V = ∏L∈L L

X(X,GL) , v = 〈e〉e∈X(X,GL),L∈L ,

with LX(X,GL) = ∏X(X,GL) L, and with existence of v guaranteed by the product preservation

by G.


Step 2. If A is locally small and has generalized equalizers (that is, limits for non-emptysets—not just pairs—of parallel morphisms) that are preserved by G, then existence of aweakly G-universal arrow (V, v) for X forces existence of a G-universal arrow (U, u) for X,as follows: let U be the limit of the diagram

t ∈ A(V, V ) | Gt · v = v // A ,

with limit projection k; hence, k : U −→ V is universal with respect to the property t · k = kfor all t with Gt · v = v. Limit preservation by G gives the morphism u : X −→ GU withGk · u = v. Weak universality of (V, v) gives weak universality of (U, u). Furthermore, ifa, b : U −→ A in A satisfy Ga · u = Gb · u, consider the equalizer c : C −→ U of a, b, andobtain w : X −→ GC with Gc · w = u from the limit preservation by G. Weak universalityof (V, v) gives d : V −→ C with Gd · v = w:

X

w

v //

u

GV

Gd

~~

GCGc

// GU

Gk

OO

Ga //

Gb// GA .

Since G(k · c · d) · v = G(k · c) ·w = Gk ·u = v, one has k · c · d · k = k by definition of k, andsince k is monic (by the uniqueness part of the limit property), one obtains c · d · k = 1U .Finally, since a · c = b · c, this implies a = b.Hence, we have proved the non-trivial part of the following result:

2.12.1 Theorem. A functor G : A −→ X of locally small categories with A small-completeis right adjoint if and only if

(1) for every object X in X one has a chosen G-solution set, and

(2) G preserves small limits.

Proof. For the necessity of the conditions, if F a G, then FX is a G-solution set forX ∈ ob X, and limit preservation by G follows from Proposition 2.11.1.

2.12.2 Example. With Grp denoting the category of groups, let us prove the existenceof a left adjoint to the forgetful functor U : Grp −→ Set, using Theorem 2.12.1. For a setX and a map f : X −→ G into a group G such that f(X) generates G, the cardinality ofG cannot exceed the cardinality of (X + 1)N (since there is a surjection (X + 1)N −→ G).Hence, as a U -solution set for X, one may choosec© a representative system of non-isomorphicgroups whose cardinality does not exceed the cardinality of (X + 1)N. Since small limitsin Grp may be formed by taking the limit in Set, and providing it with a group structure,small-limit preservation by U is clear. (Another argument for limit preservation would beU ∼= Grp(Z,−).) Hence, Theorem 2.12.1 guarantees existence of the free-group functor F .


The argument just given extends to any algebraic structure defined by a set of operationsand equations between them. That is, groups may be traded in this example for rings,R-modules, R-algebras, etc.

2.13 Kan Extensions. For functors S : A −→ B and T : A −→ X, the right Kan extensionof T along S is a F -couniversal arrow for T , where F is the whiskering functor

F = (−)S : XB −→ XA , α 7−→ αS .

Hence, the right Kan extension is given by a functor K : B −→ X and a natural transformationκ : KS −→ T such that

for all α : QS −→ T in A (with Q : B −→ X), there is precisely one naturaltransformation σ : Q −→ K with κ · σS = α.

K

Q

σ

OO KSκ // T .

QS

σS

OO

α

<<

It is common to use the notationK ∼= RanS T .

By Theorem 2.5.4op, if the right Kan extension of all functors T : A −→ X along a fixed Sexist, then (−)S has a right adjoint. However, the Kan extension is useful without theexistence of the entire right adjoint.In the previous definition, one can factor in the categories to obtain the following picturefor the right Kan extension (K,κ) of T along S:

BK

Q

A

S

OO

T// X .

σks

κ

α

z

The terminology is explained by the case where A is a full subcategory of B, as K then doesindeed extend T along the full embedding S. This extension is made to the right becauseone has a natural bijection

XA(QS, T ) ∼= XB(Q,RanS T ) .

To construct the right Kan extension, consider an object B in B. For any A in A admittinga B-morphism f : B −→ SA, one expects to obtain an X-morphism Kf : KB −→ TA. In


fact, for any commutative diagramB

f

f ′

SA Sh // SA′

in B, one wants a similar commutative diagram in X where B is replaced by KB and Shby Th. Taking into consideration the couniversal property of the right Kan extension, onedefines

KB ∼= limD

where D : (B ↓ S) −→ X is the diagram sending an object f : B −→ SA to TA, and amorphism h : (A, f) −→ (A′, f ′) to Th : TA −→ TA′:

Bf

f ′

SASh // SA′ 7−→ TA

Th // TA′ .

2.13.1 Theorem. Suppose that functors S : A −→ B and T : A −→ X are given such that thediagram D : (B ↓S) −→ X has a limit for all objects B of B. Then the right Kan extension(K,κ) of T along S exists. When S is full and faithful, the natural transformation κ is anisomorphism.Proof. The previous construction defined on objects of B extends to yield a functor K :B −→ X via the couniversal property of limits. The component at A ∈ A of the naturaltransformation κ : KS −→ T is then obtained as the component λ1SA : KSA −→ TA at 1SAof the limit cone of D : (SA↓S) −→ X. When S is full and faithful, every object SA −→ SBof the comma category (SA↓S) can be written as Sf for a unique A-morphism f : A −→ B;hence, 1SA is an initial object in this category, and the construction of KSA yields anobject isomorphic to TA.

If the right Kan extension (K,κ) of T along S is constructed as in the Theorem, so that

KB ∼= lim((B ↓S) // A T // X )for all B ∈ ob B, then the extension is called pointwise.The left Kan extension

K ∼= LanS T : B −→ Xof T : A −→ X along S : A −→ B is defined dually: Kop is the right Kan extension of T op

along Sop, so that K comes with a (−)S-universal natural transformation κ : T −→ KS.This is briefly summarized in

BK

Q

A

S

OO

T// X .

σ +3

κ

KSα

:B


Again, the extension is pointwise if

KB ∼= colim((S ↓B) // A T // X )

for all objects B in B, with the colimit cone constructed dually to 2.13.1.

2.14 Dense functors. A functor J : A −→ X is dense if (1X, 1J) is a pointwise left Kanextension of J along J ; that is, if for all X ∈ ob X, one has colimDX

∼= X, with

DX = ((J ↓X) // A J // X )

and the colimit cocone given by

λ(A,f) = f : JA −→ X

for all (A, f) ∈ ob(J ↓X). Arbitrary cocones

α : DX −→ ∆Y

correspond bijectively to natural transformations

γ : X(J−, X) −→ X(J−, Y )

viaγA(f) = α(A,f)

for A ∈ ob A, f : JA −→ X in X, naturally in Y ∈ ob X.2.14.1 Proposition. For a functor J : A −→ X of locally small categories, the followingstatements are equivalent:

(i) J is dense;

(ii) the maps X(X, Y ) −→ A(X(J−, X),X(J−, Y )), f 7−→ X(J−, f) are bijective for allX, Y ∈ ob X;

(iii) the functor J : X −→ A, φ 7−→ φJop is fully faithful.

Proof. Condition (i) means that there is a natural bijective correspondence

X −→ Y

DX −→ ∆Y ,

and condition (ii) means that there is a natural bijective correspondence

X −→ Y

X(J−, X) −→ X(J−, Y ) .

By the introductory observations, (i) and (ii) are therefore equivalent. Since X(X, Y ) ∼=X(X(−, X),X(−, Y )) by the Yoneda Lemma, (ii) and (iii) are equivalent as well.


2.14.2 Theorem. For every locally small category C, the Yoneda embedding is dense.

Proof. The Yoneda Lemma facilitates a natural bijective correspondence

F ∼= C(y−, F )

for all F : Cop −→ Set, with y : C −→ C = SetCop denoting the Yoneda embedding. Hence,for F,G ∈ ob C, there is a natural bijective correspondence

F −→ G

C(y−, F ) −→ C(y−, G) ,

showing that y satisfies condition (ii) of Proposition 2.14.1.

2.14.3 Corollary. Every functor F : Cop −→ Set is the colimit of the diagram

(1↓F )op ∼= el(F )op // C y// C

(with 1 the terminal object in Set and el(F ) the category of “elements of F”, see Exercise 2.P).

Proof. By the Yoneda Lemma, (1↓F )op ∼= (y↓F ), and the diagram becomes the same asthe one used in the defining property of density of y.

The Yoneda embedding of a locally small category C therefore embeds C into the categoryC with all small colimits, and every object in C is naturally a colimit of objects originatingfrom C.

Exercises

2.A Isomorphisms and functors. If f : A −→ B, g : B −→ C are isomorphisms in a categoryC, then so are g ·f and f−1, and (g ·f)−1 = f−1 ·g−1, (f−1)−1 = f . Every functor F : C −→ Dpreserves isomorphisms but does not generally reflect them: if f is an isomorphism, thenso is Ff , but not conversely in general; as an example, consider the forgetful functorOrd −→ Set. However, if F is fully faithful, then F reflects isomorphisms.

2.B Monomorphisms, epimorphisms and functors. Consider f : A −→ B, g : B −→ C inC. If f and g are monic, then so is g · f ; if g · f is monic, then so is f . Establish thedual statements for epimorphisms. If F : C −→ D preserves kernel pairs, then F preservesmonomorphisms; in particular, every right adjoint functor and every hom-functor does so.A faithful functor reflects both monomorphisms and epimorphisms. The monomorphismsof Set are precisely the injective maps, and the epimorphisms are the surjective maps. Notevery epimorphism in Mon is surjective, and a morphism that is both monic and epic neednot be an isomorphism.


2.C Split monomorphisms and split epimorphisms. A morphism f : A −→ B in C is a splitmonomorphism (or a section) if g · f = 1A for some g : B −→ A in C; the dual notion isthat of split epimorphism (or retraction). Every split monomorphism is monic, and it isan isomorphism precisely when it is epic. Composites of split epimorphisms are split epic,and split epimorphisms are stable under pullback. Dualize. In Set, every monomorphism∅ 6= A −→ B splits, and to say that every epimorphism splits is equivalent to the Axiom ofChoice.

2.D Regular epimorphisms. A morphism f : A −→ B in C is a regular epimorphism in C iffor any g : A −→ C with the property that f · x = f · y always implies g · x = g · y, one hasa uniquely determined morphism h : B −→ C with h · f = g. Every split epimorphism isregular, and every regular epimorphism is in fact an epimorphism. If C has kernel pairs,then f is a regular epimorphism if and only if f is the coequalizer of some pair of morphismsp, q : K −→ A; in fact, p, q may always be chosen to be the kernel pair of f . Dualize toregular monomorphisms.

2.E Full or faithful right adjoints. If Fη

ε G : A −→ X, then G is faithful if and only if

εA is epic for all A ∈ ob A, and G is full if and only if εA is a split monomorphism for allA ∈ ob A.

2.F Limits and colimits in slices of categories. For any object A of a category C, the commacategory C/A is finitely complete when C has fibred products. If C is J-cocomplete, so isC/A, and the forgetful functor C/A −→ C, (X, f) 7−→ X, is J-cocontinuous.

2.G Products and coproducts in Rel. Coproducts in Rel are formed as in Set, and also serveas products. In the category AbGrp of abelian groups, finite coproducts and finite productsmay be formed using the same abelian group.

2.H Cartesian structure of categories with finite products. In a category with finite products,establish natural isomorphisms α, λ, ρ, σ satisfying the same coherence conditions as in 1.1.

2.I Products as down-directed limits of finite products. For a family (Ai)i∈I of objects in acategory C with finite products, put AF = ∏

i∈F Ai for every finite set F ⊆ I. If G ⊆ F ,there is then an obvious morphism AF −→ AG. With F the set of finite subsets of I orderedby inclusion, this defines a diagram D : Fop −→ C, and the limit of D exists in C if and onlyif the product ∏i∈I Ai exists, and then both limits coincide up to isomorphism.

2.J Hom-functors. The metacategory CAT has products and coproducts. For every categoryC, there is a functor C : Cop × C −→ SET. Does C preserve products or coproducts?

2.K Completeness of Ord and Top. The categories Ord and Top are both small-completeand small-cocomplete. The respective forgetful functors to Set preserve all existing limitsand colimits.

2.L Representable functors. A functor H : C −→ Set (with C locally small) is representableif it is isomorphic to a hom-functor C(A,−). If H is right adjoint, then it is representable.


(Hint. If F a H, then Set(1, H(−)) ∼= C(F1,−).) Conversely, a representable functorH ∼= C(A,−) is right adjoint if and only if for every set X the coproduct X ·A := ∐

x∈X Axwith Ax = A for all x ∈ X exists in C.

2.M Equivalence of categories, irrelevance of injectivity on objects. Equivalence of categoriesis a reflexive, symmetric, and transitive property. A category equivalent to a J-completecategory is J-complete itself. Every functor F : C −→ D can be factored as

F = ( C F // D U // D )

with F injective on objects, and U an equivalence of categories. (Hint. When C 6= ∅, letob D = ob C × ob D and D((A,B), (A′, B′)) = D(B,B′).) Moreover, F is full or faithfulprecisely when F has the respective property. If F is faithful, then C is isomorphic to asubcategory C of D, so that there are inclusion functions both for objects and hom-setsforming a functor C −→ D.

2.N Formal adjointness criterion. A functor G : A −→ X is right adjoint if and only if forevery object X in X, the (not necessarily small) diagram

DX : (X ↓G) −→ A , (A, f) 7−→ A ,

has a chosen limit in A that is preserved by G.Hint. FX ∼= limDX .

2.O Limit preservation does not suffice for right adjointness. Let S be the class of all setsordered by set inclusion. Then every non-empty subclass has a least element, but there isno top element. Hence, when considered as a category, Sop is small-complete, and the onlyfunctor Sop −→ 1 trivially preserves all limits but has no left adjoint.

2.P The element construction. For a functor H : C −→ Set (with C locally small), onedefines the category el(H) of “elements of H” to have objects (A, x) with A ∈ ob C, x ∈ HA;a morphism f : (A, x) −→ (B, y) is a C-morphism f : A −→ B with Hf(x) = y. Then H isrepresentable (see Exercise 2.L) if and only if el(H) has an initial object.

2.Q Connectedness, colimits in Set revisited. A category C is connected if C 6= ∅ and,whenever C = A + B in CAT, then A = ∅ or B = ∅; equivalently, if C 6= ∅ and, for anyA,B ∈ ob C, there is a finite “zigzag” A −→ · ←− · −→ · . . . · −→ B of morphisms in C. Everycategory is a (not necessarily small) coproduct of its connected components (that is, itsmaximal connected full subcategories). Show also that the colimit of a small diagramD : J −→ Set can be taken to be the set of connected components of the category el(D) (seeExercise 2.P).

2.R General comma categories. For functors G : A −→ X, H : C −→ X, consider the category(G↓H) whose objects (A, f, C) are given by A ∈ ob A, C ∈ ob C and f : GA −→ HC in X;


a morphism (h, j) : (A, f, C) −→ (B, g,D) is formed by morphisms h : A −→ B in A andj : C −→ D in C that make

GAGh //

f

GB

g

HCHj// HD

commute in X. Composition is such that the projections

P :(G↓H) −→ A , (h, j) 7−→ h ,

Q :(G↓H) −→ C , (h, j) 7−→ j ,

become functors, andκ : GP −→ HQ , κ(A,f,B) = f

a natural transformation. Show that, given any functors R : K −→ A, S : K −→ C and anatural transformation φ : GR −→ HS, there is a unique functor T : K −→ (G↓H) with

PT = R , QT = S , κT = φ .

Observe also that the comma categories discussed in 2.7 are all special cases of the generaltype introduced here, up to isomorphism.

2.S Testing limits via Yoneda. A cone λ : ∆A −→ D in a locally small category C is a limitif and only if for all objects X in C, the cone X(X,λ(−)) : ∆X(X,A) −→ X(X,D−) is a limitin Set.

2.T Quantales freely generated by monoids. Show that the forgetful functor Qnt −→ Monhas a left adjoint (see Exercise 1.M).


3 Monads

3.1 Monads and adjunctions. A monad T = (T,m, e) on a category X is given by afunctor T : X −→ X and two natural transformations, the multiplication and unit of themonad

m : TT −→ T , e : 1X −→ T ,

satisfying the multiplication law and the right and left unit laws

m ·mT = m · Tm , m · eT = 1T = m · Te ;

equivalently, these equalities mean that the diagrams

TTT Tm //

mT

TT

m

TTm // T

T eT //

1T%%

TT

m

TTeoo

1Tyy

T

commute. A morphism of monads α : S −→ T (where S = (S, n, d)) is a natural transforma-tion α : S −→ T that preserves the monad structure:

α · n = m · (α α) , α · d = e .

Hence, comparing with the definition of a monoid given in 1.1, we observe that the set Mhas been replaced by a functor, the cartesian product by horizontal composition, and mapsby natural transformations. Since the natural isomorphisms α, λ, and β are identities, thecoherence conditions are immediately verified, so that a monad is nothing but an object ofthe functor category XX that carries a monoid structure with respect to the (horizontal)compositional structure.

Any adjunction Fη

ε G : A −→ X gives rise to the associated monad (or induced monad)

T = (GF,GεF, η) on X. Indeed, by whiskering the identities ε ·εFG = ε ·FGε, Gε ·ηG = 1G,and εF · Fη = 1F appropriately by F on the right and G on the left, we obtain thecorresponding multiplication and unit laws

GεF ·GεFGF = GεF ·GFGεF , GεF · ηGF = 1GF = GεF ·GFη .

In fact, any monad can be obtained as a monad associated to an adjunction, althoughdifferent adjunctions may yield the same monad. The “largest” of these adjunctions isdescribed in 3.2, while the “smallest” is treated in 3.6.

3.1.1 Examples.

(1) A closure operation on an ordered set X is a monad on X (considered as a category),and the closure operation c = g · f associated to an adjunction of ordered setsf a g : X −→ Y is also the associated monad.

3. MONADS 75

(2) The list monad (or the free-monoid monad) L = (L = GF,GεF, η) on Set is themonad associated to the adjunction F a G : Mon −→ Set described in 2.5.1.Similarly, from Example 2.12.2, one obtains the free-group monad, the free-abelian-group monad, the free-ring monad, etc.

(3) The covariant powerset functor P : Set −→ Set, together with the union mX : PPX −→PX and singleton eX : X −→ PX maps

mX(A) = ⋃A , eX(x) = x ,for all x ∈ X, A ∈ PPX, form the powerset monad P = (P,m, e).

(4) The contravariant powerset adjunction (P •)op a P • : Setop −→ Set (see Exam-ple 2.5.1(6)) has the double-powerset monad P2 = (P 2 = P •(P •)op,m, e) as itsassociated monad on Set, with P 2f(x ) = f [x ] defined by

B ∈ f [x ] ⇐⇒ f−1(B) ∈ x

for f : X −→ Y , and all B ∈ PY , x ∈ P 2X. The components mX : P 2P 2X −→ P 2Xare defined as in the filtered sum construction (see 1.12):

mX(A) = ∑A (A ⊆ PPPX) ,

with (A ∈ ∑A ⇐⇒ a ⊆ PX | A ∈ a ∈ A) for all A ⊆ X, and the componentseX : X −→ P 2X are given by the principal filter

eX(x) = x (x ∈ X) .

(5) The double-powerset functor P 2 : Set −→ Set, along with its monad structure can berestricted to the

· up-set functor U with UX = a ⊆ PX | ↑PX a = a,· filter functor F with FX = a ⊆ PX | a a filter on X,· ultrafilter functor β with βX = a ⊆ PX | a an ultrafilter on X,· identity functor 1Set with 1SetX = X ∼= x | x ∈ X,

and we obtain the monads U = (U,m, e), F = (F,m, e), = (β,m, e), I = (1Set, 1, 1).These monads can be thought of as being induced by the adjunctions

CopC(−,2)

//> Set

(P •)opoo

of Example 2.5.3, with successively C = Ord, SLat, Lat, and Frm. We obtain a chainof monad morphisms

I −→ −→ F −→ U −→ P2 ,

all given objectwise by inclusion maps.Note that there is also for each set X a map αX : PX −→ FX sending a subset A ofX to the principal filter A, and extending to a monad morphism α : P −→ F.


(6) On every category X, there is a unique monad morphism from the identity monadI = (1X, 1, 1) to any monad T = (T,m, e) on X, and it is given by the unit e : 1X −→ Tof T. Hence, the identity monad is an initial object in the metacategory MNDX ofmonads on X, with monad morphisms.

3.2 The Eilenberg–Moore category. Given a monad T = (T,m, e) on X, a T-algebra(or Eilenberg-Moore algebra) is a pair (X, a), where X is an object of X, and the structuremorphism a : TX −→ X satisfies

a · Ta = a ·mX and 1X = a · eX ,

diagrammatically:TTX

Ta //

mX

TX

a

TXa // X

XeX //

1X !!

TX

a

X .

A T-homomorphism f : (X, a) −→ (Y, b) is an X-morphism f : X −→ Y such that

f · a = b · Tf ,

which amounts to commutativity of the diagram

TXTf//

a

TY

b

Xf// Y .

The category of T-algebras and T-homomorphisms is denoted by XT and is also called theEilenberg–Moore category of T.The forgetful functor GT : XT −→ X has a left adjoint F T : X −→ XT:

X 7−→ (TX,mX) , (f : X −→ Y ) 7−→ (Tf : TX −→ TY ) ,

where (TX,mX) is the free T-algebra on X. The unit ηT : 1X −→ GTF T = T of thisadjunction is e, and the counit is described by its components as

εT(X,a) = a : (TX,mX) −→ (X, a) .

Since the monad associated to F T a GT gives back the original T = (T,m, e), any monadmay be obtained via an adjunction. Among such adjunctions, F T a GT is characterized asfollows:

3. MONADS 77

3.2.1 Proposition. If T = (T,m, e) is the monad associated to Fη

ε G : A // X , then

there exists a unique functor K : A −→ XT making both the inner and outer triangles of thefollowing diagram commute:

AG

K // XT

GT

X .F

]]

>

FT

@@

>

(3.2.i)

Proof. The functor K : A −→ XT defined on objects and morphisms by

A 7−→ (GA,GεA) , (f : A −→ B) 7−→ (Gf : GA −→ GB)

makes both triangles in (3.2.i) commute. If K ′ : A −→ XT is another such functor,commutativity of (3.2.i) yields K ′A = (GA, a) (for a certain T-algebra structure a :TGA −→ GA) and K ′f = Gf ; since K ′εA = GεA : K ′FGA −→ K ′A is an XT-morphism,the following diagram commutes:

TGA 1

((TηGA((

1""

TTGATGεA //

GεFGA

TGA

a

TGAGεA

// GA

which tells us that a = GεA.

A right adjoint functor G : A −→ X is monadic if the comparison functor K : A −→ XT of theprevious proposition is an equivalence, and G is strictly monadic if K is an isomorphism;one then says that A is monadic over X, or strictly monadic over X, respectively. A monadicfunctor G : A −→ X can be considered as a measure of the “algebraic character” of A over X,where the Eilenberg–Moore category XT represents the “algebraic part” of A.

3.2.2 Examples.

(1) The forgetful functor G : Mon −→ Set is monadic; in fact, the comparison functor iseven an isomorphism:

SetL ∼= Mon .Similarly, the Eilenberg–Moore category of the free-group monad is isomorphic to Grp,that of the free-abelian-group monad is isomorphic to AbGrp, etc.

(2) For the powerset monad P, the structure map a : PX −→ X of a P-algebra (X, a)defines a complete order on X via ∨

A := a(A) ,


and every P-homomorphism f : (X, a) −→ (Y, b) is then a sup-map. On the otherhand, when X is a complete lattice, the map ∨ : PX −→ X is a P-algebra structure,and sup-maps between complete lattices become P-homomorphisms. Therefore, thereis an isomorphism:

SetP ∼= Sup ,

and the forgetful functor Sup −→ Set is strictly monadic.

(3) Neither of the forgetful functors Ord −→ Set or Top −→ Set is monadic, since theEilenberg–Moore category of each of these is equivalent to Set.

(4) Any full and faithful right adjoint functor is monadic (see also Exercise 3.M).

3.3 Limits in the Eilenberg–Moore category. The forgetful functor GT : XT −→ X isvery well-behaved with respect to limits. Because GT is right adjoint, it naturally preservesall the limits that exist in XT (Proposition 2.11.1). It also reflects limits; in fact, it doesbetter than that.One says that a functor F : C −→ D creates J-limits if for every diagram D : J −→ C suchthat (A,α) is a limit of FD in D, there is a unique pair (L, λ) consisting of an object L ofC and a cone λ : ∆L −→ D satisfying (FL, Fλ) = (A,α), and moreover, (L, λ) is a limitof D. In particular, if F creates J-limits, then it reflects them. The dual notion is that ofcreation of J-colimits.

3.3.1 Proposition. The forgetful functor GT : XT −→ X creates all limits.

Proof. Let D : J −→ XT be a diagram and (X,λ) a limit of GTD in X. There is then aunique morphism a : TX −→ X with

λ ·∆a = GTεTD · Tλ .

All that needs to be shown is that (X, a) is a T-algebra, and that ((X, a), λ) serves as alimit for D in XT. For the T-algebra laws, observe that

λ ·∆(a · eX) = GTεTD · Tλ ·∆eX= GTεTD · eGTD · λ (naturality of e = ηT)= λ ,

and λ ·∆(a ·mX) = GTεTD · Tλ ·∆mX

= GTεTD ·mGTD · TTλ (naturality of m = GTεTF T)= GTεTD · TGTεTD · TTλ (naturality of εT)= GTεTD · Tλ ·∆Ta= λ ·∆(a · Ta) .

3. MONADS 79

Obviously, λ : ∆(X, a) −→ D is actually a cone in XT, and for any cone α : ∆(Y, b) −→ D,one obtains a unique morphism f : Y −→ X in X with

λ ·∆f = GTα .

That f : (Y, b) −→ (X, a) is indeed a T-homomorphism follows from

λ ·∆(f · b) = GTα ·∆GTεT(Y,b)

= GTεTD · TGTα (naturality of εT)= GTεTD · Tλ ·∆Tf= λ ·∆(a · Tf) .

3.3.2 Corollary. For a monadic functor G : A −→ X, if X is J-complete, then A is alsoJ-complete, and G preserves and reflects J-limits.

Proof. This is an immediate consequence of the previous Proposition.

3.4 Beck’s monadicity criterion. In order to characterize monadic functors, let us firstobserve that for a monad T on X, and a T-algebra (X, a), the diagram

TTXmX //

Ta// TX

a // X (3.4.i)

in addition to satisfying a ·mX = a · Ta has the property that there are splittings ηX andηTX : in general, one calls a diagram

Xf//

g// Y

h // Z

in X with h · f = h · g a split fork if there are morphisms i : Z −→ Y and j : Y −→ X (theaforementioned “splittings”) with h · i = 1Z , f · j = 1Y and g · j = i · h:

Y h //

j

1Y

Zi

1Z

Xg//

f

~~

Yh

Yh // Z .

In this case, h is a coequalizer of (f, g) which, trivially, is absolute, in the sense that it ispreserved by every functor defined on X.


The coequalizer diagram (3.4.i) has moreover the special property that it can be “lifted”along GT, that is

(TTX,mTX)mX //

Ta// (TX,mX) a // (X, a) (3.4.ii)

is a coequalizer in XT, presenting the T-algebra (X, a) as a quotient of the free T-algebraF TX. In particular, the parallel pair of morphisms (mX , Ta) in (3.4.ii) has the propertythat its GT-image is part of the split fork (3.4.i). This property turns out to be crucial, andfor a functor G : A −→ X one therefore says that morphisms s, t : A −→ B in A form a G-splitpair (s, t) if f = Gs, g = Gt belong to a split fork in X; similarly, (s, t) is a G-absolutepair if (Gs,Gt) has a coequalizer in X that is absolute. Obviously, every G-split pair isG-absolute.

3.4.1 Theorem (Beck’s Monadicity Criterion). A functor G : A −→ X is monadic ifand only if the following conditions hold:

(1) G is right adjoint;

(2) G reflects isomorphisms;

(3) A has coequalizers of G-split pairs, and G preserves them.

Condition (2) may be replaced by:

(2′) G reflects coequalizers of G-split pairs.

The criterion remains valid if “G-split” is replaced by “G-absolute” everywhere.

Proof. For the necessity of the conditions, even in the formally stronger “G-absolute”version, we remark for (2) that isomorphisms are 1-limits (then see 3.3.1), and refer toExercise 3.D for (3) and (2′). For their sufficiency, consider the adjunction F

η

ε G : A −→ X

with its induced monad T = (T,m, e). Since G = GTK, the comparison functor K alsoreflects isomorphisms, and because of Propositions 2.6.2 and 2.6.1 it suffices to establish anadjunction L

κ

λ

K : A −→ XT with κ an isomorphism.

For a T-algebra (X, a), the G-image of the pair (εFX , Fa) is (mX , Ta). Hence, condition (3)guarantees the existence of a coequalizer diagram

FGFXεFX //

Fa// FX

γ(X,a)// L(X, a) (3.4.iii)

in A that is preserved by G = GTK. Since GT reflects such coequalizers, K preservesthe coequalizers (3.4.iii) as well. Consequently, the unique morphism κ(X,a) given by the

3. MONADS 81

coequalizer property of (3.4.ii) and making the following diagram commute, must actuallybe an isomorphism:

F TTX

1

mX //

Ta// F TX

1

a // (X, a)κ(X,a)

KFTXKεFX //

KFa// KFX

Kγ(X,a)// KL(X, a) .

That κ(X,a) is K-universal is easily seen with the natural correspondence:

(X, a) f−→ KA

L(X, a) g−→ A

for all A ∈ ob A, which arises from the coequalizer property of (3.4.iii), as shown by

FGFX

FGFf

εFX //

Fa// FX

Ff

γ(X,a)// L(X, a)

g

FGFGAεFGA //

FGεA// FGA

εA // A .

This completes the proof that (1)–(3) are sufficient for monadicity of G. If (2) is replacedby (2′), one should observe that the last diagram exhibits the counit λA = g : LKA −→ A ifwe let (X, a) = KA and f = 1KA. In this case, G maps the bottom row of the diagramto a coequalizer diagram (trivial case of (3.4.i)), so under assumption (2′) the bottomrow is already a coequalizer in A, and the “comparison” morphism λA = g must be anisomorphism.

3.4.2 Example. The contravariant powerset functor P • : Setop −→ Set (see 2.2) is monadic.It certainly is right adjoint (see Example 2.5.1(6)) and reflects isomorphisms. Moreover,one easily verifies that P • transforms an equalizer diagram

E

// Xf//

g// Y

in Set into a coequalizer diagram

PYf−1(−)

//

g−1(−)// PX

(−)∩E// PE

in Set, provided that f−1(−) = P •f is surjective (that is, if f is injective). Hence, the BeckCriterion applies.


3.5 Duskin’s monadicity criterion. Coequalizers are often difficult to construct explic-itly, so the third condition in Theorem 3.4.1 can be delicate to handle. In certain categorieshowever, coequalizers can be replaced by other, more practical structures. For example,in Set a kernel pair (r, r′ : R −→ X) of a map f : X −→ Y yields the equivalence relationR ⊆ X ×X:

∀x, y ∈ X ((x, y) ∈ R ⇐⇒ f(x) = f(y)) ,and a coequalizer of (r, r′) is then simply given by the projection π : X −→ X/R. In fact,every equivalence relation in Set is the kernel pair of a split epimorphism (namely theprojection π), see Exercise 2.Cc© .For a functor G : A −→ X, a G-kernel pair is a pair (f, g : A −→ B) such that there is adiagram

XGf//

Gg// Y

h // Z (3.5.i)

where (Gf,Gg) is the kernel pair of h in X. In the case where h is a split epimorphism,this diagram becomes a split fork, so that h is a coequalizer of (Gf,Gg) (Exercise 3.C).A joint kernel pair of (f, g : A −→ B) in a category A is a pair (s, s′ : S −→ A) withf · s = f · s′, g · s = g · s′, and such that if there is a pair (t, t′ : L −→ A) with f · t = f · t′,g · t = g · t′, then there exists a unique A-morphism u : L −→ S with s · u = t and s′ · u = t′:

Ss //

s′// A

f//

g// B .

L

u

OO

t

>>

t′

>>

Since a joint kernel pair of (f, f) is simply a kernel pair of f , a category that has jointkernel pairs also has kernel pairs. Whenever the product B ×B exists, the joint kernel pairof (f, g) is simply the kernel pair of 〈f, g〉 : A −→ B ×B.

3.5.1 Theorem (Duskin’s Monadicity Criterion). Suppose that A has joint kernelpairs and that X has kernel pairs of split epimorphisms. Then a functor G : A −→ X ismonadic if and only if the following conditions hold:

(1) G is right adjoint;

(2) G reflects isomorphisms;

(3) Every G-kernel pair of a split epimorphism has a coequalizer that is preserved by G.

Proof. The necessity of these conditions is immediate from Theorem 3.4.1. Using this sameresult, the sufficiency obviously follows if one proves that every G-split pair

Xf//

g// Y

3. MONADS 83

has a coequalizer that is preserved by G. Given the hypothesis (3), it is enough to constructa G-kernel pair (f ′, g′) of a split epimorphism whose coequalizer is the coequalizer of (f, g),and such that this property is preserved by G. Thus, consider the kernel pair (r, r′) of gand the joint kernel pair (s, s′) of (f · r, f · r′); in X, we obtain a diagram

GSGs //

Gs′// GR

Gr

Gr′

K

k

k′

GAGf

//

Gg// GB

h // Z

whose bottom row is a split fork with splittings i : Z −→ GB and j : GB −→ GA (so thath · i = 1Z , Gf · j = 1GB and Gg · j = i · h), and where (k, k′) is the kernel pair of h. Since

h · (Gf ·Gr) = h ·Gg ·Gr = h ·Gg ·Gr′ = h · (Gf ·Gr′) ,

there is a uniquely determined X-morphism v : GR −→ K with k · v = Gf · Gr andk′ · v = Gf ·Gr′. This morphism is split epic: as G is right adjoint, it preserves kernel pairs,so

Gg · (j · k) = i · h · k = i · h · k′ = Gg · (j · k′)yields the existence of w : K −→ GR with j · k = Gr · w and j · k′ = Gr′ · w; hence,

j · k · v · w = j ·Gf ·Gr · w = j ·Gf · j · k = j · k

implies k·v·w = k and similarly k′ ·v·w = k′, so v·w = 1K by the universal property of (k, k′).Right adjointness of G also makes (Gs,Gs′) into a joint kernel pair of (Gf ·Gr,Gf ·Gr′),and it follows that (Gs,Gs′) is a kernel pair of the split epimorphism v. By hypothesis,(s, s′) has a coequalizer u : R −→ Q that is preserved by G, and since v : GR −→ K isalso a coequalizer of (Gs,Gs′) (Exercise 3.C), there is an isomorphism φ : K −→ GQ withGu = φ · v. By definition of the joint kernel pair (s, s′), the universal property of u yieldsA-morphisms q, q′ : Q −→ B such that q · u = f · r and q′ · u = f · r′:

Ss //

s′// R

r

r′

u // Q

q

q′

Af

//

g// B .

One has k · v = G(f · r) = Gq ·Gu = Gq · φ · v so that k = Gq · φ and similarly k′ = Gq′ · φbecause v is epic. Hence, (Gq,Gq′) is a kernel pair of the split epimorphism h, and byhypothesis, (q, q′) has a coequalizer c : B −→ C that is preserved by G. Thus, one obtains asplit fork

GAGf//

Gg// GB

Gc // GC


and Gc is a coequalizer of (Gf,Gg). To see that c is a coequalizer of (f, g), consider anA-morphism d : B −→ D such that d · f = d · g. Since

d · q · u = d · f · r = d · g · r = d · g · r′ = d · q′ · u

and u—being a coequalizer—is epic, one has d · q = d · q′ and the universal property of(C, c) follows.

3.6 The Kleisli category. The objects of the Kleisli category XT associated to the monadT = (T,m, e) on X are the objects of X, and a morphism f : X Y in XT is simply anX-morphism f : X −→ TY . The Kleisli composition of f : X Y and g : Y Z in XT, isdefined via the composition in X as

g f := mZ · Tg · f .

The identity 1X : X X in this category is just the component eX : X −→ TX of the unite.There is a functor GT : XT −→ X defined on objects and morphisms by

X 7−→ TX (f : X Y ) 7−→ (mY · Tf : TX −→ TY ) .

As for the Eilenberg–Moore category, this functor has a left adjoint FT : X −→ XT:

X 7−→ X , (f : X −→ Y ) 7−→ (eY · f : X Y ) .

The unit ηT : 1X −→ GTFT = T of this adjunction is e, and the components of the counitεT : FTGT −→ 1XT are simply the morphisms 1TX : TX −→ TX in X. Also in this case, themonad associated to this adjunction gives back the original monad T = (T,m, e), and theKleisli category may be characterized dually to XT:

3.6.1 Proposition. If T = (T,m, e) is the monad associated to Fη

ε G : A // X , then

there exists a unique functor L : XT −→ A making both the inner and outer triangles of thefollowing diagram commute:

XT

GT

L // AG

X .FT

^^

>F

AA

>

Moreover, L : XT −→ A is full and faithful.

Proof. The functor L : XT −→ A maps objects like F does, and one has Lf = εFY · Ff onmorphisms f : X Y . Uniqueness follows from the universal property of η, as does thefact that L is full and faithful.

3. MONADS 85

3.6.2 Example. For T = P the powerset monad, a SetP-morphism from X to Y is just arelation r : X −→ PY , while the Kleisli composition is the ordinary relational compositionof 1.2:

s · r = s r ,

for any relations r : X −→ PY , s : Y −→ PZ. Therefore,

SetP = Rel .

3.7 Kleisli triples. There is an alternative presentation of monads which turns out tobe very practical in verifying that given data describe a monad when there is no obviousadjunction inducing it. A Kleisli triple (T, (−)T, e) on a category X consists of

• a function T : ob X −→ ob X sending X to TX,

• an extension operation (−)T sending a morphism f : X −→ TY to a morphism fT : TX −→TY ,

• a morphism eX : X −→ TX for each X ∈ ob X,

subject to(gT · f)T = gT · fT , eTX = 1TX , fT · eX = f (3.7.i)

for all X ∈ ob X, f : X −→ TY , and g : Y −→ TZ. One can set

g f := gT · f ,

so the previous conditions are equivalent to requiring that this “Kleisli composition” isassociative, and that eX acts as an identity. A Kleisli triple morphism α : (S, (−)S, d) −→(T, (−)T, e) is given by a family of morphisms αX : SX −→ TX in X (with X runningthrough ob X) such that

αY · fS = (αY · f)T · αX , αX · dX = eX

for all f : X −→ SY . That is, a Kleisli triple morphism is a family of morphisms thatpreserve the Kleisli composition together with its unit.A Kleisli triple (T, (−)T, e) gives rise to a monad T = (T,m, e) on X simply by setting

Tf := (eY · f)T , mX := (1TX)T ,

for all X-morphisms f : X −→ Y . The facts that T defines a functor, that e and m arenatural transformations, and that the multiplication and unit laws are verified, all followfrom simple manipulations of the equalities (3.7.i). Similarly, a Kleisli triple morphismα : (S, (−)S, d) −→ (T, (−)T, e) yields a natural transformation α : S −→ T , which turns outout to be a morphism α : S −→ T between the corresponding monads.


Conversely, given a monad T = (T,m, e) on X, one obtains a Kleisli triple (T, (−)T, e) via

fT := mY · Tf ,

for all X-morphisms f : X −→ Y , and the conditions (3.7.i) readily follow by using naturalityof e and m, as well as their multiplication and unit laws. A monad morphism α : S −→ Talso yields a morphism α : (S, (−)S, d) −→ (T, (−)T, e) between the corresponding Kleislitriples.Moreover, the passages from a Kleisli triple to a monad and from a monad to a Kleislitriple are mutually inverse, so that both definitions describe the same structure on X (andthe two definitions of Kleisli composition correspond).

3.8 Distributive laws, liftings, and composite monads. Given monads S = (S, n, d)and T = (T,m, e) on X, one can consider the functor ST : X −→ X together with the naturaltransformation d e : 1X −→ ST , but there is no obvious choice in general for a naturaltransformation w : STST −→ ST making ST = (ST,w, d e) into a monad. A first steptowards solving this problem is the introduction of a distributive law of T over S, that is, anatural transformation δ : TS −→ ST making the following diagrams commute:

TSS

Tn

δS // STS Sδ // SST

nT

TSδ // ST

TTS

mS

OO

Tδ // TSTδT // STT

Sm

OO

TTd

dT

TSδ // ST .

S

eS

[[

Se

CC

(3.8.i)

Such a δ allows for a lifting of the monad S on X through GT : XT −→ X, that is, there existsa monad S = (S, n, d) on XT such that

GTS = SGT , GTn = nGT , GTd = dGT .

The first of these conditions may be used to identify the domains and codomains of thenatural transformations in the last two equalities. The latter state that the underlyingX-morphisms of n(X,a), d(X,a) are nX , dX , respectively, for any T-algebra (X, a), so themultiplication and unit laws of S are automatically satisfied. Therefore, a lifting of Sthrough GT : XT −→ X is simply provided by a functor S : XT −→ XT making

XT

GT

S // XT

GT

X S // X

3. MONADS 87

commute, and such that nX : SS(X, a) −→ S(X, a), dX : (X, a) −→ S(X, a) are XT-morphisms for all (X, a) ∈ ob(XT). Thus, if δ : TS −→ ST is a distributive law of T over S,a lifting S : XT −→ XT is obtained via

(X, a) 7−→ (SX, Sa · δX) , (f : X −→ Y ) 7−→ (Sf : SX −→ SY ) .

The fact that the components of n : SS −→ S and d : 1XT −→ S have underlying XT-morphisms then follows directly from commutativity of the diagrams in (3.8.i).In turn, a lifting of S through GT : XT −→ X yields the following composite adjunction

(XT)SGS //> XT

F Soo

GT //> X .FT

oo

Since ST = SGTF T = GTSF T = GTGSF SF T, the monad on X associated to this adjunctionis

ST = (ST,w, d e) ,where w = GTGSεSF SF T ·GTGSF SεTGSF SF T by Proposition 2.5.5 and 3.1, so that

w = GTnF T ·GTSεTSF T = nT · SGTεTSF T .

By combining this last expression of w with the lifting conditions, one easily verifies thatthe following equalities also hold:

(1) w · SeST = nT ;

(2) w · STdT = Sm;

(3) w · STSm = Sm · wT ;

(4) w · nTST = nT · Sw.

More generally, we say that a monad ST = (ST,w, d e) is a composite of S and T if itsmultiplication w satisfies (1)–(4). These conditions are useful in proofs, but there is a moreelegant and practical presentation, as the next Lemma shows. Given monads S = (S, n, d),T = (T,m, e) on X, one says that a monad ST = (ST,w, d e) satisfies the middle unit lawif the following diagram commutes:

STSedT

~~

1ST

STSTw // ST

that is, if w · (Se dT ) = 1ST .3.8.1 Lemma. For monads S = (S, n, d), T = (T,m, e), and ST = (ST,w, d e) on X, thefollowing are equivalent:

(i) ST is a composite monad;


(ii) the natural transformations Se : S −→ ST and dT : T −→ ST are monad morphisms,and ST satisfies the middle unit law.

Proof. The proof that (i) =⇒ (ii) follows from routine verifications. To show (ii) =⇒ (i),we must verify the four composite monad conditions. For (1), observe that

w · SeST = w · (STw · STSTdT · STSeT ) · SeST (middle unit law)= w · wST · STSTdT · STSeT · SeST (multiplication law)= w · STdT · wT · STSeT · SeST (naturality of w)= w · STdT · SeT · nT (Se monad morphism)= nT (middle unit law).

Condition (2) is proved in a similar way, by inserting the middle unit law wST · SeSTST ·SdTST = 1STST into w · STdT , and proceeding as above. To see (3), we write

w · STSm = w · (STw · STSTdT ) (ST applied to (2))= w · wST · STSTdT (multiplication law)= w · STdT · wT (naturality of w)= Sm · wT (by (2) again).

The last condition is proved similarly by exploiting (1) instead of (2).

Given a monad ST = (ST,w, de) on X that is a composite of S = (S, n, d) and T = (T,m, e),there is a natural transformation

δ := (TS dTSe // STSTw // ST ) ,

which turns out to be a distributive law of T over S (as can be checked routinely).As these constructions suggest, distributive laws, liftings, and composite monads areequivalent concepts:

3.8.2 Proposition. For monads S and T on X, there is a bijective correspondence between:

(i) distributive laws δ of T over S;

(ii) liftings S of S through GT : XT −→ X;

(iii) monads ST = (ST,w, d e) that are composites of S and T.

3. MONADS 89

Proof. The previous discussion has already shown that these concepts are related as in:

(i)

(iii)

?G

(ii) .ks

To verify the statement, it suffices to prove that any path of length three in this diagram isthe identity. We check this for liftings, the other verifications being similar. By definition, alifting is entirely determined by its behavior on the structure morphisms of XT-objects (X, a).Let us denote by (SX, a′) the S-image of (X, a), and by a′′ the structure corresponding tothe path (ii) =⇒ (iii) =⇒ (i) =⇒ (ii). By using that SεT(X,a) : (STX,m′X) −→ (SX, a′) is anXT-morphism, one observes

a′′ = Sa · nTX · S(GTεTSFTX

· TSeX) · dTSX= Sa · nTX · dSTX ·GTεT(STX,m′X) · TSeX (naturality of d)= Sa ·m′X · TSeX (unit law for S)= a′ · TSa · TSeX (SεT(X,a) a morphism)= a′ .

3.8.3 Corollary. The monads ST that are composites of S = (S, n, d) and T = (T,m, e)are exactly those of the form

ST = (ST, (n m) · SδT, d e) ,

where δ is a distributive law of T over S. There are also monad morphisms

Se : S −→ ST , dT : T −→ ST .

Proof. By the Proposition, any composite monad comes from a distributive law, and theresulting multiplication w is then easily seen to be (n m) · SδT . The fact that Se and dTare monad morphisms follows by a routine verification.

3.8.4 Proposition. For a composite monad ST and the corresponding lifting S (see Propo-sition 3.8.2), the comparison functor K : (XT)S −→ XST is an isomorphism.

Proof. The comparison functor sends an object ((X, t), s) of (XT)S to the ST-algebra(X, s · St), and its inverse maps (X, a) to ((X, a · dTX), a · SeX).


3.8.5 Examples.

(1) For any monad T on X, there are trivial distributive laws 1T : 1XT −→ T1X of theidentity monad I over T, and 1T : T1X −→ 1XT of T over I.

(2) A distributive law δ : LG −→ GL of the free-monoid monad L over the free-abelian-group monad G (Example 3.1.1(2)) is obtained by sending an element of LGX of theform((∑

i1=1,...,n1 α1,i1 · x1,i1

),(∑

i2=1,...,n2 α2,i2 · x2,i2

), . . . ,

(∑ik=1,...,nk αk,ik · xk,ik

))to the element of GLX given by∑

i1,i2,...,ik α1,i1α2,i2 · · ·αk,ik · (x1,i1 , x2,i2 , . . . , xk,ik) .

The resulting category of GL-algebras is the category Rng of unital rings and theirhomomorphisms (rings also occur as monoids in the category of abelian groups, seeExample 4.2.1(2)).From this example, distributive laws may be seen as a generalization of the usualnotion of distributivity (of multiplication over addition) that holds in rings.

3.9 Distributive laws and extensions. For monads S and T on X, an extension of Talong FS : X −→ XS is a monad T = (T , m, e) on XS satisfying

FST = TFS , FSm = mFS , FSe = eFS .

As is the case for liftings, the first condition insures that the last two equalities makesense. Moreover, since FS is identical on objects, one has mX = FSmX , eX = FSeX forall X-objects X, so the multiplication and unit laws are immediate. An extension of T istherefore given by a functor T : XS −→ XS making

XST // XS

XFS

OO

T // XFS

OO

commute, and such that mX = FSmX : TTX TX, eX = FSeX : X TX (X ∈ ob X)become the components of natural transformations m : T T −→ T , e : 1XS −→ T .In the presence of a distributive law δ of T over S, an extension T : XS −→ XS is obtainedvia

X 7−→ TX (f : X −→ SY ) 7−→ (δY · Tf : TX −→ STY ) .

Functoriality of T follows from naturality of δ, and commutativity of the lower diagrams in(3.8.i) of 3.8. To verify naturality of m and e in XS, one uses the other two diagrams.

3. MONADS 91

An extension of T along FS : X −→ XS yields the following composite adjunction

(XS)TGT //> XSFT

oo

GS //> X .FS

oo

One can then consider the monad ST = (ST,w, d e) as being induced by FTFS a GSGT,with

w = Sm ·GST εSFST .

The fact that this monad is a composite of S and T may then be checked directly. By 3.8,we can get back a distributive law from an extension, and we have:

3.9.1 Proposition. For monads S and T on X, there is a bijective correspondence between:

(i) distributive laws δ of T over S;

(ii) extensions T of T along FS : X −→ XS.

Proof. An easy verification shows that when an extension is obtained via a distributivelaw, the induced composite monad is the same as in Proposition 3.8.2, so we only need tocheck that the path (ii) =⇒ (i) =⇒ (ii) returns the same extension. We denote by T theoriginal extension, by T ′ the new one, and recall that εSY = 1SY . If f : X −→ SY is anXS-morphism, then

T ′f = SmY ·GST1STY ·GSFSTGSFSeY · dTSY · Tf= SmY ·GST (1STY FSGSFSeY ) · dTSY · Tf (functoriality of T )= SmY ·GST (FSeY 1SY ) · dTSY · Tf (naturality of εS)= SmY ·GSFSTeY ·GST1SY · dTSY · Tf (T an extension)= nTY · ST1SY · dTSY · Tf (unit law of T)= nTY · dSTY · T1SY · Tf (naturality of d)= T1SY FSTf (unit law of S)= T (1SY FSf) (T an extension)= T f .

3.9.2 Proposition. For a composite monad ST and the corresponding extension T, thefunctor L : (XS)T −→ XST of Proposition 3.6.1 is the identity.

Proof. The statement is immediate once the definition of the composition in (XS)T hasbeen unravelled.


Exercises

3.A Trivial monads on Set. An object (X, a) of the category SetT is trivial if X is eitherthe empty set ∅ or a singleton 1 = ?. If there exists at least one non-trivial T-algebra,then the unit eX : X −→ TX is injective for all sets X. The only examples of trivial monads(monads that admit only trivial Eilenberg–Moore algebras) are the terminal monad 1 inMNDSet, whose functor sends all sets X to 1, and the monad whose functor sends the emptyset to itself and all other sets to 1.Hint. (TX,mX) is a T-algebra for any monad T.

3.B M-actions and equivariant maps. Any monoid (M,m, e) (as in 1.1) yields a monadM = (M × (−),m, e) on Set, where mX(a, (b, x)) = (m(a, b), x) and eX(x) = (e, x). TheEilenberg–Moore category SetM of this monad is the category M -Set of M-actions andequivariant maps which is isomorphic to the functor category SetM , with M considered as aone-object category. Every monoid homomorphism f : M −→ N yields a monad morphismM −→ N; in fact, this operation describes a functor Mon −→ MNDSet.

3.C Split forks and coequalizers. A fork in a category X is a diagram

Xf//

g// Y

h // Z

such that h · f = h · g. Show that a fork splits if and only if h is a coequalizer of (f, g) andthere is j : Y −→ X with f · j = 1Y and g · j · f = g · j · g. If h is a split epimorphism and(f, g) is its kernel pair, then the fork splits.

3.D Creation of colimits by GT. If T is a monad on X, the functor GT : XT −→ X createsall colimits that exist in X and that are preserved by T and TT . Therefore, GT createscoequalizers of G-absolute pairs.

3.E Full- and faithfulness of the comparison functor. The comparison functor K : A −→ XT

(see Proposition 3.2.1) is full and faithful if and only if εA is a regular epimorphism for allA ∈ ob A; in that case, εA is coequalizer of (εFGA, FGεA) for all A ∈ ob A.

3.F Strict monadicity criterion. For a functor G : A −→ X, the following conditions areequivalent:

(i) G is strictly monadic;

(ii) G is right adjoint and creates coequalizers of G-absolute pairs;

(iii) G is right adjoint and creates coequalizers of G-split pairs.

3.G Eilenberg–Moore algebras via Kleisli triples. If (T, (−)T, e) is a Kleisli triple on acategory X, then the Eilenberg–Moore algebras associated to the monad T are those pairs

3. MONADS 93

(X, a) with X ∈ ob X and a : TX −→ X an X-morphism such that

∀f, g ∈ X(Y, TX) (a · f = a · g =⇒ a · fT = a · gT) and a · eX = 1X .

3.H Functor liftings and monad morphisms. Let Feε

G : A −→ X be an adjunction withassociated monad T = (T,m, e) on X. For a monad S = (S, n, d) on Y, a lifting of a functorR : X −→ Y through (G,GS) is a functor R : A −→ YS that makes the diagram

A R //

G

YS

GS

X R // Y

commute. Equivalently, a lifting of R through (G,GS) can be given by a natural transfor-mation α : SR −→ RT satisfying

Rm · αT · Sα = α · nR and Re = α · dR . (3.9.i)

Indeed, such an α yields a functor R : A −→ YS via

RA := (RGA,RGεA · αGA) ,

and conversely, a lifting R of R through (G,GS) returns a natural transformation α satisfying(3.9.i) via

αX := mX · SReX ,

where mX : SRTX −→ RTX denotes the Y-morphism defined by RFX = (RTX, mX).As a consequence, liftings of 1X : X −→ X through (G,GS) are in one-to-one correspondencewith monad morphisms α : S −→ T. A lifting of a monad S along GT (in the sense of 3.8)provides a lifting of the underlying functor S : X −→ X through (GT, GT).

3.I Functor extensions and monad morphisms. Consider an adjunction F eε

G : A −→ Xwith associated monad T = (T,m, e) on X. For a monad S = (S, n, d) on Y, an extensionof a functor R : Y −→ X along (FS, F ) is a functor R : YS −→ A that makes the followingdiagram commute:

YSR // A

YFS

OO

R // X .F

OO

Extensions of R along (FS, F ) are in one-to-one correspondence with natural transformationsα : RS −→ TR satisfying

mR · Tα · αS = α ·Rn and eR = α ·Rd . (3.9.ii)


Indeed, given such an α, one obtains a functor R : YS −→ A defined on Y-objects Y byRY = FRY , and on Y-morphisms f : X −→ SY by

Rf := εFRY · F (αY ·Rf) ;

conversely, an extension R of R along (FS, F ) returns a natural transformation α satisfying(3.9.ii) via

αY := GR1SY · eRSY .

In particular, an extension of 1X : X −→ X along (FS, F ) is equivalently described by amonad morphism α : S −→ T, and an extension of a monad S along FS (in the sense of 3.9)yields an extension of T : X −→ X along (FS, FS). In fact, for monads S and T on X, adistributive law α of T over S is equivalent to a natural transformation α inducing both anextension of T : X −→ X along (FS, FS) and a lifting of S : X −→ X through (GT, GT) (as inExercise 3.H).

3.J Distributive laws as monoids. A lifting α : SR −→ RT of a functor R : X −→ A through(GT, GS), where S = (S, n, d) is a monad on A and T = (T,m, e) a monad on X (seeExercise 3.H), can be seen as an arrow (R,α) : S −→ T. Composition with (L, β) : T −→ Uis given via

(L, β) (R,α) := (RL,Rβ · αL) .Liftings from S to T form the objects of a metacategory MNDLFT(S,T), whose morphismsλ : (R,α) −→ (R′, α′) are natural transformations λ : R −→ R′ satisfying α′ ·Sλ = λT ·α. Inthe same way that a monad is a monoid in a functor category (see 3.1), a distributive lawδ of T over S is equivalently described as a monoid in MNDLFT(T,T) given by the lifting(S, δ) together with two morphisms

n : (S, δ) (S, δ) −→ (S, δ) and d : (1X, 1T ) −→ (S, δ) .

A similar procedure using functor extensions (see Exercise 3.I) leads to the descriptionof a distributive law δ of T over S as a monoid ((T, δ),m, e) in MNDEXT(S,S), whereMNDEXT(S,T) denotes the metacategory of extensions from S to T.

3.K Adjoint Triangle Theorem.

(1) Show that the following assertions are equivalent for an adjunction Fη

ε G : A −→ X

with induced monad T:

(a) for every object A in A, εA is a coequalizer of εFGA, FGεA;(b) for every object A in A, εA is a regular epimorphism;(c) the comparison functor K : A −→ XT is full and faithful.

Adjunctions satisfying these equivalent conditions are said to be of descent type, inwhich case G is called premonadic.

3. MONADS 95

(2) Consider a commutative triangle of functors

A

G

K // B

J

X

with G, J right adjoint, so that one has Fη

ε G , H δ

γ J , and J premonadic. With

µ : H −→ KF determined by Jµ · δ = η, set

α := FJγ , β := εFJ · FJµJ .

Prove the equivalence of the following assertions for B ∈ ob B:

(a) there exists a K-universal arrow for B;(b) there exists a coequalizer of αB, βB in A.

(3) For functors K : A −→ B, J : B −→ X with J premonadic and coequalizers existing inA, the functor K has a left adjoint if and only if JK has a left adjoint.

3.L Cocompleteness of the Eilenberg–Moore category. For a monadic functor G : A −→ X,if A has coequalizers and X is J-cocomplete, then A is also J-cocomplete. In particular,for a monad T on a category X with coproducts, XT is small-cocomplete whenever it hascoequalizers.Hint. Consider the commutative diagram

A ∆ //

G

AJ

GJ

X ∆ // XJ

and apply 3.K(2).

3.M Idempotent monads. A monad T = (T,m, e) on X is idempotent if m : TT −→ T is anisomorphism. For any adjunction

Fη

ε G : A // X

inducing T, the following statements are equivalent:

(i) T is idempotent;

(ii) Tη = ηT ;

(iii) any one of Gε, ηG, εF , Fη is an isomorphism;


(iv) for all T-algebras (X, a), the structure a is an X-isomorphism;

(v) GT : XT −→ X restricts to an isomorphism of XT with the full subcategory of Xcontaining those X with ηX an isomorphism;

(vi) FGT is left adjoint to the comparison functor K : A −→ XT and the unit κ of thisadjunction satisfies GTκ = ηGT.

These conditions hold in particular when G or F is full and faithful.

4. MONOIDAL AND ORDERED CATEGORIES 97

4 Monoidal and ordered categories

4.1 Monoidal categories. A monoidal category is a category C together with

(1) a functor (−)⊗ (−) : C× C −→ C,

(2) a distinguished object E in C,

(3) natural isomorphisms

αA,B,C : A⊗ (B ⊗ C) −→ (A⊗B)⊗ C , λA : E ⊗ A −→ A , ρA : A⊗ E −→ A

making the following diagrams commute for all objects A,B,C,D:

A⊗ (B ⊗ (C ⊗D))1A⊗αB,C,D

αA,B,C⊗D// (A⊗B)⊗ (C ⊗D)

αA⊗B,C,D// ((A⊗B)⊗ C)⊗D

A⊗ ((B ⊗ C)⊗D)αA,B⊗C,D

// (A⊗ (B ⊗ C))⊗D

αA,B,C⊗1D

OO

A⊗ (E ⊗B)

1A⊗λB &&

αA,E,B// (A⊗ E)⊗B

ρA⊗1Bxx

A⊗B

and satsifying λE = ρE.

The monoidal category C is symmetric if there are also

(4) natural isomorphisms σA,B : A⊗B −→ B ⊗ A with σB,A · σA,B = 1A⊗B such that thefollowing diagrams commute:

A⊗ (B ⊗ C)1A⊗σB,C

αA,B,C// (A⊗B)⊗ C

σA⊗B,C// C ⊗ (A⊗B)αC,A,B

A⊗ (C ⊗B)αA,C,B

// (A⊗ C)⊗BσA,C⊗1B

// (C ⊗ A)⊗B

E ⊗ A

λA

σE,A// A⊗ E

ρA

A .

Since λE = ρE, the commutative triangle gives σE,E = 1E,E.

This somewhat cumbersome definition is best illustrated in terms of examples.

4.1.1 Examples.

(1) Set, with its cartesian structure ⊗ = × (see 1.1) is symmetric monoidal. In fact, everycategory C with finite products can be considered a symmetric monoidal category (seeExercise 2.H).


(2) The prototype of a symmetric monoidal category is the category AbGrp of abeliangroups with its usual tensor product. (For abelian groups A,B, the abelian groupA⊗ B comes with a bilinear map ⊗ : A× B −→ A⊗ B such that any bilinear mapf : A × B −→ C factors as f = g · ⊗, with a uniquely determined homomorphismg : A⊗B −→ C.) One puts E = Z and produces α, λ, ρ, σ with the universal propertyof the tensor product. More generally, such a construction works for the categoryModR of R-modules, where R is a commutative unital ring.

(3) Sup has a symmetric monoidal structure which may be constructed analogously to thetensor product of AbGrp. Indeed, for X, Y, Z ∈ ob Sup, call a mapping f : X×Y −→ Za bimorphism if f(x,−) : Y −→ Z and f(−, y) : X −→ Z are morphisms of Sup for allx ∈ X, y ∈ Y . Then one can construct the universal bimorphism ⊗ : X×Y −→ X⊗Y(so that every bimorphism f factors as f = g·⊗, with a unique sup-map g : X⊗Y −→ Z)similarly to the tensor product of abelian groups: on the free sup-lattice P (X×Y ) (seeExample 2.5.1(5)), consider the least compatible congruence relation ∼ that makesthe composite map

X × Y eX×Y// P (X × Y ) proj

// P (X × Y )/∼

a bimorphism; this composite map is universal, hence X ⊗ Y = P (X × Y )/∼.

Many of the monoidal categories considered in this book are strict, that is: α, λ, and ρ canall be taken to be identity morphisms, so that the coherence conditions (expressed by thefirst two commutative diagrams) are trivially satisfied. These strict monoidal categories,however, typically fail to be symmetric:

(4) For a set X, consider the ordered set Rel(X,X) of relations on X, seen as a category.Relational composition defines a strict monoidal structure on Rel(X,X) (see 1.2).

(5) More generally, let V be an ordered set, considered as a category. V becomes strictmonoidal precisely when it carries a monoid structure whose multiplication V×V −→ Vis monotone. In particular, every quantale V is a strict monoidal category.

(6) For a category C the functor category CC becomes a strict monoidal metacategory,with composition of functors as tensor product.

(7) If C is a monoidal category, then so is Cop (and similarly for a symmetric monoidalcategory C).

A homomorphism of monoidal categories C and D is a functor F : C −→ D which comeswith natural isomorphisms

δA,B : FA⊗D FB −→ F (A⊗C B) , ε : ED −→ FEC


for all A,B ∈ ob C, which commute with the natural isomorphisms defining the structureof the monoidal categories C and D:

FA⊗ (FB ⊗ FC)1⊗δ

α // (FA⊗ FB)⊗ FCδ⊗1

FA⊗ F (B ⊗ C)δ

F (A⊗B)⊗ FCδ

F (A⊗ (B ⊗ C)) Fα // F ((A⊗B)⊗ C)

E ⊗ FAε⊗1

λ // FA

FE ⊗ FA δ // F (E ⊗ A)Fλ

OO FA⊗ E1⊗ε

ρ// FA

FA⊗ FE δ // F (A⊗ E) .

Fρ

OO

Observe that naturality of δ makes F commute with ⊗, up to isomorphism, not just forobjects but also for morphisms:

FA⊗ FBδ

Ff⊗Fg// FC ⊗ FD

δ

F (A⊗B) F (f⊗g)// F (C ⊗D)

4.2 Monoids. Let C be a monoidal category as in 4.1. A monoid M in C is a C-objecttogether with two morphisms

m : M ⊗M −→M , e : E −→M

such that the diagrams

M ⊗ (M ⊗M)1M⊗m

αM,M,M// (M ⊗M)⊗M m⊗1M //M ⊗M

m

M ⊗M m //M

E ⊗M

λM''

e⊗1M //M ⊗Mm

M ⊗ E

ρMww

1M⊗eoo

M

commute. A homomorphism of monoids f : (M,m, e) −→ (N, n, d) is a C-morphism makingthe diagrams

M ⊗Mm

f⊗f// N ⊗N

n

Mf

// N

Ee

d

Mf

// N


commute. The resulting category is denoted by MonC. A comonoid M in C is simply amonoid M in Cop.

4.2.1 Examples.

(1) For Set with its cartesian structure, MonSet = Mon.

(2) A unital ring R is an abelian group that is also a monoid in which the distributive lawshold, that is, the multiplication R×R −→ R is Z-bilinear and is therefore equivalentlydescribed as a homomorphism R ⊗ R −→ R. Hence, unital rings are precisely themonoids in AbGrp (with its usual tensor product), and MonAbGrp = Rng, the categoryof unital rings and their homomorphisms.

(3) A quantale V is a complete lattice with a monoid operation ⊗ : V × V −→ V that pre-serves suprema in each variable; with the tensor product in Sup (see Example 4.1.1(3)),the monoid operation may equivalently be considered a morphism V ⊗ V −→ V in Sup.This way, one shows MonSup = Qnt.

(4) Monoids in Rel(X,X) (see Example 4.1.1(4)) are precisely orders on the set X (seealso 1.3).

(5) A monoid in a quantale V (see Example 4.1.1(5)) is simply an idempotent elementv ∈ V (so that v ⊗ v = v) with k ≤ v. A comonoid in V is an idempotent elementv ∈ V with v ≤ k. A frame is thus an example of a commutative quantale in whichevery element is a comonoid.

(6) For a category C, monoids of the monoidal category CC (see Example 4.1.1(6)) areprecisely monads on C, and homomorphisms of monoids are precisely morphisms ofmonads (see 3.1).

(7) A topology on a set X may be defined by a monoid in SLat(PX,PX) (see Exercise 1.F),where PX is considered as a join-semilattice.

4.3 Actions. Let C be a monoidal category, and M = (M,m, e) a monoid in C. A leftM-action (or simply an M-action) is an object A in C that comes with a C-morphism

a : M ⊗ A −→ A

such that (with the notations of 4.1 and 4.2) the following diagrams commute:

M ⊗ (M ⊗ A)(m⊗1A)·αM,M,A

1M⊗a //M ⊗ Aa

M ⊗ A a // A

E ⊗ A e⊗1M //

λA&&

M ⊗ Aa

A .


A C-morphism f : A −→ B between M -actions (A, a) and (B, b) is equivariant if

M ⊗ Aa

1M⊗f //M ⊗Bb

Af

// B

commutes. Similarly, a right M -action is an object A in C with a C-morphism a : A⊗M −→ Amaking the corresponding diagrams commute.Any monoid (M,m, e) in a monoidal category C gives rise to a monad M = (M ⊗ (−),m, e)on C, where

mA = (m⊗ 1A) · αM,M,A and eA = (e⊗ 1A) · λ−1A

for all A ∈ ob C (using the notations of 4.2). The Eilenberg–Moore category CM of thismonad is the category of M -actions and equivariant C-morphisms.

4.3.1 Examples.

(1) If C = Set the category of M -actions is the category M -Set of M -actions and equiv-ariant maps (see Exercise 3.B).

(2) The monoidal structure of AbGrp is given by the tensor product over Z (Exam-ple 4.1.1(2)), and a monoid R in AbGrp is a ring (Example 4.2.1(2)). Hence, AbGrpR(with R the monad induced by the monoid R) is the usual category of left R-modules.

(3) Given a quantale V = (V,⊗, k), that is, a monoid in Sup (Example 4.2.1(3)), thecategory SupV is described as follows. A V-action X in Sup is a complete lattice Xtogether with a bimorphism (−) · (−) : V ×X −→ X in Sup (Example 4.1.1(3)) suchthat

(u⊗ v) · x = u · (v · x) , k · x = x ,

for all v ∈ V , x ∈ X, and a sup-map f : X −→ Y is equivariant whenever

f(v · x) = v · f(x)

for all v ∈ V, x ∈ X. Objects in SupV are also known as V-modules (see [Krumland Paseka, 2008]), but here we reserve this name for a generalization of the termintroduced in 1.4 (see III.1.3).

4.4 Monoidal closed categories. An object A in a monoidal category C (as in 4.1) is⊗-exponentiable if the functors

A⊗ (−) , (−)⊗ A : C −→ C

have right adjointsA (−) , (−) A : C −→ C ,


respectively, that is: if for all objects B,C in C there are bijections

A⊗B −→ C

B −→ (A C)B ⊗ A −→ C

B −→ (C A)

which are natural in B. The functor A (−) is the right internal hom-functor ofA, and (−) A is the left internal hom-functor. Obviously, when C is symmetric,(A (−)) ∼= ((−) A), and there is no need for distinction between right and left, soeither of the two notations may be used. The monoidal category C is closed if every objectis ⊗-exponentiable.If the monoidal structure of C is the cartesian structure of C (given by finite direct products),then one says exponentiable instead of ×-exponentiable and usually writes

BA

for the internal hom-object A B ∼= B A, also called an exponential, and one says thatC is cartesian closed if it is closed.

4.4.1 Examples.

(1) Set, Ord, and Cat are cartesian closed. For Set, see Example 2.5.1(4). Essentially thesame proof can be used to prove cartesian closure of Ord, with

BA = Ord(A,B)

provided with its pointwise order. In Cat, the functor category BA serves as internalhom-object, see Exercise 4.A.

(2) AbGrp, and more generally ModR (for a commutative unital ring R) are monoidalclosed, with

A B = homR(A,B) = ModR(A,B) .

(3) Sup is monoidal closed, with

X Y = Sup(X, Y ) ,

provided with the pointwise structure.

(4) Rel(X,X), considered as a monoidal category as in Example 4.1.1(4), is monoidalclosed, and one has

x (r s) y ⇐⇒ ∀z ∈ X ( y r z =⇒ x s z ) ,x (s r) y ⇐⇒ ∀z ∈ X ( z r x =⇒ z s y ) ,

for all x, y ∈ X.


(5) The internal hom-objects in a quantale V have been described in 1.10; hence V ismonoidal closed. As mentioned in Example 4.1.1(5), any ordered monoid (that is, anyordered set with a monotone monoid operation) can be considered a monoidal category.In particular, every monoid V , provided with the discrete order, is a monoidal category,and it is closed precisely when it is a group. In that case, for u, v ∈ V , one has

u v = u−1v , v u = vu−1 ,

which provides some justification for the internal hom notation in general.

(6) For a category C, the monoidal category CC of Example 4.1.1(6) is rarely closed.For example, when C is a small discrete category, CC is the monoid Set(X,X) withX = ob C which, according to the previous example, is monoidal closed only when itis a group, that is, when |X| ≤ 1.

4.4.2 Proposition. In a locally small monoidal closed category C, one has natural iso-morphisms

(1) (A⊗B) C ∼= B (A C);

(2) C (A⊗B) ∼= (C B) A;

(3) (A C) B ∼= A (C B).

If C is cartesian closed, then CA×B ∼= (CA)B ∼= (CB)A.

Proof. For all objects X, one has

X −→ ((A⊗B) C)(A⊗B)⊗X −→ C

A⊗ (B ⊗X)−→ C

B ⊗X −→ (A C)X −→ (B (A C)) ,

which implies (1) (because the Yoneda embedding reflects isomorphisms, see Corollary 2.4.2and Exercise 2.A). Point (2) is proved analogously to (1). For (3), we note

X −→ ((A C) B)X ⊗B−→ (A C)

A⊗ (X ⊗B)−→ C

(A⊗X)⊗B−→ C

A⊗X −→ (C B)X −→ (A (C B)) .


Given an object C of a monoidal closed category C, there is a functor

(−) C : Cop −→ C , (f : A −→ B) 7−→ (f 1C : (B C) −→ (A C)) ,

with f 1C the unique C-morphism that makes the diagram

A⊗ (A C)evAC // C

A⊗ (B C)

1A⊗(f1C)

OO

evBC ·(f⊗1BC)

99

commute (here evA : A⊗ (A (−)) −→ 1C denotes the counit of the adjunction A⊗ (−) a(−)A). One can naturally define a functor C (−) that will be isomorphic to (−) Cwhenever C is symmetric. In this case, the functor is also right adjoint:4.4.3 Proposition. For a symmetric monoidal closed category C, the functor (−) C :Cop −→ C is self-adjoint (for all C-objects C).

Proof. By couniversality of the component at C of the counit evAC : (A C)⊗ ((AC) (−)) −→ 1C, there is a unique C-morphism uA : A −→ ((A C) C) making thefollowing diagram commute:

(A C)⊗ ((A C) C)evACC // C .

(A C)⊗ A

1AC⊗uA

OO

evAC ·σAC,A

44

The morphism uA is a ((−) C)-universal arrow for A. Indeed, given a C-morphismf : A −→ (B C), we define g : B −→ (A C) as the unique C-morphism such that

A⊗ (A C)evAC // C

A⊗B

1A⊗gOO

evBC ·(1B⊗f)·σA,B

66

commutes; since

evBC ·(1B ⊗ ((g 1C) · uA)) = evBC ·(1B ⊗ (g 1C)) · (1B ⊗ uA) ((−)⊗ (−) functor)= evACC ·(g ⊗ 1(AC)C) · (1B ⊗ uA) (definition of g 1C)= evACC ·(1AC ⊗ uA) · (g ⊗ 1A) ((−)⊗ (−) functor)= evAC ·σAC,A · (g ⊗ 1A) (definition of uA)= evAC ·(1A ⊗ g) · σB,A (σ natural)= evBC ·(1B ⊗ f) (definition of g),


one can conclude that (g 1C) · uA = f by couniversality of evBC . The displayed identitiesalso show that g is uniquely determined, so that uA is universal as claimed. To see that((−) C)op : C −→ Cop is the required left adjoint, it suffices to verify that the morphisms uA(with A ∈ C) form a natural transformation u : 1C −→ (((−) B) B) (Theorem 2.5.4);by couniversality of evBC , this follows from the identities

evBCC ·(1BC ⊗ (uB · f)) = evBC ·σBC,B · (1BC ⊗ f)= evBC ·(f ⊗ 1BC) · σBC,A= evAC ·(1A ⊗ (f C)) · σBC,A= evAC ·σAC,A · ((f C)⊗ 1A)= evACC ·(1AC ⊗ uA) · ((f C)⊗ 1A)= evACC ·((f C)⊗ 1(AC)C) · (1BC ⊗ uA)= evBCC ·(1BC ⊗ (((f C) C) · uA))

(for all C-morphisms f : A −→ B).

4.4.4 Example. When C = Set with its cartesian closed structure and C = 2, for every setX the internal hom-object 2X = X 2 is the powerset PX. The functor 2(−) = (−) 2then describes the contravariant powerset functor P • of Example 2.5.1(6).

4.5 Ordered categories. An ordered category C is a category C with each hom-classC(A,B) carrying an order (that is, a reflexive and transitive relation ≤), such that thecomposition maps

C(A,B)× C(B,C) −→ C(A,C) , (f, g) 7−→ g · f ,

are monotone. This is already guaranteed if composition from either side preserves theorder, that is: if for all f, f ′ : A −→ B, h : B −→ C, g : D −→ A, one has

f ≤ f ′ =⇒ h · f · g ≤ h · f ′ · g .

Hence, an ordered category is a special case of a 2-category (see [Borceux, 1994a]), namelya 2-category whose local categories are thin (i.e., ordered classes). Alternatively, readersfamiliar with enriched category theory (see [Kelly, 1982]) may want to think of an orderedcategory as a category enriched over the metacategory ORD of ordered classes; see also 4.10below.

4.5.1 Examples.

(1) Rel is an ordered category, with Rel(X, Y ) ordered by inclusion for all sets X, Y .

(2) Mod is, like Rel, an ordered category.

(3) Ord is an ordered category, with Ord(X, Y ) carrying the pointwise order.


(4) Sup is, like Ord, an ordered category.

(5) Every category can be considered an ordered category when provided with the discreteorder.

Note that for any object A of an ordered category C, C(A,A) can be considered a strictmonoidal category, with the tensor product given by composition in C (see Examples 4.1.1(4),(5), and (6)).If C is an ordered category, then its dual Cop (obtained by “turning arrows around”) is alsoordered. Moreover, one can form the conjugate ordered category Cco that leaves the arrowsintact but turns around the order:

Cco(A,B) = (C(A,B),≥) = (C(A,B))op .

There is only one combination of the two dualization processes, since

Cco op = Cop co .

4.6 Lax functors, pseudo-functors, 2-functors, and their transformations. A laxfunctor F : C −→ D of ordered categories is given by functions

F : ob C −→ ob D and FA,B : C(A,B) −→ D(FA, FB)

for all A,B ∈ ob C (with FA,B usually written as F ), such that

(1) FA,B is monotone;

(2) Fg · Ff ≤ F (g · f);

(3) 1FA ≤ F1A;

for all A,B,C ∈ ob C, f : A −→ B, g : B −→ C in C. The lax functor F is a pseudo-functorif the inequalities “≤” in (2) and (3) may be replaced by “'” (see 1.3), and F is a 2-functorif they may be replaced by “=”.An oplax functor F : C −→ D must satisfy condition (1) and, instead of (2) and (3),

(2∗) F (g · f) ≤ Fg · Ff ;

(3∗) F1A ≤ 1FA.

Hence, pseudo-functors are precisely the simultaneously lax and oplax functors.This terminology coincides with the one used in the more general context of 2-categories (see[Borceux, 1994a]). If one thinks of C and D as ORD-enriched categories, then 2-functors areprecisely ORD-functors (see [Kelly, 1982]). For ordinary categories, considered as discrete


ordered categories, all the functor notions just introduced coincide with the ordinary notionof functor.For lax or oplax functors F,G : C −→ D, a lax transformation α : F −→ G is given by anob C-indexed family of D-morphisms αA : FA −→ GA such that for all f : A −→ B in C,

(4) Gf · αA ≤ αB · Ff ,

while an oplax transformation satisfies

(4∗) αB · Ff ≤ Gf · αA

instead. A pseudo-natural transformation is a simultaneously lax and oplax transformation,and the prefix “pseudo” may be dropped if the order of the hom-class of D is separated.Lax transformations can be vertically composed and lead to a lax transformation again,via (β · α)A = βA · αA (for α : F −→ G, β : G −→ H); oplax transformations composesimilarly. Horizontal composition is more delicate. Here we only mention that for a laxor oplax functor S : B −→ C and a lax transformation α : F −→ G, one obtains a laxtransformation αS : FS −→ GS; one can proceed analogously with oplax transformations.However, “whiskering” from the other side with T : D −→ E requires a pseudo-functor inorder to obtain from α again a lax or oplax transformation Tα : TF −→ TG.

4.7 Maps. A morphism f : A −→ B in an ordered category C is a map if there is amorphism g : B −→ A in C with

1A ≤ g · f and f · g ≤ 1B ;

one writes f a g in this situation, and calls f the left adjoint and g the right adjoint of theadjunction.For C = Ord, being a map means being left adjoint (see 1.5). The map terminology ishowever better motivated by the example C = Rel: a relation r : A −→7 B of sets A,B is amap in Rel precisely when it is the graph of a morphism r : A −→ B in Set. Indeed, theexistence of a relation s : B −→7 A with

1A ≤ s · r means ∀x ∈ A ∃y ∈ B (x r y & y s x) ,

whiler · s ≤ 1B means ∀x ∈ A ∀y, y′ ∈ B (y s x & x r y′ =⇒ y = y′) .

Hence, the expression (r(x) = y ⇐⇒ x r y) defines a unique Set-map r : A −→ B. Onecan also easily see that necessarily s = r (see Exercise 4.B).Of course, in any ordered category (or more generally, in a 2-category), a right adjoint g ofa map f as above is uniquely determined up to “'”; we often write f ∗ (or fa) for g:

1A ≤ f ∗ · f and f · f ∗ ≤ 1B .


This notation is coherent with the fact that for a monotone map f : A −→ B of orderedsets one has f∗ a f ∗ in Mod (see 1.4). Adjunctions in C can be detected by “hom-ing” intoORD:

4.7.1 Proposition. For morphisms f : A −→ B and g : B −→ A in an ordered category C,one has f a g if and only if C(T, f) a C(T, g) in ORD for all T ∈ ob C.

Proof. C(T, f) a C(T, g) means (by Proposition 1.5.1) that

C(T, f)(x) ≤ y ⇐⇒ x ≤ C(T, g)(y)

for all x ∈ C(T,A), y ∈ C(T,B); that is:

f · x ≤ y ⇐⇒ x ≤ g · y .

But this equivalence (for all x, y) means exactly f a g in C (for which one may re-employthe proof of Proposition 1.5.1).

4.8 Quantaloids. A quantaloid is a category C with each hom-class being a completelattice (although not necessarily small), and with composition preserving suprema on eitherside:

g · (∨i∈I fi) = ∨i∈I(g · fi) , (∨i∈I gi) · f = ∨

i∈I(gi · f) ,

for all C-morphisms f, fi : A −→ B, g, gi : B −→ C (i ∈ I). A quantaloid is therefore anordered category. However, C is not just ORD-enriched, but SUP-enriched (where SUPdenotes the metacategory of sup-complete classes and sup-maps). A homomorphism ofquantaloids F : C −→ D is simply a functor that preserves suprema:

F (∨i∈I fi) = ∨i∈I Ffi .

4.8.1 Examples.

(1) The ordered categories Rel and Sup are quantaloids, but Ord is not. An abstractcategory provided with the discrete order is not a quantaloid either, unless it is anordered class (so that its hom-classes are at most singleton sets).

(2) Quantales are precisely the one-object quantaloids. Indeed, one-object quantaloidshave a monoid structure that can be considered as the tensor product of the quantale(see 1.10). A homomorphism of one-object quantaloids is precisely a homomorphismof quantales.

(3) For every quantaloid C, its opposite category Cop is a quantaloid as well, but Cco isgenerally not so: composition will preserve infima from either side, but not supremain general.


A biproduct of a family (Ai)i∈I of objects in a quantaloid C is given by an object P togetherwith morphisms

Ppi //

Aieioo (i ∈ I)

such that

pi · ej =

1Ai if i = j

⊥ otherwiseand ∨

i∈I ei · pi = 1P .

The terminology explains itself with the following statement.

4.8.2 Proposition. For a biproduct (P, pi, ei)i∈I of (Ai)i∈I in a quantaloid, (P, pi)i∈I is aproduct and (P, ei)i∈I is a coproduct of (Ai)i∈I .

Proof. For fi : B −→ Ai, (i ∈ I), f := ∨i∈I ei · fi is the only morphism with pi · f = fi

(i ∈ I). Likewise, given gi : Ai −→ B (i ∈ I), g := ∨i∈I gi · pi is the only morphism with

g · ei = gi (i ∈ I).

4.8.3 Example. In Rel, the biproduct of (Ai)i∈I can be taken to be the coproduct (thatis, the disjoint union in Set) (ei : Ai −→ C)i∈I with pi = ei (i ∈ I).

For a morphism f : A −→ B in a quantaloid C, the sup-map

C(C, f) = f · (−) : C(C,A) −→ C(C,B)

has, for all objects C, a right adjoint, denoted by f (−) and characterized by

f · g ≤ h

g ≤ f h

Af//

≤B .

C

g

OO

h

??

In this situation, f h is called a lifting of h along f . Dually, for f : B −→ A in C, thesup-maps

C(f, C) = (−) · f : C(A,C) −→ C(B,C)have right adjoints, denoted by (−) f and characterized by

g · f ≤ h

g ≤ h f

B

h

f// A

g

≥

C .

One calls h f an extension of h along f .For A = B = C, the notations used here coincide with those introduced in 4.4 when themonoidal category is taken to be C(A,A).


Expanding on Proposition 4.7.1, one can prove the following.4.8.4 Proposition. For morphisms f : A −→ B and g : B −→ A in a quantaloid C, thefollowing are equivalent:

(i) f a g in C;

(ii) C(T, f) a C(T, g) in SUP for all T ∈ ob C;

(iii) C(g, T ) a C(f, T ) in SUP for all T ∈ ob C;

(iv) C(T, g) = f (−) in SUP for all T ∈ ob C;

(v) C(f, T ) = (−) g in SUP for all T ∈ ob C.

Proof. Note that, when deriving (i) from any of (ii)–(v), the choices T = A and T = Bsuffice. Further details are left to the reader.

4.9 Kock–Zoberlein monads. A monad T = (T,m, e) on an ordered category C is ofKock–Zoberlein type (or simply Kock–Zoberlein) or lax idempotent if T is a 2-functor, andfor every C-object X there is a chain of adjunctions:

TeX a mX a eTX .

This condition can be replaced by any of the following equivalent expressions:

∀X ∈ ob C (TeX ≤ eTX) ⇐⇒ ∀X ∈ ob C (TeX a mX) ⇐⇒ ∀X ∈ ob C (mX a eTX) .

Indeed, if any one of the previous conditions holds, then the two others can be obtained byusing the following implications:

TeX ≤ eTX =⇒ TeX ·mX = mTX · TTeX ≤ mTX · TeTX = 1TTX ,

T eTX ·mTX ≤ 1TTTX =⇒ 1TTX = TmX · TeTX ·mTX · eTTX ≤ TmX · eTTX = eTX ·mX ,

1TTX ≤ eTX ·mX =⇒ TeX ≤ eTX ·mX · TeX = eTX .

The study of the Eilenberg–Moore algebras is facilitated in the case of a Kock–Zoberleinmonad: if a : TX −→ X is a C-morphism with a · eX ' 1X , then one has

1TX ' Ta · TeX ≤ Ta · eTX = eX · a ,

so that a a eX . The adjunctions Ta a TeX , a a eX and mX a eTX yield

a · Ta a TeX · eX and a ·mX a eTX · eXand therefore a · Ta ' a ·mX because each side is a left adjoint of TeX · eX = eTX · eX .4.9.1 Proposition. Let T = (T,m, e) be a monad on an ordered category C. The followingare equivalent when T is a 2-functor:

(i) T is of Kock–Zoberlein type;


(ii) any C-morphism a : TX −→ X with a · eX ' 1X is left adjoint to eX and satisfiesa · Ta ' a ·mX .

Proof. The proof of (i) =⇒ (ii) is given in the previous discussion. For (ii) =⇒ (i), noticethat if (ii) holds, then mX · eTX = 1TX implies mX a eTX , so that T is Kock–Zoberlein.

4.9.2 Corollary. The Eilenberg–Moore category CT of a Kock–Zoberlein monad T =(T,m, e) on a separated ordered category C may be described as follows: objects are thoseC-objects X for which eX is a section; morphisms are those C-morphisms that commute inthe usual way with the retractions of the unit.

Proof. When the order on the hom-sets of C is separated, “'” becomes an equality inProposition 4.9.1, and left adjoints are uniquely determined.

4.9.3 Example. The down-set functor Dn : Ord −→ Ord (see 2.2) with the union maps∨DnX = ∨ : DnDnX −→ DnX (where ∨A = ⋃

A∈AA for all A ∈ DnDnX) and thedown-set maps ↓X : X −→ DnX form the down-set monad

Dn = (Dn ,∨Dn , ↓)

on Ord. As ∨DnX is left adjoint to the down-set map ↓DnX and the down-set functor ismonotone, the monad Dn is of Kock–Zoberlein type. The up-set monad Up = (Up ,∧Up , ↑)is dual Kock–Zoberlein (that is, of dual Kock–Zoberlein type or colax idempotent), as it isKock–Zoberlein with the hom-sets equipped with their dual order (the infimum map isright adjoint to the up-set one); note that because Up X is ordered by reverse inclusion(see 1.7), the map ∧Up X : Up Up X −→ Up X is given by set-theoretic union.The down-set and the up-set monads on Ord restrict to separated ordered sets, so Proposi-tion 4.9.1 and 1.7 yield 2-isomorphisms

OrdDnsep∼= Sup and OrdUpsep

∼= Inf .

4.10 Enriched categories. A locally small ordered category C has its hom-sets andcomposition operation live in the cartesian closed category Ord. More generally, one mayconsider any monoidal category V instead of Ord and, keeping the notations of 4.1, define aV-category C to be given by a class ob C of objects, a hom-object C(A,B) ∈ ob V for eachpair A,B ∈ ob C, composition and identity operations in V

mA,B,C : C(A,B)⊗ C(B,C) −→ C(A,C) and eA : E −→ C(A,A) ,


for all A,B,C ∈ ob C, subject to the associativity and identity laws which require that thediagrams

C(A,B)⊗ (C(B,C)⊗ C(C,D))1⊗m

α // (C(A,B)⊗ C(B,C))⊗ C(C,D) m⊗1// C(A,C)⊗ C(C,D)

m

C(A,B)⊗ C(B,D) m // C(A,D)

E ⊗ C(A,B)

λ))

e⊗1// C(A,A)⊗ C(A,B)

m

C(A,B)

C(A,B)⊗ C(B,B)m

C(A,B)⊗ E

ρuu

1⊗eoo

C(A,B)

commute in V.For V = Set and V = Ord, the definition of a V-category returns the ordinary notions oflocally small category and of locally small ordered category, respectively. A 2-categoryis simply a CAT-category, where again the monoidal structure is given by the cartesianproduct. Starting with Chapter III, a major part of this book is devoted to the study ofV-categories when V is merely a quantale, and generalizations thereof which involve theconsideration of a monad on Set. If V is AbGrp, the prototype of a monoidal category, thenV-categories are called additive categories.The general theory of V-categories can be found in [Kelly, 1982]. For the reader’s convenience,we mention here the definitions of V-functors and V-natural transformations. A V-functorF : C −→ D of V-categories C, D is given by a function F : ob C −→ ob D and morphisms

FA,B : C(A,B) −→ D(FA, FB)

in V for all A,B ∈ ob C, subject to the commutativity of the diagrams

C(A,B)⊗ C(B,C)m

F⊗F// D(FA, FB)⊗ D(FB,FC)

m

C(A,C) F // D(FA, FC) .

E

e

gg

e

55

For V-functors F,G : C −→ D, a V-natural transformation τ : F −→ G is given by morphisms

τA : E −→ D(FA,GA)


in V such that the diagrams

C(A,B)⊗ E F⊗τB // D(FA, FB)⊗ D(FB,GB)m

**

C(A,B)

ρ−166

λ−1((

D(FA,GB)

E ⊗ C(A,B) τA⊗G // D(FA,GA)⊗ D(GA,GB)m

44

commute. Again, for V = Set, one obtains the ordinary categorical notions, and for V = Ord,a V-functor is a 2-functor of ordered categories (see 4.6), while a V-natural transformationis simply a natural transformation.

Exercises

4.A Cartesian closedness of Cat. In Cat, the functor category BA serves as internal hom-object.

4.B Relational adjoints to maps. In Rel, if r : A −→7 B is a map and s : B −→7 A is its rightadjoint, then one has s = r.

4.C Tensor commutes with coproducts. In a closed monoidal category, one has the rules

A⊗∐i∈I Bi∼=∐i∈I A⊗Bi , A⊗ 0 ∼= 0 ,

whenever the coproduct on the left exists. Likewise,

A∏i∈I Bi

∼=∏i∈I(A Bi) , (A 1) ∼= 1 ,

whenever the product on the left exists. Thus, in a cartesian closed category, one has inparticular

(A×B)C ∼= AC ×BC , 1C ∼= 1 .

4.D Further isomorphisms for the internal hom-functors. If C is a monoidal closed categorywith (−) B : Cop −→ C right adjoint (for example, if C is symmetric monoidal, seeProposition 4.4.3), then for all objects A,B,C, there are natural isomorphisms

(E A) ∼= A and ∏i∈I(Ai B) ∼= (∐i∈I Ai) B .

In the case where B (−) : Cop −→ C is right adjoint, one has for all objects A,B,C,

(A E) ∼= A and ∏i∈I(B Ai) ∼= B

∐i∈I Ai .


4.E Adjunctions and equivalent maps. If f a g : B −→ A and f ′ a g′ : B −→ A in an orderedcategory, then f ≤ f ′ if and only if g′ ≤ g. In particular, if f ≤ f ′, and g ≤ g′, then f ' f ′

and g ' g′.

4.F Trivial extensions and liftings. For a morphism f : A −→ B in a quantaloid, one has

1B f = f and f 1A = f .

4.G Adjoints in quantaloids. A morphism f : A −→ B in a quantaloid has a right adjointg if and only if (f 1B) · f = (f f); in this case, g = (f 1B). Dually, g : B −→ Ahas a left adjoint f if and only if g · (1B g) = (g g), and in this case, f = (1B g).Moreover, one has the following rules:

(1) h · (φ ψ) = (h · φ) ψ if h is right adjoint;

(2) (φ ψ) · f = φ (g · ψ) if f a g;

(3) (φ · p) ψ = φ (ψ · q) if p a q.

4.H Existence of biproducts. For a family (Ai)i∈I of objects in a quantaloid C, the followingare equivalent:

(i) the biproduct of (Ai)i∈I exists;

(ii) the product of (Ai)i∈I exists;

(iii) the coproduct of (Ai)i∈I exists.

4.I V as a V-category. For V a symmetric monoidal closed category, and all objects A,B,Cin V, there are morphisms

(A B)⊗ (B C) −→ (A C) and E −→ (A A)

which render V a V-category, with the same object class and (A B) as the internal hom.Hint. For the first morphism, construct A⊗ (A B) −→ B, and tensor with (B C).

4.J Categories as monoids.

(1) Let O be a set. A directed graph G = (M,d : M −→ O, c : M −→ O) over O (thevertices of G) is given by a set M (the edges of G) and maps d, c (which assign to anedge its domain and codomain). A morphism f : (M,d, c) −→ (N, b, a) in the categoryGph(O) of graphs over O is a map f : M −→ N with b · f = d, a · f = c. Show thatGph(O) becomes a monoidal category when one puts

(M,d, c)⊗ (N, b, a) = (M ×O N, d · b′, a · c′)


using the pullback diagramM ×O N

b′

c′ // N

b

Mc // O .

However, the tensor product fails to be symmetric.

(2) Show that the category of monoids in Gph(O) is equivalent to the category Cat(O) ofsmall categories with object set O and functors mapping O identically.


5 Factorizations, fibrations and topological functors

5.1 Factorization systems for morphisms. We denote respectively by Iso C, Epi C,and Mono C the classes of all isomorphisms, epimorphisms, and monomorphisms in C.An orthogonal factorization system for morphisms (or simply a factorization system) in acategory C is given by a pair (E ,M) of morphism classes in C such that:

(1) Iso C · E ⊆ E , M · Iso C ⊆M: that is, E and M are closed under composition withisomorphisms from the left and the right, respectively;

(2) mor C = M · E : every morphism factors into an E-morphism followed by an M-morphism; in fact, we tacitly assume that there is a fixed choice for these factorizations;

(3) E ⊥M: every E-morphism e is orthogonal to everyM-morphism m, so that for everycommutative solid-arrow square

·e

u // ·m

·

w@@

v // ·there is a unique morphism w with w · e = u, m · w = v; one writes e ⊥ m in thissituation.

The system (E ,M) is called proper if E ⊆ Epi C and M ⊆ Mono C; in that case, any wwith w · e = u or m · w = v (where m · u = v · e) satisfies both equations and is triviallyuniquely determined. The notion of factorization system is selfdual in the sense that (M, E)is a factorization system in Cop if (E ,M) is one in C (and the same is true for properfactorization systems).An immediate consequence of the unique diagonalization property (3) is that (E ,M)-fac-torizations of a morphism are unique, up to a unique isomorphism: if m · e = m′ · e′ withe, e′ ∈ E , m,m′ ∈ M, then j · e = e′, m′ · j = m for a unique isomorphism j. (This factfollows also from (1) and (2) below.)5.1.1 Proposition.

(1) For a factorization system (E ,M) in C one has:

E = ⊥M := e ∈ mor C | ∀m ∈M : e ⊥ m ,M = E⊥ := m ∈ mor C | ∀e ∈ E : e ⊥ m .

In particular, E and M determine each other uniquely, and

E ∩M = Iso C .

(2) Any class M = E⊥ (for some E ⊆ mor C) satisfies:

(a) Iso C ⊆M and M ·M =M (M is closed under composition);

5. FACTORIZATIONS, FIBRATIONS AND TOPOLOGICAL FUNCTORS 117

(b) if g · f , g ∈M, then f ∈M (M is weakly left-cancellable);(c) M is stable under pullbacks (see 2.8), so that for any pullback diagram

·f ′

g′// ·f

· g

// ·

f ∈M implies f ′ ∈M;(d) M is stable under multiple pullbacks, so that when (A, f) is a product of (Ai, fi)

in C/B, one has f ∈M whenever all fi ∈M;(e) M is closed under limits, so that when µ : D −→ E is a natural transformation

that is componentwise in M (with D,E : J −→ C), the induced morphism limµ :limD −→ limE also lies inM (provided that the needed limits exist); in particular,M is closed under products:

∀i ∈ I (mi ∈M) =⇒ ∏i∈I mi ∈M .

Any class E = ⊥M (for some M⊆ mor C) satisfies the properties dual to (a)–(e).

(3) A factorization system (E ,M) in C with E ⊆ Epi C satisfies:

(a) ExtMono C ⊆M, where a monomorphism f is an extremal monomorphism in Cif f = g · e with e epic only if e is an isomorphism;

(b) SplitMono C ⊆M (see Exercise 2.C);(c) g · f ∈M implies f ∈M (M is left-cancellable).

Conversely, if C has finite products, any of (a), (b), (c) implies E ⊆ Epi C. The dualassertions hold as well.

Proof. (1), (2) are shown by standard arguments. (It is less standard to show that properties(c) and (d) in (2) actually follow from (e); see [Dikranjan and Tholen, 1995].) (3) First assumeE ⊆ Epi C. For (a), (E ,M)-factoring f ∈ ExtMono C gives f = m · e with e epic, so that emust be an isomorphism and f ∈M. (b) follows trivially since SplitMono C ⊆ ExtMono C.For (c), let g · f ∈ M. In order to show f ∈ M = E⊥, assume f · u = v · e with e ∈ E .Since (g · f) · u = (g · v) · e with g · f ∈ M there is w with w · e = u. Thus, E ⊆ Epi Cimplies that w is unique and also satisfies f · w = v.Conversely, in order to show E ⊆ Epi C under hypotheses (a), (b), or (c), assume e ∈ E andf ·e = g ·e. With A = cod f = dom f , h = 〈f, g〉, δ = 〈1A, 1A〉 one obtains the commutativediagram

·e

f ·e// A

δ

· h // A× A .


Since pi · δ = 1A ∈ M (with p1, p2 the projections of A× A), δ ∈ M follows under eitherhypothesis (a), (b) or (c), so that e ⊥ δ gives w with δ · w = h, hence f = p1 · h = w =p2 · h = g.

Because of (1) one calls E a left factorization class and M a right factorization class ifthey belong to a factorization system (E ,M), and then E is the left companion of M andM the right companion of E . Of course, one talks about extremal epimorphisms in thesituation dual to (3)(a).

5.1.2 Examples. Every category C has both trivial factorization systems (Iso C,mor C)and (mor C, Iso C). The class of monomorphisms is a right factorization class in the categoriesSet, Mon, Grp, Ord, Sup, and Top; its left companion E is given by the surjective morphismsin Set, Mon, Grp and Sup, and by the quotient morphisms in Ord and Top (see Exercise 5.N);E is precisely the class of regular epimorphisms in all six categories, which only in Setand Grp coincide with the class of all epimorphisms. The class of epimorphisms is a leftfactorization class with an easily determined right companion M both in Ord and Top,given by the fully faithful injections (see 1.3) and the subspace embeddings, respectively;M is the class of regular monomorphisms in both categories. The following two diagramsdepict these two fundamental factorizations in Ord and Top:

X/∼mono

!!

X

reg. epi==

f// Y

f(X) qreg. mono

""

X

epi<<

f// Y

where ∼ is the equivalence relation induced by f .

For a class E of morphisms in C, a subcategory B of C is called E-reflective if all B-reflections lie in E ; accordingly, epi-reflective, regular epi-reflective, etc., refers to the casesE = epimorphisms, regular epimorphisms, etc. Bireflective means epi-reflective andmono-reflective. The dual notions are M-coreflective, mono-coreflective, regular mono-coreflective, bicoreflective, etc.

5.1.3 Proposition. For a factorization system (E ,M) in C, a replete reflective subcategoryB of C is E-reflective if and only if for all m : A −→ B in M, one has A ∈ ob B whenB ∈ ob B.

Proof. For the necessity of the condition, factor m : A −→ B in M with B ∈ ob B throughthe B-reflection ρA : A −→ RA and obtain t : RA −→ A with t · ρA = 1A from ρA ⊥ m.Now, (ρA · t) · ρA = ρA given ρA · t = 1RA, by universality of ρA. For the sufficiency of thecondition, consider an (E ,M)-factorization ρA : (A e // B n // RA) of the B-reflectionof A ∈ ob C. Since B ∈ ob B, there is s : RA −→ B with s · ρA = e, hence (n · s) · ρA = ρAand n · s = 1RA. Now e ⊥ n implies s · n = 1B.


5.2 Subobjects, images and inverse images. For a class M⊆ Mono C containing allisomorphisms and being closed under composition with them, and for every A ∈ ob C, oneforms the full subcategory

subA = subMA =M/A

of the comma category C/A, given by the objects m : M −→ A with m ∈ M. (If M isclosed under composition and is weakly left-cancellable, one may also consider M/A as acomma category of the category M with obM = ob C.) Of course subA is just an orderedclass (with m ≤ n if n · l = m for some morphism l). Its objects are calledM-subobjects (orsimply subobjects) of A. Most authors reserve this term for isomorphism classes of objectsin M/A. The category C is said to have inverse M-images (or inverse images) if for allf : A −→ B in C and all n : N −→ B in M there is a pullback diagram

f−1(N)f−1(n)

f ′// N

n

Af

// B

(5.2.i)

with f−1(n) in M (for convenience, the choice of such a diagram is supposed to be fixed).One then has the inverse-image functor

f−1(−) : subB −→ subA .

5.2.1 Proposition. Let C have inverse M-images. Then M is a right factorization classof C if and only if M is closed under composition and f−1(−) has a left adjoint f(−),for every morphism f in C. In that case, the M-image f(m) of m : M −→ A in Munder f : A −→ B may be constructed by the (E ,M)-factorization of f ·m (with E the leftcompanion of M):

M

m

e∈E// f(M)

f(m)

Af

// B .One has:

(1) m ≤ f−1(f(m)) for all m ∈ subA, with m ∼= f−1(f(m)) when f ∈M;

(2) f lies in E if and only if f(1A) ∼= 1B;

(3) f(f−1(n)) ≤ n for all n ∈ subB, while n ∼= f(f−1(n)) for all n precisely when flies M-hereditarily in E, that is, when f ′ ∈ E in every pullback diagram (5.2.i) withn ∈M.

Proof. All verifications are routine, except the proof that M is a right factorization classwhen M ·M ⊆M and there is f(−) a f−1(−) for all f . To establish a factorization of f ,


one puts m := f(1A) and lets e be the upper horizontal composite arrow in

A

1A

// ·f−1(f(1A))

// C

f(1A)

Af

// B .

The universal property of the adjunction amounts to the fact that for every commutativesolid-arrow diagram

A

e

u // ·n

C

w

??

m // B

with n ∈M there is a unique “diagonal” w. Applying the same factorization procedure toe in lieu of f gives e = m′ · e′ and the above solid arrow diagram with n = m ·m′ and u = e′.Since m ·m′ ∈ M, by hypothesis one obtains w with m ·m′ · w = m, hence m′ · w = 1C .Consequently, m′ is an isomorphism so that e must have the universal property symbolizedby

A

e

// ·n

C

??

1C // C .

This is sufficient to verify e ∈ ⊥M, since the search for a diagonal for an arbitrary square

·e

u // ·n

· v // ·

can be reduced to the case v = 1 by pulling n back along v.

It follows that the image and inverse-image functors associated with a factorization system(E ,M) with M⊆ Mono C preserve existing joins and meets, respectively:

5.2.2 Corollary.

f(∨i∈I mi) '∨i∈I f(mi) , f−1(∧i∈I mi) '

∧i∈I f

−1(mi) .

Proof. Left adjoints preserve colimits, and right adjoints limits.

5.3 Factorization systems for sinks and sources. A source in a category C is simplya discrete cone in C; hence, it is given by an object A and a family (gi : A −→ Bi)i∈I ofmorphisms in C (I may be empty and is not required to be small); the dual notion is sink.


One says that the sink (fi : Ai −→ B)i∈I is orthogonal to the morphism m : C −→ D if forevery solid-arrow diagram

Ai

fi

ui // C

m

B

w

??

v // D (i ∈ I)there is a unique morphism w with w · fi = ui (i ∈ I) and m · v = v; one writes (fi)i∈I ⊥ min this case. A factorization system for sinks (also called an orthogonal factorization systemfor sinks) is a pair (E,M) consisting of a collection E of sinks and a class M of morphismsin C such that:

(1) E and M are closed under composition with isomorphisms from the left and right,respectively;

(2) every sink in C factors into an E-sink followed by an M-morphism;

(3) E ⊥M.

We can leave it to the reader to formulate the appropriate sink generalizations of thestatements of Proposition 5.1.1. For example, closure of E under composition meansthat when all (fi,j : Aij −→ Bi)j∈Ji and (gi : Bi −→ C)i∈I lie in E, the composite sink(gi · fi,j)i∈I, j∈Ji also lies in E. Conversely, weak right-cancellation of E stipulates that whenthe composite sink and all (fi,j)j∈Ji lie in E, then (gi)i∈I also lies in E.Of course, a factorization system (E,M) for sinks gives in particular a factorization system(E ,M) for morphisms, but not vice versa. For example, there cannot be a factorizationsystem (E,mor C) for sinks in any category C containing morphisms that are not monic, aswe show next:5.3.1 Lemma. For a factorization system (E,M) for sinks, one has M⊆ Mono C.

Proof. For f : A −→ B in M, assume f · x = f · y = h with x, y : D −→ A, and considerthe constant sink (h)i∈I , with I := ob(C/A). We obtain an (E,M)-factorization h = m · ei(i ∈ I) and consider

J := w | ∀i ∈ I : w · ei ∈ x, y .The diagrams

D

ei

ui // A

f

C

w

>>

m // B

show J 6= ∅ (take ui := x for all i); therefore, the inclusion map τ : J → I has a retractionσ : I −→ J (so that σ · τ = 1J). For i ∈ I, consider

ui :=

x if σ(i) · ei = y

y if σ(i) · ei = x,


and obtain w ∈ J as above. Then, with σ(i0) = w, one has

σ(i0) · ei0 = x ⇐⇒ σ(i0) · ei0 = y ,

which is possible only if x = y.

5.3.2 Theorem. A class of morphisms M belongs to a factorization system (E,M) forsinks in C if and only if:

(1) Iso C ⊆M ⊆ Mono C and M ·M ⊆M,

(2) C has inverse M-images,

(3) C has M-intersections, that is, multiple pullbacks of families of M-morphisms existin C, and M is stable under them.

Proof. The Lemma takes care of the only delicate part in the proof of the necessity ofcondition (1). For the necessity of (2), let f : A −→ B be in C, n : N −→ B in M andconsider the class

J := k ∈M | ∃xk : f · k = n · xk .The sink (k)k∈J has an (E,M)-factorization m · ek = k, with m ∈ M and (ek)k∈J in E.Since (ek) ⊥ n, one finds f ′ making

· xk

##ek

&&

k

· f ′//

m

N

n

Af// B

commute for all k ∈ J . Using the fact that we have in particular an (E,M)-factorizationsystem for morphisms, one easily verifies that the square of the diagram is a pullback.Condition (3) is shown similarly.For the sufficiency of conditions (1)–(3), let us consider a sink (fi : Ai −→ B)i∈I , and form

J := k ∈M | ∀i ∈ I ∃xi,k : k · xi,k = fi .

The universal property of the multiple pullback

m = ∧k∈J k

shows m ∈ J , that is: there are morphisms ei with fi = m · ei (i ∈ I). It remains to beshown that (ei) ⊥ t for all t ∈M. But if

Ai

ei

ui // T

t

Cv // D


for all i ∈ I, then t′ = v−1(t) is a factor of each ei, so that m · t′ ∈ M is a factor of eachfi. Consequently, m · t′ ∈ J , and one has that t′ is an isomorphism by construction of m.Hence, w = v′ · (t′)−1 is the needed diagonal for the square above, with v′ the pullback of valong t.

With M⊆ Mono C, a category C is M-wellpowered if, for every A ∈ ob C, the separatedreflection of subMA (see 1.3) is small; that is, every object has only a set of isomorphismtypes of M-subobjects. The prefix M is omitted when M = Mono C.

5.3.3 Corollary. Let C be small-complete and M-wellpowered, for a class M with Iso C ⊆M ⊆ Mono C. The following are equivalent:

(i) M is a right factorization class for morphisms;

(ii) M is a right factorization class for sinks;

(iii) M is closed under composition, and stable under pullbacks, as well as under multiplepullbacks.

Proof. The equivalence follows from Proposition 5.1.1 and the Theorem.

A pair (E ,M) is a factorization system for sources in C if (M, E) is a factorization systemfor sinks in Cop. Similarly, C is E-cowellpowered if Cop is E-wellpowered. We leave theformulation of the statements dual to 5.3.1–5.3.3 to the reader.For C any of the categories Set, Mon, Grp, Ord, Sup, and Top, Mono C is not just a rightfactorization class for morphisms, but also for sinks. For C = Set, the left companion Econtains precisely the jointly surjective sinks (fi : Ai −→ B)i∈I , so that B = ⋃

i∈I fi(Ai).In Mon, Grp, and Sup, B is only generated by ⋃i∈I fi(Ai) for (fi)i∈I ∈ E. In Ord and Top,sinks in E are still jointly surjective but must carry the appropriate structure that will bediscussed more generally in Examples 5.6.1.Epi C is a left factorization class for sources, with an easily described companion M forC = Set or Grp: M contains precisely the point-separating sources (fi : A −→ Bi)i∈I , so thatfor x, y : X −→ A, one has x = y when fi · x = fi · y for all i ∈ I. In a general category,sources with this property are called mono-sources (the dual notion is that of an epi-sink).In Ord and Top, the domain A must carry the appropriate structure, the description ofwhich will appear more generally in Examples 5.6.1.

5.4 Closure operators. Let M⊆ Mono C be a right factorization class, hence part of afactorization system (E ,M) for morphisms. As in 5.2, for every object A in C, we considerthe ordered class subA = M/A and have, for every morphism f : A −→ B, the imagefunction

f(−) : subA −→ subB


(constructed as in Proposition 5.2.1) which, if C has inverse M-images, is left adjoint tof−1(−). An M-closure operator (or simply closure operator) on C is a family

c = (cA : subA −→ subA)A∈ob C

of functions which are

(1) extensive: m ≤ cA(m),

(2) monotone: if m ≤ m′, then cA(m) ≤ cA(m′),

(3) and which satisfy the continuity condition: f(cA(m)) ≤ cB(f(m)),

for all f : A −→ B in C and m,m′ ∈ subA. In the presence of inverse images, it is easy tosee that the continuity condition may be equivalently formulated as

(3′) cA(f−1(n)) ≤ f−1(cB(n)),

for all f : A −→ B in C and n ∈ subB. Extensivity yields for every m : M −→ A in M afactorization

cA(M)cA(m)

M

jm>>

m // A

with c(m) = cA(m) ∈M and a uniquely determined morphism jm ∈M. Monotonicity andcontinuity ensure that the passage from m to its c-closure cA(m) is functorial as follows:for every commutative diagram

Mf ′//

m

N

n

Af// B

in C with m,n ∈ C, there is a unique morphism f ′′ making the following diagram

Mf ′

//

jm

N

jn

cA(M) f ′′//

cA(m)

cB(N)cB(n)

Af

// B

commute (see Exercise 5.G for a specification of the functoriality claim).A subobject m is c-closed if jm is an isomorphism, and it is c-dense if c(m) is an isomorphism.More generally, a morphism f : A −→ B is c-dense if f(1A) : f(A) −→ B is c-dense. The


closure operator c is idempotent if c(m) is c-closed (so that c(c(m)) ∼= c(m) for all m ∈M),and c is weakly hereditary if jm is c-dense for all m ∈M.

5.4.1 Proposition. The following conditions on a closure operator c are equivalent:

(i) c is idempotent and weakly hereditary;

(ii) c is idempotent, and the class Mc of c-closed subobjects in C is closed under composi-tion;

(iii) the class Mc is a right factorization class for morphisms in C.

If these conditions hold, the left companion of Mc is the class Ec of all c-dense morphismsin C. The (Ec,Mc)-system factors a morphism f : A −→ B through cB(f(A)).

Proof. The statement is easily proved by using functoriality as described above.

It is also a straightforward exercise to show that for any closure operator c the class Mc isclosed under limits, in particular stable under pullbacks and under multiple pullbacks.These latter properties are characteristic in the following sense:

5.4.2 Theorem. LetM be a right factorization class for sinks in C, and consider a subclassK ⊆M. Then K =Mc for an idempotent closure operator c if and only if K is stable underpullbacks and multiple pullbacks. In fact, the closure operator c is uniquely determined byK.Moreover, c is weakly hereditary if and only if K is closed under composition.

Proof. See Exercise 5.H.

A closure operator is called hereditary if for all m : M −→ A, k : A −→ B in M, the lowerrectangle in

M1M //

jm

M

jk·m

cA(M) //

cA(m)

cB(M)cB(k·m)

A k // B

(5.4.i)

is a pullback diagram; that is, if cA(m) ∼= k−1(cB(k ·m)). Exploitation of this propertywith m = jn and k = cB(n) for any N −→ B in M yields

ccB(N)(jn) ∼= cB(n)−1(cB(n)) ∼= jn ,

so that jn is c-dense. Hence, heredity implies weak heredity. The following result describesthe extent to which heredity is stronger than weak heredity.


5.4.3 Proposition. A closure operator c is hereditary if and only if c is weakly hereditaryand satisfies the following cancellation condition: for all m, k ∈M, if k ·m is c-dense, thenm is also c-dense.

Proof. The cancellation condition is certainly necessary for heredity of c since when cB(k ·m)is an isomorphism, any pullback of it is also an isomorphism. Conversely, let the weaklyhereditary closure operator c satisfy the cancellation condition. We want to see that thecanonical morphism t : cA(M) −→ k−1(cB(M)) induced by the commutative lower rectangleof (5.4.i) is an isomorphism. But weak heredity makes

jk·m = k−1(cB(k ·m)) · t · jmc-dense, so that s := t · jm : M −→ k−1(cB(M)) is c-dense by hypothesis. Functoriality of cthen yields the inverse of t:

M1M //

js

M

jm

· //

∼=

cA(M)cA(m)

k−1(cB(M)) // A .

5.4.4 Examples.(1) We may think of an idempotent closure operator c as a family (cA)A∈ob C of closure

operations cA on subA (in the sense of 1.6) which collectively must satisfy thecontinuity condition. (The notion of c-closedness defined here then coincides with theone defined in 1.6.)

(2) The down- and up-closure for subsets of ordered sets define idempotent and hereditaryclosure operators of Ord (with its (Epi,RegMono)-factorization system, see 1.7, whereRegMono refers to RegMono C, the class of all regular monomorphisms in C). Likewise,Kuratowski closure defines an idempotent and hereditary closure operator of Top(with its (Epi,RegMono)-factorization system, see Exercise 1.F).

(3) In terms of Theorem 5.4.2, the Kuratowski closure operator of Top corresponds to theclass K of closed subspace injections in Top. The class O of open subspace injectionsin Top still induces a closure operator θ via

θX(M) =⋂O ⊆ X |M ⊆ O and O open=x ∈ X | ∀V neighborhood of x : M ∩ V 6= ∅

(with V the Kuratowski closure of V in X). But failure of O to be closed undermultiple pullbacks (that is, intersections) makes θ fail to be idempotent, and θ is notweakly hereditary either.


5.5 Generators and cogenerators. A class G of objects in a category C is generating iffor every object A in C the family C(G, A) of all morphisms with codomain A and domain inG forms an epi-sink. Hence, for all f, g : A −→ B in C, one has f = g whenever f · x = g · xfor all x : G −→ A, G ∈ G. Equivalently, G is generating if the generalized hom-functor

C(G,−) : C −→ SetG , A 7−→ (C(G,A))G∈G

is faithful (where C is assumed to be locally small). A class G is strongly generating in C ifC(G, A) is an extremal epi-sink for all objects A, so that one has the additional propertythat for a monomorphism m : B −→ A, one can have x = m · hx for all x in C(G, A) only ifm is an isomorphism (the dual notion is that of an extremal mono-source). Equivalently,G is strongly generating if C(G,−) is faithful and reflects isomorphisms. We call a smallgenerating class a generator of C, and a single object G in C a generator of C if G is one.Similarly, one says that a class is a strong generator if it is a small strongly generating class,and a single object G can similarly be called a strong generator . Note that one commonlyuses separator as an alternative name for generator. The terms cogenerating, cogenerator ,or coseparator are used in the dual situation.When G is small, the functor C(G,−) has a left adjoint F if and only if all coproducts

FX = ∐G∈G XG ·G

(with X = (XG)G∈G an object in SetG) exist; here XG ·G denotes the coproduct of XG-manycopies of G in C. The counits

εA : ∐G∈G C(G,A) ·G −→ A

are the canonical morphisms with εA · ix = x for all x : G −→ A, G ∈ G, with ix denoting acoproduct injection.

5.5.1 Proposition. The following conditions on a set G of objects in a locally smallcategory C with coproducts are equivalent:

(i) G is generating;

(ii) for all objects A, the canonical morphisms εA are epimorphisms;

(iii) for every object A, there is some epimorphism∐i∈I Gi −→ A

with a small family (Gi)i∈I of objects in G.

The same equivalence holds if one specializes to a strongly generating set G in (i), andextremal epimorphisms in (ii) and (iii).

Proof. The equivalence follows from Exercise 5.I.


A singleton set (in fact, every non-empty set) is a single-object strong generator of Set. Theterminal object is also a generator in Ord or Top, but it is not strong. The free algebra overa singleton set is a strong generator in every Eilenberg–Moore category over Set (see 3.2).Hence, the additive group Z is a generator of both Grp and AbGrp.A two-element set is a single-object strong cogenerator of Set. Provided with the indiscretestructure, it is a cogenerator in both Ord and Top, but it is not strong. The “rational circle”Q/Z is a strong cogenerator in AbGrp but Grp has no cogenerator at all.

5.6 U-initial morphisms and sources. For a functor U : A −→ X, a source (gi : A −→Bi)i∈I of A-morphisms is U-initial if, for every source (hi : C −→ Bi)i∈I in A and everyX-morphism s : UC −→ UA with Ugi · s = Uhi, there is exactly one morphism t : C −→ Ain A with Ut = s and gi · t = hi for all i ∈ I:

A

C

t

OO UAUgi // UBi .

UC

s

OO

Uhi

::

Of course, uniqueness of t as well as gi · t = hi (i ∈ I) follow from Ut = s when U is faithful.Hence, for U faithful, U -initiality of (gi)i∈I simply means that any X-morphism s : UC −→UA can be lifted to an A-morphism t : C −→ A along U whenever all Ugi · s : UC −→ UBi

can be lifted to A-morphisms hi : C −→ Bi along U .If the given source consists of a single morphism f : A −→ B (hence, if |I| = 1), it is morecustomary to say that f is U-cartesian instead of U -initial, and we shall do so especiallywhen U is not necessarily faithful.In the case where I = ∅, the source (gi)i∈I is given by an object A, and we say that A isU-indiscrete when A is U -initial as an empty source. The dual notions for sinks are thoseof U -final sink, U -cocartesian morphism, and U -discrete object, with the universal propertydepicted by

UAiUfi //

Uki $$

UB

s

UC

B

t

C .(We note a certain contradiction in the common terminology here, since U -final sinks withdomain (Ai)i∈I are characterized as initial objects in a certain category; but the first twoexamples below give some justification.)

5.6.1 Examples.

(1) For the forgetful functor U : Ord −→ Set, a source (gi : A −→ Bi)i∈I is U -initial preciselywhen

x ≤ y ⇐⇒ ∀i ∈ I (gi(x) ≤ gi(y))


for all x, y ∈ A. A sink (fi : Ai −→ B)i∈I is U -final precisely when, for all z 6= w in A,

z ≤ w ⇐⇒ ∃i0, . . . , in ∈ I ∃x0 ≤ y0 in Ai0 , x1 ≤ y1 in Ai1 . . . , xn ≤ yn in Ain :z = fi0(x0), fi0(y0) = fi1(x1), . . . , fin−1(yn−1) = fin(xn), fin(yn) = w .

Briefly, B carries the least order making all fi monotone.

(2) For the forgetful functor U : Top −→ Set, a source (gi : A −→ Bi)i∈I is U -initial preciselywhen g−1

i (V ) | i ∈ I, V ⊆ Bi open is a generating system of open sets for A. A sink(fi : Ai −→ B)i∈I is U -final precisely if V ⊆ B is open whenever all f−1

i (V ) ⊆ Ai areopen.

(3) Let T be a monad on a category X. Then every mono-source in XT is initial withrespect to the forgetful functor GT : XT −→ X.

(4) For a category C, the functor category C2 (with 2 = ⊥,> considered as a category)can be thought of as having the morphisms of C as its objects, and a morphism(u, v) : f −→ g in C2 is given by a commutative square

·f

u // ·g

· v // ·

(5.6.i)

in C. The evaluation functor at > ∈ ob 2 now appears as the codomain functor

cod : C2 −→ C , f 7−→ cod f , (u, v) 7−→ v .

The morphism (u, v) : f −→ g is cod-cartesian if and only if (5.6.i) is a cartesian squarein C, that is, a pullback diagram in C.

For U : A −→ X, we denote the classes of U -initial and U -final morphisms in A by

IniU and FinU ,

respectively, and list some easily established properties for them.5.6.2 Proposition.

(1) IniU contains all isomorphisms, is closed under composition, and is weakly left-can-cellable; it is even left-cancellable when U is faithful.

(2) IniU is stable under pullbacks and multiple pullbacks when U preserves them, and isclosed under those limits in A that are preserved by U .

Proof. The statements follow by routine verifications.

We leave it to the reader to formulate the corresponding generalized statements for sources,as well as their dualizations for morphisms and sinks.


5.7 Fibrations and cofibrations. A functor U : A −→ X is a fibration (more precisely, acloven fibration) when, for all f : X −→ UB in X with B ∈ ob A, there is a (tacitly chosen)U-cartesian lifting, i.e., a morphism g ∈ IniU with Ug = f :

Ag//

_

B

UX

f// UB .

It is easy to show that U is a fibration if and only if the functors

UB : A/B −→ X/UB , (A, g) −→ (UA,Ug) ,

have right adjoints ΓB such that the counits are identity morphisms. Hence, UBΓB = 1X↓UBand ΓB is a full embedding for all B ∈ ob A.A functor U : A −→ X is a cofibration (also called an opfibration) when Uop : Aop −→ Xop is afibration; explicitly, when for all f : UA −→ Y in X with A ∈ ob A, there is a U -cocartesianlifting (again, tacitly chosen) g : A −→ B in FinU :

Ag//

_

B

UUA

f// Y .

For X ∈ ob X, let U−1X denote the fibre of U over X: its objects are the A-objects Awith UA = X, and a morphism t : A −→ A′ in U−1X is an A-morphism t with Ut = 1X(and composition as in A). When U is a fibration, any f : X −→ Y in X gives rise toa functor f ∗ : U−1Y −→ U−1X which assigns to B ∈ ob(U−1Y ) the domain A of thechosen U -cartesian lifting of f : X −→ UB. Dually, when U is a cofibration, there is afunctor f∗ : U−1X −→ U−1Y which assigns to A ∈ ob(U−1X) the codomain B of the chosenU -cocartesian lifting of f : UA −→ Y . One easily checks that there is an adjunction

U−1Yf∗

//> U−1X ,f∗

oo

when U is a fibration and a cofibration.This adjunction describes best the “transport of structure along f”. For example, for theforgetful functor U : Top −→ Set and a set X, the fibre U−1X is the lattice of topologies onX, and for a map f : X −→ Y of sets and a topology τ ∈ U−1X, the topology σ = f∗(τ) isdescribed by (V ∈ σ ⇐⇒ f−1(V ) ∈ τ). Likewise, for σ ∈ U−1Y , the topology τ = f ∗(σ)is given by (V ∈ τ ⇐⇒ ∃W ∈ σ : V = f−1(W )). This way, one sees that U is a fibrationand a cofibration.The forgetful functor Ord −→ Set is another example of a fibration and a cofibration. The(non-faithful) object-functor ob : Cat −→ Set is a fibration, but not a cofibration (seeExercise 5.R).


A useful generalization of the notions of fibration and cofibration is that of anM0-fibrationand E0-fibration, for classes of morphisms M0 and E0 in X. For an M0-fibration, onerequires the existence of a U -cartesian lifting of f : X −→ UB in X only when f ∈ M0;dually, for an E0-fibration, U -cocartesian liftings are only required for morphisms in E0.For example, the forgetful functor Ordsep −→ Set of the category of separated ordered setsis not a fibration, but it is a mono-fibration (that is, a Mono Set-fibration). Similarly, theforgetful functor Haus −→ Set from the category Haus of Hausdorff topological spaces is amono-fibration. (A topological space X is Hausdorff if, for all x, y ∈ X with x 6= y, thereexist open subsets A,B ⊆ X with x ∈ A, y ∈ B and A ∩ B = ∅.) The following resultoffers a general explanation of these examples.

5.7.1 Proposition. For a functor U : A −→ X and a factorization system (E0,M0) in X,let

E = U−1E0 = e ∈ mor A | Ue ∈ E0 , M = U−1M0 ∩ IniU .

(1) If U is an M0-fibration, then (E ,M) is a factorization system in A.

(2) If U is a fibration and B is an E-reflective subcategory of A, then U |B is an M0-fibration.

Proof. (1) In order to (E ,M)-factorize f : A −→ B in A, let m0 · e0 = Uf be an (E0,M0)-factorization in X, and consider a U -cartesian lifting m : C −→ B of m0 : Z −→ UB. Thene : A −→ C is the only morphism with Ue = eo and m · e = f . The fact that E ⊥M followsfrom routine diagram chasing. (2) follows from (1) and Proposition 5.1.3.

5.7.2 Corollary. If U : A −→ X is a fibration, then U−1(Iso X) is a left factorization classin A.

Proof. Apply Proposition 5.7.1 with E0 = Iso X.

5.8 Topological functors. A functor U : A −→ X is topological if every source (fi : X −→UBi)i∈I with a family (Bi)i∈I of A-objects admits a U-initial lifting (once more, assumedto be chosen), that is, a U -initial source (gi : A −→ Bi)i∈I with UA = X and Ugi = fi forall i ∈ I:

Agi //_

Bi

UX

fi// UBi .

Hence, in generalization of the corresponding statement for fibrations, U is topological iffor every discrete category I and every object B = (Bi)i∈I ∈ ob AI , the induced functor

UB : (∆A ↓B) −→ (∆X ↓UB)

has a right adjoint ΓB such that the counits are identity morphisms; here ∆A : A −→ AI isas in 2.8.


Obviously, every topological functor is a fibration (|I| = 1) and has a full and faithfulright adjoint (I = ∅; in this case, for the only object B of A∅, one has (∆A ↓B) ∼= A and(∆X ↓UB) ∼= X).The forgetful functor U : Ord −→ Set is topological. For a source (gi : X −→ Bi)i∈I withordered sets Bi, make the set into an ordered set A by providing it with the appropriateorder described in Example 5.6.1(1). Similarly, U : Top −→ Set is topological: provide Xwith the topology generated by the sets g−1

i (V ) (i ∈ I, V ⊆ Bi open), see Example 5.6.1(2).Topological functors possess extremely good lifting properties. We already saw that, asfibrations, they lift factorization systems for morphisms (see Proposition 5.7.1(1)). In fact,for a collection M0 of sources in X, we can say that U : A −→ X is M0-topological by requiringU -initial liftings only for sources (fi : X −→ UBi)i∈I in M0. One then has analogously toProposition 5.7.1:

5.8.1 Proposition. For a functor U : A −→ X and a factorization system (E0,M0) forsources in X, let

E = U−1E0 , M = (fi)i∈I | (fi)i∈I U-initial & (Ufi)i∈I ∈ M0 .

(1) If U is M0-topological, then (E ,M) is a factorization system for sources in A.

(2) If U is topological and B is an E-reflective subcategory of A, then U |B is M0-topological.

Proof. The proof of Proposition 5.7.1 can be adapted to the present situation.

Next, we show how to “lift” limits. For that, one first proves a lemma similar to 5.3.1. (Infact, there is a common generalization for both lemmata, see [Borger and Tholen, 1978].)

5.8.2 Lemma. A topological functor U : A −→ X is faithful.

Proof. For x, y : A −→ D in A, assume Ux = Uy = h, and consider the constant source(h)i∈I with I := morU−1(UA). Let (ei : C −→ D)i∈I be a U -initial lifting of (h)i∈I , andconsider

J := w ∈ I | ∀i ∈ I : ei · w ∈ x, y .With a retraction σ : I −→ J , consider for i ∈ I

ui :=

x if ei · σ(i) = y

y if ei · σ(i) = x,

and derive x = y as in Lemma 5.3.1.

5.8.3 Proposition. For a faithful functor U : A −→ X, the U -initial lifting of a limit coneλ : ∆L −→ UD in X (with D : J −→ A) yields a limit cone α : ∆A −→ D in A.

Proof. Faithfulness of U makes sure that the U -initial lifting (αi : A −→ Di)i∈ob J of thesource (λi : L −→ UDi)i∈ob J does give a cone in A. Its limit property follows routinely.


5.8.4 Corollary. For a topological functor U : A −→ X, if X is J-complete, then so is A,and U preserves J-limits.

Proof. This follows immediately from the Proposition.

We note that a topological functor preserves in fact all limits since it has a left adjoint (seeTheorem 5.9.1 below).The Proposition and its Corollary fully explain how to construct limits in Top. Indeed,one just needs to provide the limit of the underlying sets with the U -initial structure withrespect to the limit projections.The previous assertions hold analogously for colimits since topologicity of a functor is aself-dual concept, as we show next.

5.9 Selfdual characterization of topological functors. A functor U : A −→ X istransportable if for every isomorphism f : X −→ UB in X with B ∈ ob A, there is a (chosen)isomorphism g : A −→ B in A with Ug = f . Since U -cartesian liftings of isomorphisms areisomorphisms, transportability of U means precisely that U is an Iso X-fibration.When U is faithful, all fibres U−1X (X ∈ ob X) are ordered classes. We call them large-complete if the infimum (or supremum) of any subclass exists.

5.9.1 Theorem. The following conditions are equivalent for a functor U : A −→ X:

(i) U is topological;

(ii) every sink (fi : UAi −→ Y )i∈I admits a U-final lifting (gi : Ai −→ B)i∈I ;

(iii) U is a faithful fibration and a cofibration with large-complete fibres;

(iv) U is faithful and transportable with a fully faithful left adjoint, and U−1(Iso X) is aleft factorization class for sources in A;

(v) U is faithful and transportable with a fully faithful right adjoint, and U−1(Iso X) is aright factorization class for sinks in A.

Proof. (i) =⇒ (iii): An infimum of a family (Bi)i∈I in U−1Y is obtained from a U -initiallifting of the source (1Y : Y −→ UBi)i∈I . Hence, the fibres of U are large-complete. Itremains to be shown that U is a cofibration. For f : UA −→ Y in X with A ∈ ob A, oneconsiders all morphisms gi : A −→ Bi in A with Ugi = f (i ∈ I), and then a U -initial lifting(ei : B −→ Bi)i∈I of the source (1Y : Y −→ UBi)i∈I . There is then a morphism g : A −→ Bwith Ug = f , and we should check that it is U -final. Hence, let k : A −→ C in A ands : UB −→ UC in X with s · Ug = Uk. With a U -initial lifting t : D −→ C of s, one obtainsg′ : A −→ D with Ug′ = f , so that g′ = gi for some i ∈ I. Now, t · ei : B −→ C satisfiesU(t · ei) = s, as desired.


(iii) =⇒ (i): Given a source (fi : X −→ UBi)i∈I in X with Bi ∈ ob A for every i ∈ I, letgi : Ai −→ Bi be a U -initial lifting of fi : X −→ UBi, and let (ei : A −→ Ai)i∈I representan infimum in U−1X. We need to show that (gi · ei : A −→ Bi)i∈I is U -initial. Hence, weconsider s : UC −→ X in X and ki : C −→ Bi in A with fi · s = Uki for all i ∈ I. Now,U -initiality of every gi gives ti : C −→ Ai in A with Uti = s, and every ti factors throughthe U -final lifting t : C −→ D of s : UC −→ X. The infimum property of A in U−1X thenyields a morphism j : D −→ A in U−1X, and we have U(j · t) = s, as desired.Since (iii)op = (iii) and (i)op = (ii), we have established the equivalence of (i), (ii), (iii). Inparticular, a topological functor has both a fully faithful left adjoint and a fully faithful rightadjoint, and it is faithful and transportable. Furthermore, for a source (fi : A −→ Bi)i∈I in A,one obtains a (U−1(Iso), U -initial sources)-factorization by U -initially lifting the source(Ufi : UA −→ UBi)i∈I . The unique diagonalization property follows as in the morphismcase. Hence, (i) =⇒ (iv) is shown.(iv) =⇒ (i): Let D a U with unit η an isomorphism. A source (fi : X −→ UBi)i∈I in Xgives rise to a source (gi : DX −→ Bi)i∈I , for which there is then a morphism e : DX −→ Cin U−1(Iso X) and a U -initial source (mi : C −→ Bi)i∈I with mi · e = gi for all i ∈ I. TheX-isomorphism Ue · ηX : X −→ UC may be lifted to an A-isomorphism j : A −→ C withUj = Ue · ηX . Hence, (mi · j)i∈I is the desired U -initial lifting of (fi)i∈I . Since (iv)op = (v),this completes the proof.

The factorization needed in (v) may be constructed with Theorem 5.3.2. For this, we saythat X has small connected limits if every diagram D : J −→ X with J small and connected(see Exercise 2.Q) has a chosen limit in X.

5.9.2 Corollary.c© Let X have small connected limits, and let U : A −→ X have small fibres.The functor U is topological if and only if the following conditions hold:

(1) U is faithful and transportable;

(2) A has small connected limits, and U preserves them;

(3) U has a fully faithful right adjoint.

Proof. For the necessity of the conditions, see Proposition 5.8.3 and Theorem 5.9.1. Fortheir sufficiency, after 5.9.1(v) we must only show that M = U−1(Iso X) satisfies theconditions of Theorem 5.3.2. Certainly, M satisfies Iso A ⊆M ⊆ Mono A, is closed undercomposition, and also stable under pullbacks since U preserves them. The only delicatepoint is the existence of (not necessarily small) intersections of morphisms in M. Hence,consider mi : Ai −→ B in M (i ∈ I). Transportability gives isomorphisms fi : Ai −→ Bi

with Ufi = Umi (i ∈ I), and

Bi ∈ ob X | i ∈ I ⊆ U−1(UB)

is just a set. Furthermore, if Bi = Bj, then (Ai,mi) ∼= (Aj,mj) in A/B for all i, j ∈ I.Consequently, in order to form the multiple pullback of (mi)i∈I in A, it suffices to form the


multiple pullback of a small subfamily c©, which exists and is preserved by U , so that it lies inM again.

5.10 Epireflective subcategories. In Proposition 5.1.3, we noted that a replete reflectivesubcategory B of a category C with a factorization system (E ,M) for morphisms has itsreflection morphisms in E if and only if B is closed under M-morphisms in C, that is, ifm : A −→ B in M with B ∈ ob B implies A ∈ B. More generally, for a collection M ofsources in C, one says that B is closed under M-sources in C if (mi : A −→ Bi)i∈I in M withBi ∈ ob B implies A ∈ ob B, and one proves the following result.

5.10.1 Proposition. In a category C with a factorization system (E ,M) for sources, a fullreplete subcategory B of C is E-reflective if and only if B is closed under M-sources in C.

Proof. For the “only if” part, one proceeds as in Proposition 5.1.3. For the “if” part, givenC ∈ ob C, one considers the source (fi : C −→ Ai)i∈I of all morphisms with domain C andcodomain in B, and one (E ,M)-factors it as fi = gi · e with e : C −→ B in E . Since B ∈ ob Bby hypothesis and E ⊆ Epi C (by the dual of Lemma 5.3.1), e is a B-reflection morphismfor C.

5.10.2 Corollary. c©Let C have products and a factorization system (E ,M) for morphisms,with E ⊆ Epi C, and suppose that C is E-cowellpowered. Then a full replete subcategory Bof C is E-reflective if and only if B is closed under products and M-morphisms in C.

Proof. After Proposition 5.10.1, it suffices to guarantee the existence of a factorizationsystem (E ,M) for sources. Given a source (fi : A −→ Bi)i∈I in C, one may (E ,M)-factoreach fi = mi · ei, and then choose c©a small representative family (ej : A −→ Cj)j∈J (withJ ⊆ I) among the morphisms ei ∈ E (i ∈ I). By (E ,M)-factoring the induced morphismg : A −→ ∏

j∈J Cj with pj · g = ej (for j ∈ J) as g = m · e, one obtains

fj = (mj · pj ·m) · e (j ∈ J)

with e ∈ E and d ⊥ (mj · pj ·m)j∈J for all d ∈ E . Since each ei (i ∈ I) is isomorphic tosome ej (j ∈ J), this factorization extends to the original source.

In the presence of a topological functor U : C −→ X, we may apply Proposition 5.10.1 inparticular to the (U−1(Iso X),M)-factorization system for sources in C, where now M consistsof all U -initial sources, and obtain at once the equivalence (i) ⇐⇒ (ii) in the followingTheorem.

5.10.3 Theorem. Let U : C −→ X be a topological functor. The following assertions for afull replete subcategory B of C are equivalent:

(i) B is U−1(Iso X)-reflective in C;

(ii) B is closed under U-initial sources in C;


(iii) U |B is topological, and U |B-initial sources in B are also U-initial in C.

Proof. From the comment preceding the Theorem, we are left to verify (ii)⇐⇒ (iii), butthis is immediate.

Let us finally discuss an important sufficient condition for the inclusion functor B → Cto preserve initiality, as in (iii). One says that a full replete subcategory B is U-finallydense (or simply finally dense) if every object C in C is the codomain of some U -final sink(fi : Ai −→ C)i∈I with all Ai ∈ ob B; the dual notion is that of a U-initially dense (or justinitially dense) subcategory. Without the topologicity assumption on U , one can still prove:

5.10.4 Proposition. Let U : C −→ X be a functor and B a full replete subcategory of Cthat is finally dense in C. Then U |B-initial sources in B are U-initial in C, and when U isfaithful and U |B is topological, B is U−1(Iso X)-reflective in C.

Proof. In order to show U -initiality of a U |B-initial source (fi : A −→ B)i∈I in B, consider(gi : C −→ Bi)i∈I in C and s : UC −→ UA in X with Ufi · s = Ugi for all i ∈ I, and let(hj : Aj −→ C)j∈J be U -final with all Aj ∈ ob B. U |B-initiality then gives, for every j ∈ J , aunique tj : C −→ Aj with Utj = s · Uhj and fi · tj = gi · hj for all i, j ∈ I, and U -finalityyields a unique t : C −→ A in B with Ut = s and t · hj = tj for all j ∈ J . By U |B-initialityagain, t also satisfies fi · t = gi for all i ∈ I, and one easily sees that t is the only morphism“over s” satisfying these equations:

Afi // Bi

Aj

tj

??

hj// C

t

OO

gi

??UA

UC .s=Ut

OO

For the reflexivity assertion, one proceeds as in the proof of Proposition 5.10.1 and, givenC ∈ ob C, one considers the source (fi : C −→ Ai)i∈I of all morphisms with domain C andcodomain in B. Let (gi : B −→ Ai)i∈I be a U |B-initial lifting of (Ufi : UC −→ UAi)i∈I . Since(gi)i∈I must be even U -initial, there is e : C −→ B in C with Ue = 1UC which is easily seento serve as the B-reflection morphism for C.

5.10.5 Example. There is a full coreflective embedding E : Ord −→ Top which providesan ordered set (X,≤) with the topology of open sets generated by the down-sets ↓x, forx ∈ X. Its right adjoint S provides a topological space with its underlying order (see 1.9).A topological space X is in the image of E precisely when it is an Alexandroff space, thatis, when arbitrary intersections of open sets in X are open. By the dual of Theorem 5.10.3,Ord is closed under U -final sinks in Top (with the forgetful functor U : Top −→ Set), andU |Ord-final sinks in Ord are U -final in Top. In fact, Ord is even initially dense in Top: for atopological space X, consider the source of characteristic functions of all open sets of Xinto the two-chain. But Ord is not closed under U -initial sources, not even under products,as no infinite power in Top of the two-chain is Alexandroff.


5.11 The Taut Lift Theorem. We consider a commutative diagram of functors

AU

G // BV

X J // Y

such that J has a left adjoint Hγ

δ

J . Our goal is to find a left adjoint Fη

ε G when U

has good lifting properties.

5.11.1 Theorem (Taut Lift Theorem). Let U be a topological functor. Then in theprevious displayed diagram, G has a left adjoint F with UF = HV if and only if G mapsU-initial sources to V -initial sources.

Proof. For the necessity of the stated condition, see Exercise 5.T. For its sufficiency, weconsider an object B in B and the source of all morphisms t : B −→ GAt with At ∈ A. Thesource of all

HV B HV t // HVGAt = HJUAtδUAt // UAt

has a U -initial liftingFB

ft// At

so that Uft = δUAt · HV t. As JHV B = JUFB, the codomain of γV B is the domain ofJUft, and

JUft · γV B = JδUAt · JHV t · γV B = JδUAt · γJUAt · V t = V t .

By hypothesis G transforms the source into a V -initial source, so the diagram

JUFB = V GFBJUft=V Gft

// JUAt = V GAt

V B

γV B

OO

V t

66

produces a morphism ηB : B −→ GFB with V ηB = γV B and Gft · ηB = t for all t. Byadjointness of J and faithfulness of U , the factorization V t = JUft · γV B determines ftuniquely.

5.11.2 Example. Consider the diagram of forgetful functors

TopGrp //

Top

Grp // Set


where TopGrp is the category of topological groups (that is, groups G with a topologythat makes the group operations G × G −→ G, (x, y) 7−→ x · y, and G −→ G, x 7−→ x−1

continuous), and of continuous group homomorphisms. It is easy to see that TopGrp −→ Grpis topological with initial structures formed as for Top −→ Set. Hence, the free-group functorF : Set −→ Grp may be lifted to a left adjoint functor Top −→ TopGrp. In lieu of groups, theexample generalizes to any algebraic structures defined by a set of operations and equationsbetween them, and that admit free functors (see Example 2.12.2).

Exercises

5.A More cancellation rules. LetM be a pullback-stable class of morphisms in C. ThenMsatisfies the cancellation property: if g · f ∈M with g monic implies f ∈M. IfM belongsto a factorization system (E ,M), then g ·f ∈M with a split epimorphism f implies g ∈M.The family (fi : A −→ Bi)i∈I is a mono-source if (gij · fi)i∈I,j∈J is a mono-source for sources(gij : Bi −→ Cij)j∈Ji , i ∈ I.

5.B More on proper systems. For a factorization system (E ,M) in a category C withequalizers, one has E ⊆ Epi C if and only if RegMono C ⊆M.

5.C Extremal, strong and regular epimorphisms. The class of strong epimorphisms in C is

StrongEpi C := Epi C ∩ ⊥(Mono C) .

One hasRegEpi C ⊆ StrongEpi C ⊆ ExtEpi C ,

where RegEpi C and ExtEpi C are the classes of regular and extremal epimorphisms, respec-tively (see 5.1 for the latter). If C has pullbacks, then StrongEpi C = ExtEpi C. Of course,the dual statements hold for strong monomorphisms. Furthermore, each of the followingstatements implies the next, and all are equivalent when C has kernel pairs and coequalizersof kernel pairs:

(i) Mono C · RegEpi C = mor C;

(ii) (RegEpi C,Mono C) is a factorization system in C;

(iii) RegEpi C = ExtEpi C;

(iv) RegEpi C is closed under composition.

5.D Regular monomorphisms in a topological category. If U : A −→ X is topological (orjust a faithful functor with both a left and a right adjoint), an A-morphism f is a regularmonomorphism if and only if it is U -initial and Uf is a regular monomorphism. The samestatement holds if “regular” is replaced by “extremal”.


5.E Functoriality of factorizations. For every factorization system (E ,M), there is a functorF : C2 −→ C which assigns to a morphism f in C the object domm = cod e for a chosen(E ,M)-factorization f = m · e.

5.F Wellpowered and cowellpowered categories over Set. The category Set and every topo-logical category over Set is wellpowered and cowellpowered. Every monadic category overSet is wellpowered and E-cowellpowered with E the class of regular epimorphisms. However,a monadic category over Set is not necessarily cowellpowered: Frm is not cowellpowered(see [Johnstone, 1982]).

5.G Closure operators as functors. A class M of morphisms in C can be considered as afull subcategory of C2 (see 5.6.1(4)). Show that an M-closure operator c defines a functorc : M −→ M, together with a natural transformation j : 1M −→ c. If there is a mono-fibration U : C −→ Set such that M = U−1 Mono∩ IniU (see 5.9) and C is M-wellpowered,then c defines a functor

c : C −→ Cls , X 7−→ (UX, cX) .

5.H Closure operators and closed subobjects. Let M be a right factorization class for sinksin C, and let K ⊆M be stable under pullbacks. Then

cA(m) =∧k ∈ subA | k ∈ K, k ≤ m

defines an idempotent M-closure operator of C, and one has Mc = K if and only if Kis stable under multiple pullbacks. Furthermore, as an idempotent closure operator, c isuniquely determined by the condition M⊆ K, and c is weakly hereditary if and only if Kis closed under composition.

5.I Adjunctions, epimorphisms and generating classes. For an adjunction Fη

ε G : A // X ,

one has that if G is faithful and H is generating in X, then

FH = FH | H ∈ H

is generating in A. Furthermore, the following conditions are equivalent:

(i) G is faithful;

(ii) G reflects epimorphisms;

(iii) the counits εA are epimorphisms;

(iv) the class FX | X ∈ X is generating in A.

The same equivalence holds if one specializes to a faithful functor that reflects isomorphismsin (i), extremal epimorphisms in (ii) and (iii), and a strongly generating class in (iv).


5.J Special Adjoint Functor Theorem. Suppose that A is locally small, small-complete,has a cogenerator G, and is wellpowered, so that for every A ∈ ob A there is a chosen setJA ⊆ ob A such that

if m : B −→ A is a monomorphism, then there is J ∈ JA with J ∼= B.

Then a functor G : A −→ X into a locally small category X is right adjoint if and only if itpreserves small limits.Hint. Use Proposition 5.5.1op and the sets JA to construct a G-solution set for everyX ∈ ob X.

5.K Special Adjoint Functor Theorem for a class of monomorphisms. For some class M ofmonomorphisms closed under composition with isomorphisms, let the locally small categoryA have pullbacks of morphisms in M (along arbitrary morphisms) and intersections ofarbitrarily large families of morphisms in M, and suppose that both belong to M again.Furthermore, let A have an M-cogenerator, that is, a set G of objects in A such that theproduct PA = ∏

G∈G GA(G,A) exists for all A ∈ ob A, and the canonical morphism PA −→ A

lies in M. Then a functor G : A −→ X has a left adjoint if and only if G preserves all limitswhose existence is guaranteed by the hypotheses.

5.L Dense generators. A class G of objects in a locally small category C is denselygenerating in C if C(G, A) is a strict epi-sink for all objects A, that is: whenever a sink(hx : Gx −→ B)x∈C(G,A) has the property

x · a = y · b =⇒ hx · a = hy · b

for all a : D −→ Gx, b : D −→ Gy, x, y ∈ C(G, A), then hx = f · x for all x, with a uniquelydetermined morphism f : A −→ B. Show that G is densely generating if and only if thefunctor

yG : C −→ SetGop, A 7−→ (C(−, A) : Gop −→ Set)

(which has the Yoneda embedding as a factor) is full and faithful; here, and differentlyfrom 5.5, we consider G as a full subcategory of C. A densely generating set is stronglygenerating, but not vice versa: a one-point space is a strong generator of CompHaus, thecategory of compact Hausdorff spaces and continuous maps, but it is not dense (CompHaushas in fact no dense generator, see [Gabriel and Ulmer, 1971]).

5.M M-injective objects. An object C in a category C is M-injective for a class M ofmorphisms in C if for any m : A −→ B with m ∈ M and f : A −→ C, there exists a mapg : B −→ C extending f , that is, such that the following diagram commutes:

A m //

f

B

g

C


(the dual notion is that of an E-projective object in C). If r : C −→ D is a retraction from anM-injective object C, then D is also M-injective. Also, a product of M-injective objectsis an M-injective object.

5.N U-cocartesian liftings of regular epimorphisms. Let A have kernel pairs, and supposethat U : A −→ X preserves them. Then a U -cocartesian lifting of a regular epimorphismf : UA −→ Y in X is a regular epimorphism in A. Morphisms in FinU ∩U−1(RegEpi X) arealso called quotient morphisms of A (with respect to U).

5.O Fibrations and equivalences. A fibration U : A −→ X which reflects isomorphisms is anequivalence, provided that A has a terminal object and U preserves it. The provision isessential: for any category C and T ∈ ob C, the domain functor dom = domT : C/T −→ C isa fibration and reflects isomorphisms, but it is not an equivalence, unless T is terminal in C.

5.P The codomain functor as a fibration. The functor cod : C2 −→ C of Example 5.6.1(4)is a fibration if and only if C has pullbacks. It is topological if and only if C has multiplepullbacks (of any size).

5.Q The domain functor is not necessarily topological. Let C be locally small and small-complete. For any T ∈ ob C, the domain functor dom : C/T −→ C satisfies all conditionsof Corollary 5.9.2, except that the right adjoint of dom generally fails to be fully faithful.Thus, in general, dom fails to be topological.

5.R The set-of-objects functor as a fibration. The set-of-objects functor ob : Cat −→ Set isa fibration, but not a cofibration. Hence, it is not topological.

5.S Topological restrictions of topological functors. Let U : C −→ X be topological and Aa full replete subcategory of C, and set J := U−1(Iso X). The following conditions areequivalent:

(i) U |A is topological;

(ii) there is a full J -reflective subcategory B of C which contains A as a J -coreflectivesubcategory;

(iii) there is a full J -coreflective subcategory B of C which contains A as a J -reflectivesubcategory;

(iv) there is a functor R : C −→ A with (U |A)R = U and R|A = 1A.

Hint. For (i) =⇒ (ii), take for ob B all C-objects C for which there exist a U -initial sourcewith domain C and all codomains in A.

5.T Preservation of initiality. For a commutative diagram of functors

AU

G // BV

X J // Y


such that there are adjunctions Fη

ε G and H

γ

δ

J , with the canonical transformationHV −→ UF an isomorphism, show that G maps U -initial sources to V -initial sources. (Notethat no topologicity assumption for U or V is required.)

5.U Generalized Taut Lift Theorem. In the commutative diagram of functors

AU

G // BV

X J // Y

let A have an (E ,M)-factorization system for sources such that G maps sources in M toV -initial sources. If U and J have left adjoints, then G has also a left adjoint.

5.V Grothendieck construction versus faithful fibrations and topological functors.

(1) For a faithful cloven fibration U : A −→ X one obtains (in the notation of 5.7) apseudo-functor

U−1 : Xop −→ ORDinto the ordered metacategory of ordered classes; it assigns to f : X −→ Y in X themonotone function f ∗ : U−1Y −→ U−1X. If U is topological, ORD may be replaced byINF, where objects have all infima and whose morphisms preserve them.

(2) For a pseudo-functor T : Xop −→ ORD, let the objects of the category XT be pairs(X, τ) with X ∈ ob X, τ ∈ TX; a morphism f : (X, τ) −→ (Y, σ) is an X-morphismf : X −→ Y with τ ≤ Tf(σ). The forgetful functor

UT : XT −→ X , (X, τ) −→ X

is a faithful fibration. When T takes values in INF, the functor UT is topological.

(3) For a faithful fibration U : A −→ X and T = U−1, there is an isomorphism G : A −→ XTwith UTG = U .

(4) For a pseudo-functor T : Xop −→ ORD, there is a pseudo-natural isomorphism γ : T −→(UT )−1.

5.W Equivalence of topological functors and INF-valued pseudo-functors.

(1) Let the objects of the metacategory FFIB be faithful fibrations, and let morphisms(F,G) : U −→ V be commutative diagrams

A G //

U

BV

X F // Y


in CAT, such that G transforms U -initial morphisms into V -initial morphisms. Theobjects of the metacategory CAT//ORD are contravariant pseudo-functors with valuesin ORD; a morphism

(F, γ) : T −→ S

is given by a diagramXop F op

//

T

Yop

S~~

ORD

γ+3

with a functor F : X −→ Y and a pseudo-natural transformation γ : T −→ SF op, andits composite with (H, δ) : S −→ R is given by

(H, δ)(F, γ) = (HF, δF op · γ) .

Show that FFIB and CAT//ORD are equivalent metacategories.

(2) Let TOPFUN be the (non-full) subcategory of FFIB formed by those (F,G) : U −→ Vwith U , V topological and G transforming U -initial sources into V -initial sources.Prove that TOPFUN is equivalent to CAT// INF.

(3) Formulate (1) and (2) for cofibrations instead of fibrations.


Notes on Chapter II

Most of the topics presented in this chapter may be found in more elaborate form in the standard books oncategory theory: [Mac Lane, 1971], [Adamek, Herrlich, and Strecker, 1990], [Borceux, 1994a,b,c]. Each ofthese books contains remarks or brief sections on the foundations of category theory and its connections withlogic or set theory, a topic that has been adressed in many articles, including [Lawvere, 1966], [Mac Lane,1969], [Feferman, 1969, 1977], [Benabou, 1985], as well as in books on topos theory, such as [Mac Laneand Moerdijk, 1994]. Beginners in category theory will enjoy the books [Lawvere and Rosebrugh, 2003]and [Awodey, 2006] while advanced readers are referred to [Kelly, 1982] as the standard text on enrichedcategory theory, and to [Johnstone, 2002a,b] as a rich resource for a broad range of categorical topics.Readers looking for further reading on order and quantale theory as it pertains to Section 1 of this chapterare referred to [Johnstone, 1982], [Rosenthal, 1990], [Wood, 2004].We highlight some particular aspects that distinguish this chapter from the literature mentioned thus farand give some additional references. The chapter covers some aspects of monad theory in greater detailthan the standard texts on category theory, as it includes Duskin’s Monadicity Criterion (see Theorem 3.5.1,originally established in [Duskin, 1969]) and a discussion of Beck’s distributive laws [Beck, 1969] (see also[Manes and Mulry, 2007]), as well as a treatment of Kock-Zoberlein monads (see [Kock, 1995] and referencestherein), albeit in the simplified context of ordered categories. For further reading on monads and theirconnection with algebraic theories, we refer the reader to [Manes, 1976], [Barr and Wells, 1985], [MacDonaldand Sobral, 2004] and [Adamek, Rosicky, and Vitale, 2011]. In a higher-order context, they are treatedin [Street, 1974] and [Lack and Street, 2002]. In this chapter, however, we touch upon higher-categoricalstructures only to a minimal level, with 2-cells most often given simply by order, the notion of quantaloidbeing an important example as treated in Stubbe’s articles [2005; 2006; 2007].

Another non-standard emphasis of this chapter concerns factorization systems and topological functorswhich, unlike in [Adamek, Herrlich, and Strecker, 1990], are presented in concert with fibrations (see[Grothendieck, Verdier, and Deligne, 1972], [Benabou, 1985], [Streicher, 1998–2012]), as highlighted byTheorem 5.9.1. A predecessor of its Corollary 5.9.2 first appeared in Hoffmann’s thesis [1972]. Wyler’s TautLift Theorem 5.11.1 was first proved in [Wyler, 1971], and presented in a general categorical context in[Tholen, 1978]. The key existence theorem on factorization systems (Theorem 5.3.2) appeared in generalizedform in [Tholen, 1979]; see also [Tholen, 1983]. For a categorical treatment of closure operators, the readeris referred to [Dikranjan and Tholen, 1995].

III Lax AlgebrasDirk Hofmann, Gavin J. Seal, Walter Tholen

For a quantale V and a monad T on Set, laxly extended to the category V-Rel of sets andV-valued relations, this chapter introduces the key category of interest to this book, thecategory (T,V)-Cat whose objects, depending on context, may be called (T,V)-algebras,(T,V)-spaces, or (T,V)-categories. After a first introduction to the V-relational settingand the required lax monad extension, the guiding examples (ordered sets, metric spaces,topological spaces, approach spaces) are presented in full detail, followed by the basicproperties of the category (T,V)-Cat, like its topologicity over Set and its embeddabilityinto the quasitopos of (T,V)-graphs. The seemingly “technical” lax extendability of T tothe “syntactical” category V-Rel permits us to consider T as a monad on V-Cat and even(T,V)-Cat, the Eilenberg–Moore algebras of which lead to the consideration of objects thatcombine relevant ordered, topological and metric structures in a very natural way.

1 Basic Concepts

1.1 V-relations. Recall from II.1.2 that a relation r from a set X to a set Y associatesto every pair (x, y) ∈ X × Y a truth value in 2 = false, true which tells us whether xand y are r-related or not. In order to model situations where quantitative information isavailable, r can be allowed to take values in any quantale V = (V ,⊗, k) rather than just in2 = (2,∧,>). (The quantale V is associative and unital, as defined in II.1.10.) A V-relationr : X −→7 Y from X to Y is therefore presented by a map r : X × Y −→ V . As for ordinaryrelations, a V-relation r : X −→7 Y can be composed with another V-relation s : Y −→7 Z via“matrix multiplication”

(s · r)(x, z) =∨

y∈Yr(x, y)⊗ s(y, z)

(for all x ∈ X, z ∈ Z) to yield a V-relation s · r : X −→7 Z. This composition is associative(see Exercise 1.C), and the V-relation 1X : X −→7 X that sends every diagonal element (x, x)to k, and all other elements to the bottom element ⊥ of V serves as the identity morphismon X. Thus, sets and V-relations form a category, denoted by

V-Rel .

145

146 CHAPTER III. LAX ALGEBRAS

The set V-Rel(X, Y ) of all V-relations from X to Y inherits the pointwise order induced byV : given r : X −→7 Y and r′ : X −→7 Y , we have

r ≤ r′ ⇐⇒ ∀(x, y) ∈ X × Y (r(x, y) ≤ r′(x, y)) .

Since the order on V is complete, so is the pointwise order on V-Rel(X, Y ), and since thetensor in V distributes over suprema, V-relational composition preserves suprema in eachvariable:

s · ∨i∈I ri = ∨i∈I(s · ri) and ∨

i∈I ri · t = ∨i∈I(ri · t)

for V-relations ri : X −→7 Y (i ∈ I), s : Y −→7 Z, and t : W −→7 X. Thus, V-Rel is not justan ordered category, but even a quantaloid (see II.4.5 and II.4.8).The canonical isomorphism X × Y ∼= Y × X induces a bijection between V-Rel(X, Y )and V-Rel(Y,X), so that for every V-relation r : X −→7 Y one has the opposite (or dual)V-relation r : Y −→7 X defined by

r(x, y) = r(y, x)

for all x ∈ X, y ∈ Y . This operation preserves the order on V-Rel(X, Y ):

r ≤ r′ =⇒ r ≤ (r′) ,

and one has 1X = 1X as well as r = r. Let us also note that the equality

(s · r) = r · s

holds whenever V is commutative.

1.1.1 Examples.

(1) As mentioned above, a 2-relation is just an ordinary relation. For relations r : X −→7 Yand s : Y −→7 Z, the previous “matrix multiplication” formula specializes to the usualrelational composition:

x (s · r) z ⇐⇒ ∃y ∈ Y (x r y & y s z) .

Therefore, the category of 2-relations is just the category of relations:

2-Rel ∼= Rel .

(2) For V = P+ (see II.1.10), a P+-relation is a “distance function” r : X × Y −→ P+, andcomposition with s : Y × Z −→ P+ yields

s · r(x, z) = infr(x, y) + s(y, z) | y ∈ Y

for all x ∈ X and z ∈ Z. Therefore, P+-Rel can be seen as the category of sets andmetric relations.

1. BASIC CONCEPTS 147

(3) For 22 = ⊥, u, v,> the diamond lattice of Exercise II.1.H, a 22-relation is a “choicerelation” that chooses between the truth values u and v, taking value ⊥ if none isselected, and > if both are. Each 22-relation r can therefore be considered as a pairof relations (ru, rv), and 22-Rel as the category of sets and birelations:

X × Y r // 22 ∼= 2× 2X × Y ru // 2 , X × Y rv // 2 .

1.2 Maps in V-Rel. There is a functor from Set to V-Rel that interprets the graph of aSet-map f : X −→ Y as the V-relation f : X −→7 Y given by

f(x, y) =

k if f(x) = y,⊥ otherwise.

The functor(−) : Set −→ V-Rel

is faithful if and only if ⊥ < k in V . Therefore, from now on, the quantale V is assumed tobe non-trivial, so that V is not reduced to a singleton (see Exercise 1.A). To keep notationssimple, we usually write f : X −→ Y instead of f : X −→7 Y to designate a V-relationinduced by a map.The formula for V-relational composition becomes considerably easier if one of the V-relations comes from a Set-map:

s · f(x, z) = s(f(x), z) , h · s(y, w) = s(y, h(w)) ,g · r(x, z) = ∨

y∈g−1(z) r(x, y) , t · f (y, z) = ∨x∈f−1(y) t(x, z) ,

for all maps f : X −→ Y , g : Y −→ Z, h : W −→ Z, V-relations r : X −→7 Y , s : Y −→7 Z,t : X −→7 Z, and elements w ∈ W , x ∈ X, y ∈ Y , z ∈ Z. In particular, one should keep inmind that the pointwise expression of a relation of the form h · s · f is

h · s · f(x,w) = s(f(x), h(w)) ,

as this formula will be used systematically from now on. We note that, without anycommutativity assumption on V , composition of V-relations with Set-maps is also compatiblewith the involution (−):

(s · f) = f · s and (g · r) = r · g .

The formulas for “relation-with-map composition” imply at once that every Set-mapf : X −→ Y satisfies the inequalities

1X ≤ f · f and f · f ≤ 1Y


in V-Rel, so that f is a map in the sense of II.4.7, thus providing further justification forthe identification f = f . Given Set-maps f : X −→ Y and g : Y −→ Z, we therefore obtainthe adjunction rules:

g · r ≤ t ⇐⇒ r ≤ g · t and t · f ≤ s ⇐⇒ t ≤ s · f (1.2.i)

for V-relations r : X −→7 Y , s : Y −→7 Z, and t : X −→7 Z (see Proposition II.4.7.1).For V = 2, every map (that is, every left-adjoint morphism) in V-Rel = Rel is given by aSet-map: whenever r a s : Y −→7 X, one has r = f, s = f for a uniquely determined mapf : X −→ Y (see II.4.7). We briefly discuss to which extent this fact holds more generallyand call the quantale V integral if k = >, that is, if the top element is the neutral elementof the tensor. In an integral quantale V, one has u ⊗ v ≤ u ∧ v for all u, v ∈ V (sinceu⊗ v ≤ u⊗> = u⊗ k = u and likewise u⊗ v ≤ v). We say that V is lean if

(u ∨ v = > and u⊗ v = ⊥) =⇒ (u = > or v = >)

for all u, v ∈ V. In an integral and lean quantale, > and ⊥ are the only complementedelements (that is, elements u for which there is v with u∨v = >, u∧v = ⊥). The quantales2, P+, Pmax are integral and lean, 3 and P× are lean but not integral, and 22 is integral butnot lean.

1.2.1 Proposition. For an integral quantale V, all left-adjoint V-relations are Set-mapsif and only if V is lean.

Proof. Let V be integral and lean, and assume r a s : Y −→7 X in V-Rel. If X = ∅, thenr : ∅ → Y is the inclusion map. Hence, one can consider x ∈ X. Since

⊥ < k = (s · r)(x, x) =∨

y′∈Yr(x, y′)⊗ s(y′, x) ,

there is some y ∈ Y with u := r(x, y)⊗ s(y, x) > ⊥, and we can write k = > = u∨ v, wherev := ∨

y′ 6=y r(x, y′)⊗ s(y′, x). From r · s(y, y′) ≤ ⊥ one obtains s(y, x)⊗ r(x, y′) = ⊥ for ally′ 6= y, and therefore

u⊗ v =∨y′ 6=y r(x, y)⊗ (s(y, x)⊗ r(x, y′))⊗ s(y′, x) = ⊥ .

Consequently, u = r(x, y)⊗ s(y, x) = > (since v = > = k would force u = u⊗ v = ⊥), sothat r(x, y) = > = s(y, x) because > = r(x, y) ⊗ s(y, x) ≤ r(x, y) ∧ s(y, x). Since u = >and v = ⊥, we have shown that for every x ∈ X there is precisely one y := f(x) in Y withr(x, y) ⊗ s(y, x) > ⊥, and then r(x, y) = s(y, x) = k. Consequently, f ≤ r and f ≤ s,which actually forces r = f :

r = r · 1X ≤ r · f · f ≤ r · s · f ≤ 1Y · f = f .

Conversely, for maps in V-Rel to be Set-maps we must show that V is necessarily lean. Ifu∨ v = > and u⊗ v = ⊥, with X := u, v we may define r : 1 = ? −→7 X by r(?, x) = x


and claim r a r. Indeed, since k = > and (r · r)(u, v) = u⊗ v = ⊥, one has r · r ≤ 1Y ;for 11 ≤ r · r we first observe

u = u⊗ k = u⊗ (u ∨ v) = (u⊗ u) ∨ (u⊗ v) = u⊗ u

and likewise v = v ⊗ v which implies

(r · r)(?, ?) = (u⊗ u) ∨ (v ⊗ v) = u ∨ v = k .

By hypothesis, r is then given by the maps ? 7−→ u or ? 7−→ v, which means u = k or v = k.

The adjunction f a f detects injectivity and surjectivity of the Set-map f by the equivalentconditions f · f = 1X and f · f = 1Y , respectively. In fact, one can easily prove a moregeneral fact:

1.2.2 Proposition.

(1) The Set-maps fi : X −→ Yi (i ∈ I 6= ∅) form a mono-source if and only if ∧i∈I f i ·fi =1X .

(2) The Set-maps g : Xi −→ Y (i ∈ I) form an epi-sink if and only if ∨i∈I gi · gi = 1Y .

Proof. The statements follow from

(∧i∈I f i · fi)(x, x′) = ∧i∈I 1Y (fi(x), fi(x′)) = k ⇐⇒ ∀i ∈ I (fi(x) = fi(x′))

for all x, x′ ∈ X, and

(∨i∈I gi · gi )(x, x′) = ∨i∈I∨x∈g−1

i y k = k ⇐⇒ ⋃i∈I g

−1i y 6= ∅

for all y ∈ Y .

1.2.3 Remarks.

(1) The empty source with domain X is a mono-source in Set if and only if |X| ≤ 1.Hence, the assertion of Proposition 1.2.2(1) remains valid also in the case I = ∅precisely when either X = ∅, or when |X| = 1 and > = k. Consequently, for anintegral quantale V , the restriction I 6= ∅ of Proposition 1.2.2(1) may be dropped.

(2) The functor (−) : Set −→ V-Rel has a right adjoint V-Rel −→ Set that sends a setX to the set VX of all maps φ : X −→ V, and a V-relation r : X −→7 Y to the maprPV : VX −→ VY defined by

rPV (φ)(y) = ∨x∈X φ(x)⊗ r(x, y)

for all φ ∈ VX and y ∈ Y . The monad PV induced by this adjunction is the V-powersetmonad—whose Kleisli category is V-Rel (see Exercise 1.D).


1.3 V-categories, V-functors and V-modules. We introduced V-categories and V-functors for a monoidal category V in II.4.10. Here we recall the definition in the highlysimplified case where V = (V,⊗, k) is a quantale.A V-relation a : X −→7 X is transitive if a ·a ≤ a and reflexive if 1X ≤ a. A V-category (X, a)is a set X with a transitive and reflexive V-relation a. A V-functor f : (X, a) −→ (Y, b) ofV-categories is given by a map f : X −→ Y with f · a ≤ b · f or, equivalently, a ≤ f · b · f .Hence, in pointwise notation, the characteristic conditions for a V-category read as

a(x, y)⊗ a(y, z) ≤ a(x, z) and k ≤ a(x, x) ,

and for a V-functor asa(x, y) ≤ b(f(x), f(y))

for all x, y, z ∈ X. Since identity maps and composites of V-functors are V-functors,V-categories and V-functors form a category

V-Cat .

1.3.1 Examples.

(1) For V = 2 = true, false, writing x ≤ y for a(x, y) = true, the transitivity andreflexivity conditions read as expected:

(x ≤ y & y ≤ z =⇒ x ≤ z) and x ≤ x

for all x, y, z ∈ X. Thus, a 2-category (X,≤) is just an ordered set. (Recall from II.1.3that we do not require an order to be antisymmetric.) A 2-functor f : (X,≤) −→ (Y,≤)is a map f : X −→ Y with

x ≤ y =⇒ f(x) ≤ f(y)

(for all x, y ∈ X), so the category of 2-categories is just the category of ordered sets:

2-Cat = Ord .

(2) For V = P+ (see Example II.1.10.1(3)), a transitive and reflexive P+-relation isequivalently described as a metric on X, that is, a map a : X ×X −→ P+ such that

a(x, y) + a(y, z) ≥ a(x, z) and 0 = a(x, x) ,

for all x, y, z ∈ X, and we say that X is a metric space. Whenever we require anyof the other traditionally assumed conditions, namely symmetry (a(x, y) = a(y, x)),separation (a(x, y) = 0 = a(y, x) =⇒ x = y), and finiteness (a(x, y) < ∞),we will say so explicitly, thus calling X a symmetric, separated or finitary metric


space, respectively. A P+-functor f : (X, a) −→ (X, b) is a map f : X −→ Y that isnon-expansive:

a(x, y) ≥ b(f(x), f(y))(for all x, y ∈ X). Hence, the category of P+-categories is equivalently described asthe category Met of metric spaces in the generalized sense as specified above:

P+-Cat = Met .

It contains the full subcategories Metsym and Metsep of symmetric and separated metricspaces, respectively.

We sketch some general procedures for creating V-categories, starting with V itself: theV-valued binary operation on V with

v ⊗ t ≤ w ⇐⇒ t ≤ v w

defines a V-relation, and one obtains:

1.3.2 Proposition. The quantale V endowed with the V-relation is a V-category.

Proof. Since v ⊗ k ≤ v, one has k ≤ v v, and from v w ≤ v w, one obtainsv ⊗ (v w) ≤ w. Consequently,

v ⊗ (v w)⊗ (w z) ≤ w ⊗ (w z) ≤ z ,

which yields transitivity: (v w)⊗ (w z) ≤ v z.

Note that the more general situation of this Proposition where V is not just a quantale buta monoidal category was sketched in Exercise II.4.I.For V = 2, returns the order of 2 (false < true), while for V = P+, is the truncateddifference: v w = w − v if v ≤ w <∞, v w = 0 if w ≤ v, and v ∞ =∞ if v <∞(see II.1.10).Let now X = (X, a) and Y = (Y, b) be V-categories. One defines a V-relation [−,−] onV-Cat(X, Y ) = f : X −→ Y | f is a V-functor by

[f, g] = ∧x∈X b(f(x), g(x)) ,

and a V-relation a⊗ b on X × Y by

(a⊗ b)((x, y), (x′, y′)) = a(x, x′)⊗ b(y, y′)

for all x, x′ ∈ X, y, y′ ∈ Y .

1.3.3 Proposition. Let X = (X, a), Y = (Y, b) be V-categories.

(1) [X, Y ] = (V-Cat(X, Y ), [−,−]) is a V-category.


(2) If V is commutative, then X ⊗ Y := (X × Y, a⊗ b) is a V-category.

Proof. Straightforward verifications.

1.3.4 Example. For V = P+, [f, g] = supx∈X b(f(x), g(x)) is the usual “sup-metric” onthe function space [X, Y ], and (a ⊗ b)((x, y), (x′, y′)) = a(x, x′) + b(y, y′) endows X × Ywith the usual “+-metric”.

1.3.5 Remark. The structure on [X, Y ] may be rewritten as

[f, g] = ∧x,x′∈X a(x, x′) b(f(x), g(x′)) . (1.3.i)

Indeed, since g : X −→ Y is a V-functor, for all x, x′ ∈ X, one has

b(f(x), g(x))⊗ a(x, x′) ≤ b(f(x), g(x))⊗ b(g(x), g(x′)) ≤ b(f(x), g(x′))

and then b(f(x), g(x)) ≤ a(x, x′) b(f(x), g(x′)), which proves “≤” of (1.3.i). “≥” followsfrom

a(x, x) b(f(x), g(x)) ≤ k b(f(x), g(x)) ≤ b(f(x), g(x))for all x ∈ X.

1.3.6 Theorem. For a commutative quantale V, V-Cat is a symmetric monoidal closedcategory.

Proof. We prove thatZ

φ−→ [X, Y ]Z ⊗X φ−→ Y

with φ(z)(x) = φ(z, x) for all z ∈ Z, x ∈ X establishes a bijective correspondence ofV-functors φ and φ when X = (X, a), Y = (Y, b), Z = (Z, c) are V-categories. In fact, byRemark 1.3.5, V-functoriality of φ means equivalently

c(z, z′) ≤ a(x, x′) b(φ(z)(x), φ(z)(x′))

for all z, z′ ∈ Z, x, x′ ∈ X, which may be equivalently rewritten as

(c⊗ a)((z, x), (z′, x′)) = c(z, z′)⊗ a(x, x′) ≤ b(φ(z, x), φ(z, x′)) ,

and this last inequality describes the V-functoriality of φ. We note that this inequality alsoentails V-functoriality of φ(z) : X −→ Y for all z ∈ Z, since

a(x, x′) = k ⊗ a(x, x′) ≤ c(z, z)⊗ a(x, x′) ≤ b(φ(z)(x), φ(z)(x′)) .

The correspondence is obviously bijective and natural in Z, which shows monoidal closure,and symmetry (X ⊗ Y ∼= Y ⊗X) holds trivially.


1.3.7 Examples. For V = 2, Theorem 1.3.6 confirms that Ord is cartesian closed and, forV = P+, that Met is monoidal closed. But note that the tensor product X ⊗ Y in Met mustnot be confused with the product X × Y : while the structure of X ⊗ Y is given by the“+-metric”, X × Y carries the “max-metric”; see Exercise 1.G.

The notion of module for ordered sets (see II.1.4) extends naturally from the case V = 2to the arbitrary case: for V-categories (X, a), (Y, b) one calls a V-relation r : X −→7 Y aV-module (also V-bimodule, V-profunctor or V-distributor) if

r · a ≤ r and b · r ≤ r .

Since the reversed inequalities always hold, these are in fact equalities:

r · a = r and b · r = r .

We writer : (X, a) −→ (Y, b)

if the V-relation r is a V-module. The module inequalities are stable under V-relationalcomposition, and

a : (X, a) −→ (X, a)serves as an identity morphism in the category

V-Mod

whose objects are V-categories and morphisms are V-modules. This category is ordered,with the order inherited from V-Rel; in fact, V-Mod is a quantaloid, with suprema in itshom-sets formed as in V-Rel.There is now a structured version of the functors

Set (−)// V-Rel Setop ,(−)

oo

as follows. For a V-functor f : (X, a) −→ (Y, b) one defines V-modules

f∗ : (X, a) −→ (Y, b) and f ∗ : (Y, b) −→ (X, a)

byf∗ := b · f and f ∗ := f · b ,

that is:f∗(x, y) = b(f(x), y) and f ∗(y, x) = b(y, f(x))

for all x ∈ X, y ∈ Y . One easily verifies the V-module conditions and functoriality:

V-Cat (−)∗// V-Mod (V-Cat)op .(−)∗

oo


In particular,1(X,a) = a = (1X)∗ = (1X)∗ ,

so that we can simply write 1∗X for the identity V-module on (X, a). Moreover, there is alsoa structured version of the adjunction f a f :1.3.8 Proposition. For a V-functor f : (X, a) −→ (Y, b), one has f∗ a f ∗ in V-Mod.

Proof. One writesf∗ · f ∗ = b · f · f · b ≤ b · 1Y · b ≤ b = 1∗Y ,

and1∗X = a ≤ a · 1X · a ≤ a · f · f · a ≤ f · b · b · f = f ∗ · f∗ ,

which proves the claim.

Let us finally observe that if V is commutative, then for every V-category X = (X, a), thepair

Xop := (X, a)is also a V-category, called the dual of X. For every V-functor f : (X, a) −→ (Y, b) one hasa V-functor f op : Xop −→ Y op given by f , so there is a functor

(−)op : V-Cat −→ V-Cat .Furthermore,

(f op)∗ = (f ∗) and (f op)∗ = (f∗) .

1.4 Lax extensions of functors. Subsection 1.2 shows that the category V-Rel of V-relations can be seen as an extension of Set. For a given monad T = (T,m, e) on Set, wenow consider extensions of T to V-Rel. For this, we first concentrate on the underlyingSet-functor T ; the natural transformations e and m will be considered afterwards, in 1.5.1.4.1 Definition. For a quantale V and a functor T : Set −→ Set, a lax extension T :V-Rel −→ V-Rel of T to V-Rel is given by functions

TX,Y : V-Rel(X, Y ) −→ V-Rel(TX, TY )

for all sets X, Y (with TX,Y simply written as T ), such that

(1) r ≤ r′ =⇒ T r ≤ T r′,

(2) T s · T r ≤ T (s · r),

(3) Tf ≤ T f and (Tf) ≤ T (f ),

for all sets X, Y, Z, V-relations r, r′ : X −→7 Y , s : Y −→7 Z, and maps f : X −→ Y . Bysetting TX = TX for all sets X, and observing that condition (3) yields 1TX ≤ T1X , onecan define a lax extension of a Set-functor T equivalently as a lax functor T : V-Rel −→ V-Rel(see II.4.6) that agrees with T on objects of V-Rel and satisfies the extension condition (3).


1.4.2 Examples.

(1) The identity functor on Set has a lax extension given by the identity functor on V-Rel.

(2) For V = 2, the covariant powerset functor P : Set −→ Set has lax extensions P , P :Rel −→ Rel given by

A (P r) B ⇐⇒ A ⊆ r(B) ⇐⇒ ∀x ∈ A ∃y ∈ B (x r y)A (P r) B ⇐⇒ B ⊆ r(A) ⇐⇒ ∀y ∈ B ∃x ∈ A (x r y)

for every relation r : X −→7 Y , and all A ⊆ X, B ⊆ Y .

(3) Every functor T on Set admits a largest lax extension to V-Rel given by

T>r : TX × TY −→ V , (x , y ) 7−→ >

for all V-relations r : X −→7 Y .

Although a lax extension T preserves composition of V-relations only up to inequality, itoperates more strictly on composites of V-relations with Set-maps, as the Corollary to thefollowing Proposition shows.

1.4.3 Proposition. Given functions TX,Y : V-Rel(X, Y ) −→ V-Rel(TX, TY ) that satisfythe conditions (1) and (2) of 1.4.1, the following are equivalent:

(i) Tf ≤ T f and (Tf) ≤ T (f ) for all f : X −→ Y (this is condition 1.4.1(3));

(ii) Tf ≤ T f and T (s · f) = T s · Tf for all f : X −→ Y and s : Y −→7 Z;

(iii) (Tf) ≤ T (f ) and T (f · r) = (Tf) · T r for all f : X −→ Y and r : Z −→7 Y .

The next condition is a consequence of any of the previous ones, and is equivalent to eachof them if T also satisfies 1TX ≤ T1X :

(iv) T (s · f) = T s · Tf and T (f · r) = (Tf) · T r for all f : X −→ Y and r : Z −→7 Y ,s : Y −→7 Z.

Proof. For (i) =⇒ (ii), we observe

T s · Tf ≤ T s · T f ≤ T (s · f) ≤ T (s · f) · (Tf) · Tf≤ T (s · f) · T (f ) · Tf ≤ T (s · f · f ) · Tf ≤ T s · Tf ,

so these inequalities are all equalities, and (i) =⇒ (iii) is shown in the same way. For(ii) =⇒ (iv), we see that 1TX ≤ T1X follows immediately from Tf ≤ T f ; moreover, oneobserves

Tf · T (f · r) ≤ T f · T (f · r) ≤ T (f · f · r) ≤ T r ,


so T (f · r) ≤ (Tf) · T r follows by the first adjunction rule in (1.2.i); for the otherinequality, we apply T to 1X ≤ f · f to obtain 1X ≤ T (f ) · Tf thanks to the hypothesis,and (Tf) ≤ T (f ) by the second adjunction rule, so that

(Tf) · T r ≤ T (f ) · T r ≤ T (f · r) .

The implication (iii) =⇒ (iv) is proved similarly. Finally, for (iv) =⇒ (i), we observe that1X ≤ f · f yields

1TX ≤ T (1X) ≤ T (f · f) = T (f ) · Tf ,which implies (Tf) ≤ T (f ), and Tf ≤ T f follows in a similar way.

1.4.4 Corollary. For a lax extension T : V-Rel −→ V-Rel of a Set-functor T one has

T (s · f) = T s · T f = T s · Tf , T (f · r) = T (f ) · T r = (Tf) · T r

for all maps f : X −→ Y and V-relations r : Z −→7 Y , s : Y −→7 Z.

Proof. This follows immediately from the Proposition since a lax extension is a lax functor,and T s · T f ≤ T (s · f) = T s · Tf ≤ T s · T f ; the other equalities are obtained in the sameway.

A lax extension T of T is flat if

T1X = T1X = 1TX ,

that is, if both diagrams

V-Rel T // V-Rel

Set(−)

OO

T // Set(−)

OO V-Rel T // V-Rel

Setop

(−)OO

T op// Setop

(−)OO

commute. Indeed, if T is flat, by the previous Proposition one obtains

T f = T1Y · Tf = Tf and T (f ) = (Tf) · T1X = (Tf)

for all f : X −→ Y in Set. Note that, of all the Examples 1.4.2, only the given lax extensionof the identity functor is flat.

1.5 Lax extensions of monads. Let us now turn our attention to the natural trans-formations e and m that we wish to extend from Set to V-Rel together with the functorT .1.5.1 Definition. A triple T = (T ,m, e) is a lax extension of the monad T = (T,m, e) if Tis a lax extension of T which makes both m : T T −→ T and e : 1V-Rel −→ T oplax (see II.4.6),that is:

(4) mY · T T r ≤ T r ·mX ,


(5) eY · r ≤ T r · eX ,

for all V-relations r : X −→7 Y .

By using both adjunction rules (1.2.i) of 1.2 for the maps mX and eX , we obtain thefollowing equivalent formulations of (4) and (5):

(4) T T r ·mX ≤ mY · T r,

(5) r · eX ≤ eY · T r.

Similarly, these conditions are equivalent to:

(4′) T T r ≤ mY · T r ·mX ,

(5′) r ≤ eY · T r · eX .

These inequalities then yield the following pointwise expressions:

(4∗) T T r(X ,Y ) ≤ T r(mX(X ),mY (Y )),

(5∗) r(x, y) ≤ T r(eX(x), eY (y)),

for all x ∈ X, y ∈ Y , X ∈ TTX, Y ∈ TTY , and V-relations r : X −→7 Y .One says that a lax extension T = (T ,m, e) of the monad T is flat if the lax extension T ofthe functor T is flat.The construction of lax extensions in a general setting can be rather technical, and ispostponed until Chapter IV. In this Chapter, however, we will describe especially importantextensions in 1.10 and 2.4. Here, we restrict ourselves to reconsidering the easy examplesof 1.4.2. These demonstrate in particular that a monad on Set may generally admit morethan one lax extension to V-Rel.

1.5.2 Examples.

(1) The identity monad I on Set can be extended to the identity monad I on V-Rel, andunless otherwise stated, this is the flat lax extension that will be used from now onfor this monad.

(2) The lax extensions P , P of 1.4.2 provide non-flat lax extensions P, P of the powersetmonad P (see Example II.3.1.1(3)) to Rel.

(3) Every monad T on Set admits a largest lax extension T> to V-Rel. It fails to be flat.


1.6 (T,V)-categories and (T,V)-functors. Let V be a quantale, and T = (T ,m, e) alax extension to V-Rel of a monad T = (T,m, e) on Set. A (T,V)-relation a : TX −→7 X istransitive if it satisfies

a · T a ·mX ≤ a or equivalently a · T a ≤ a ·mX

by adjunction (see 1.2). In pointwise notation, this transitivity condition becomes

T a(X , y )⊗ a(y , z) ≤ a(mX(X ), z)

for all X ∈ TTX, y ∈ TX, and z ∈ X. A (T,V)-relation a : TX −→7 X is reflexive if itsatisfies

eX ≤ a or equivalently 1X ≤ a · eX .

In pointwise notation, a : TX −→7 X is reflexive if and only if

k ≤ a(eX(x), x)

holds for all x ∈ X.

1.6.1 Definition. A (T,V)-category, depending on context also referred to as a lax algebra,a (T,V)-algebra or a (T,V)-space, is a pair (X, a) consisting of a set X and a transitive andreflexive (T,V)-relation a : TX −→7 X; that is, it is a set X with a V-relation a : TX −→7 Xsatisfying the two laws for an Eilenberg–Moore algebra laxly:

TTX

mX

T a //

≥

TX_ a

TX a

// X

XeX //

1X

≤

!!

TX_ a

X .

Note that the notion depends in fact not just on T but also on T, hence, whenever needed,we will refer to a (T,V)-category more precisely as a (T,V , T)-category.

We already considered an important special type of (T,V)-categories in 1.3. When T is theidentity monad I identically extended to V-Rel, an (I,V)-category is simply a V-category.Hence, in what follows we consider easy examples with other choices of T. Further exampleswill follow in Section 2.

1.6.2 Examples.

(1) With V = 2 and T = P laxly extended by P (Example 1.5.2(2)), a transitive andreflexive relation a : PX −→7 X must satisfy the conditions

(A ⊆ a(B) & B a z =⇒ (⋃A) a z) and x a x


for all x ∈ X, B ⊆ X, A ⊆ PX. Since x a y may be re-written asx

⊆ a(y),

byx ≤ y ⇐⇒ x a y

one defines an order on X. We claim that this order completely determines a, since

A a y ⇐⇒ ∀x ∈ A (x a y) ⇐⇒ A ⊆ ↓ y . (1.6.i)

Indeed, when A a y and x ∈ A one hasx

⊆ a(A), hence x a y by transitivity;

when x a y for all x ∈ A one usesx | x ∈ A

⊆ a(y) to obtain A a y by

transitivity.Conversely, starting with an order ≤ on X, (1.6.i) defines a (P, 2, P)-category structurea on X which reproduces the original order.

(2) Trading P for P (Example 1.5.2(2)), for a transitive and reflexive relation a : PX −→7 Xwe may define a closure operation c on PX by

x ∈ c(A) ⇐⇒ A a x .

(For idempotency of c, consider A =c(x) | x ∈ A

, where A ⊆ X.) Conversely,

given c, this definition yields a (P, 2, P)-category structure a on X.

(3) For arbitrary V and the largest lax extension T> of a monad T (Example 1.5.2(3)),the only (T,V ,T>)-category structure t on a set X is given by t(x , y) = > for allx ∈ TX, y ∈ X. Indeed, for any (T,V ,T>)-category structure a on X, one has

a = a · eTX ·mX ≤ eX · T>a ·mX ≤ a · T>a ·mX ≤ a ,

soa(x , y) = eX · T>a ·mX(x , y) =

∨X∈m−1

X (x ) T>a(X , eX(y)) = >

since eTX(x ) ∈ m−1X (x ) 6= ∅.

1.6.3 Definition. A map f : X −→ Y between (T,V)-categories (X, a) and (Y, b) is a(T,V)-functor if it satisfies

f · a ≤ b · Tf .

Diagrammatically this means that f is a lax homomorphism of lax algebras:

TXTf//

_a

≤

TY_ b

Xf// Y .


We can transcribe this condition equivalently as a ≤ f · b ·Tf which, in pointwise notation,reads as

a(x , x) ≤ b(Tf(x ), f(x))for all x ∈ TX and x ∈ X.

The identity map 1X : (X, a) −→ (X, a) is a (T,V)-functor, and so is the composite of(T,V)-functors. Hence, (T,V)-categories and (T,V)-functors form a category, denoted by

(T,V)-Cat .

Of course, this category depends on the lax extension T of T , but we will always assume thatsuch an extension is given beforehand and will therefore write more precisely (T,V , T)-Cat inlieu of (T,V)-Cat only if there is a danger of ambiguity. When T = I is identically extendedto V-Rel, an (I,V)-functor is simply a V-functor. Hence,

(I,V)-Cat = V-Cat .

1.6.4 Examples.

(1) For a map between (P, 2, P)-categories to be a (P, 2)-functor means equivalently thatthe map must be monotone with respect to the induced orders (1.6.2(1)). As aconsequence, one obtains an isomorphism

(P, 2, P)-Cat ∼= Ord

which leaves underlying sets invariant.

(2) Similarly, with 1.6.2(2) one has an isomorphism

(P, 2, P)-Cat ∼= Cls .

(3) Because of 1.6.2(3) there is an isomorphism

(T,V ,T>)-Cat ∼= Set

for every monad T on Set and every quantale V .

A T-algebra (X, a) (where a is a map satisfying a · eX = 1X and a ·Ta = a ·mX) is generallynot a (T,V)-algebra (X, a) (which requires a · T a ≤ a ·mX), as one may see already in thecase T = P with its extensions P or P. There is, of course, no problem when T is flat:

1.6.5 Proposition. If T is a flat lax extension of T to V-Rel, then a T-algebra (X, a :TX −→ X) is also a (T,V)-category. In this case, morphisms of T-algebras yield (T,V)-functors between the corresponding (T,V)-categories, and there is a full embedding

SetT → (T,V)-Cat .


Proof. The only non-obvious fact in the statement of the Proposition is that the embeddingof SetT in (T,V)-Cat is full. To see this, consider T-algebras (X, a) and (Y, b) with a(T,V)-functor f : (X, a) −→ (Y, b), that is, a map f : X −→ Y satisfying

f · a ≤ b · Tf

in V-Rel. As f · a and b ·Tf are really Set-maps, the inequality means that the graph of thefirst is contained in the second, but an inclusion of graphs of Set-maps with same domainis an equality.

1.7 Kleisli convolution. The relations representing (T,V)-category structures are of theform a : TX −→7 X. More generally, a (T,V)-relation is a V-relation r : TX −→7 Y , alsodenoted by r : X −7 Y . In order to compose such relations, we introduce the Kleisliconvolution of (T,V)-relations as a variation of the Kleisli composition presented in II.3.6.Let us emphasize that associativity of this operation turns out to depend on the monad laxextension, so that sets with V-relations r : TX −→7 Y only form a category in particularcases. In 1.8, we will provide a context in which the Kleisli convolution allows for identitymorphisms, and in 1.9 we will introduce a category that has the Kleisli convolution as itscomposition.

1.7.1 Definition. Given a lax extension T = (T ,m, e) of a Set-monad T = (T,m, e), theKleisli convolution s r : X −7 Z of (T,V)-relations r : X −7 Y and s : Y −7 Z is the(T,V)-relation defined by

s r := s · T r ·mX ,

an operation that may be depicted as

( TX r // Y , TY s // Z ) // ( TX mX // TTX Tr // TY s // Z ) .

When T = I, thens r = s · r

is just the relational composition of V-relations.The set of all (T,V)-relations from X to Y inherits the order of V-Rel(TX, Y ):

r ≤ r′ ⇐⇒ ∀(x , y) ∈ TX × Y (r(x , y) ≤ r′(x , y)) ,

and the Kleisli convolution preserves this order in each variable:

r ≤ r′, s ≤ s′ =⇒ r s ≤ r′ s′ ,

for all r, r′ : X −7 Y and s, s′ : Y −7 Z. The (T,V)-relation eX : X −7 X is a lax identityfor this composition: one has

eY r = eY · T r ·mX ≥ r · eTX ·mX = r ,


with equality holding if e = (eX)X : T −→ 1 is a natural transformation, and

r eX = r · T (eX) ·mX ≥ r · (TeX) ·mX = r ,

with equality holding if T is flat. In particular, eX eX ≥ eX , but generally this inequalityis strict when T fails to be flat. It turns out that eX can be modified to become idempotentwithout loss of its lax identity properties. To this end, we first prove the following result.

1.7.2 Lemma. For a lax extension T = (T ,m, e) to V-Rel of a monad T = (T,m, e) onSet one has:

T1X = T (eX) ·mX .

Proof. On one hand, we can exploit 1TX = 1TX = (mX · TeX) = (TeX) ·mX to obtain

T1X = T1X · 1TX (1TX = 1TX)= T1X · (TeX) ·mX (1TX = (TeX) ·mX)≤ T (1X) · T (eX) ·mX ((TeX) ≤ T (eX))≤ T (eX) ·mX (T lax functor).

On the other hand,

T (eX) ·mX ≤ T (eX · T1X) ·mX (1TX ≤ T1X)= (TeX) · T T1X ·mX (Corollary 1.4.4)≤ (TeX) ·mX · T1X (m oplax)= T1X (1TX = (TeX) ·mX),

which concludes the proof.

We set1]X := eX eX ,

hence1]X = eX · T1X

by the Lemma, and we can prove:

1.7.3 Proposition. If T = (T ,m, e) is a lax extension of the monad T = (T,m, e) toV-Rel, then

r eX = r · T1X = r 1]X and eY r = 1]Y r

for all (T,V)-relations r : X −7 Y . In particular, 1]X 1]X = 1]X , so that (X, 1]X) is a(T,V)-algebra.


Proof. We first observe that

r 1]X = r · T (eX · T1X) ·mX= r · (TeX) · T T1X ·mX (Corollary 1.4.4)≤ r · (TeX) ·mX · T1X (m oplax)= r · T1X (1X = (TeX) ·mX)= r · T (eX) ·mX (Lemma 1.7.2)= r eX .

This inequality suffices to prove the first set of equalities, since eX ≤ 1]X implies reX ≤ r1]X .The other equality follows directly from Corollary 1.4.4, as

1]Y r = eY · T (1Y ) · T r ·mX = eY · T r ·mX = eY r .

Finally,1]X 1]X = 1]X · T1X = eX · T1X · T1X = eX · T1X = 1]X .

We call (X, 1]X) the discrete (T,V)-category over X, see 3.2 below.

1.8 Unitary (T,V)-relations. Our candidates for the identities of the Kleisli convolutionare the (T,V)-relations 1]X , but the array of (T,V)-relations from X to Y that are leftinvariant by composition with these identities must still be determined.

1.8.1 Definition. A (T,V)-relation r : X −7 Y is right unitary if it satisfies

r eX ≤ r ,

while it is left unitary ifeY r ≤ r

holds. In terms of the relational composition, these conditions amount to

r · T1X ≤ r and eY · T r ·mX ≤ r ,

respectively. The (T,V)-relation r is unitary if it is both left and right unitary.

The (T,V)-relation eX itself is not unitary in general, but Proposition 1.7.3 shows that wecan replace it in the previous definitions by 1]X . It also follows from the discussion precedingLemma 1.7.2 that the inequalities appearing in the left and right unitary conditions are infact equalities. Hence, a (T,V)-relation r is right unitary, respectively left unitary, if

r 1]X = r , respectively 1]Y r = r .


Let us examine (T,V)-categories and (T,V)-functors in the light of Kleisli convolution andunitary (T,V)-relations. By definition, a (T,V)-category structure is a relation a : TX −→7 Xsuch that

a a ≤ a and eX ≤ a .

These conditions imply that such a (T,V)-relation a : X −7 X is always unitary:

a 1]X = a = 1]X a ,

since a eX ≤ a a ≤ a and eX a ≤ a a ≤ a. As a consequence, a : TX −→7 X is a(T,V)-category structure if and only if

a a = a and 1]X ≤ a . (1.8.i)

Indeed, the first condition follows from transitivity: a ≤ a eX ≤ a a ≤ a, and the secondcondition follows from reflexivity: 1]X = eX eX ≤ a eX ≤ a (the converse resulting fromeX ≤ 1]X). Hence, a (T,V)-algebra structure a can also be seen as a monoid in the orderedset of unitary (T,V)-relations from X to X, considered as a category that is provided withthe -operation. But we recall that associativity of is guaranteed only under additionalhypotheses (see 1.9 below), a property that is needed to consider as a monoidal structure.By definition, a (T,V)-functor f : (X, a) −→ (Y, b) is a map f : X −→ Y satisfying

f · a ≤ b · Tf .

Since a (T,V)-category structure b is right unitary, it satisfies b · T1Y = b eY = b byProposition 1.7.3; one then obtains by Proposition 1.4.3 the equalities b · T f = b · T1Y ·Tf =b · Tf . Hence, the (T,V)-functor condition can equivalently be expressed by using the laxextension of T :

f · a ≤ b · T f .

Settingf ] := f · 1]Y : Y −7 X ,

one can also express (T,V)-functoriality of a map f : X −→ Y via Kleisli convolution as

a f ] ≤ f ] b ,

see Exercise 1.M.

1.9 Associativity of unitary (T,V)-relations. With respect to the Kleisli convolution,the unitary (T,V)-relation 1]X serves as an identity for all unitary (T,V)-relations composablewith 1]X . In general however, unitary (T,V)-relations do not compose associatively, evenwhen T is the identity monad (see Proposition 1.9.7 below).


1.9.1 Definition. A lax extension T to V-Rel of a monad T = (T,m, e) on Set is associativewhenever the Kleisli convolution of unitary (T,V)-relations is associative. Explicitly, a laxextension T is associative whenever

t (s r) = (t s) r ,

or equivalently,t · T (s · T r ·mX) ·mX = t · T s ·mY · T r ·mX (1.9.i)

for all unitary (T,V)-relations r : X −7 Y , s : Y −7 Z, and t : Z −7 W .

For T associative, unitary (T,V)-relations are closed under Kleisli convolution:

(s r) 1]X = s (r 1]X) = s r and 1]X (s r) = (1]X s) r = s r .

Hence, in the presence of an associative lax extension T, we can form the category

(T,V)-URel

whose objects are sets, and whose morphisms are unitary (T,V)-relations that composevia Kleisli convolution. We note that, like (T,V)-Cat, also (T,V)-URel depends on the laxextension T; we write (T,V , T)-URel whenever this dependency needs to be emphasized.When the hom-sets (T,V)-URel(X, Y ) are equipped with the pointwise order induced by V ,

r ≤ r′ ⇐⇒ ∀(x , y) ∈ TX × Y (r(x , y) ≤ r′(x , y)) ,

(T,V)-URel becomes an ordered category.Condition (1.9.i) appears daunting to verify directly, so we postpone examples of associativelax extensions until after Proposition 1.9.4 which presents more practical conditions. Tothis end, we introduce the unitary (T,V)-relation

r] := eY · T r : TX −→7 Y

associated to a V-relation r : X −→7 Y (see Exercise 1.N). Notice that (1X)] = 1]X as definedin 1.7.

1.9.2 Lemma. Let T be a lax extension to V-Rel of a monad T = (T,m, e) on Set. Then

T (s] · T r) ·mX = T (s · r)

for all V-relations r : X −→7 Y and s : Y −→7 Z. In particular,

T (s]) ·mY = T s

for all V-relations s : Y −→7 Z.


Proof. The first stated equality follows from

T (s · r) = T (s · r) · T (eX) ·mX (Lemma 1.7.2)≤ T (s · r · eX) ·mX (T lax functor)≤ T (eZ · T s · T r) ·mX = T (s] · T r) ·mX (e lax natural)≤ T (eZ · T (s · r)) ·mX (T lax functor)= (TeZ) · T T (s · r) ·mX (Corollary 1.4.4)≤ (TeZ) ·mZ · T (s · r) (m lax natural)= T (s · r) .

The particular case is obtained by setting r = 1Y .

1.9.3 Lemma. Let T be a lax extension to V-Rel of a monad T = (T,m, e) on Set. Then

mX · T1X = T1TX ·mX · T1X = T T1X ·mX · T1X .

Proof. Since 1TTX ≤ T1TX ≤ T T1X , we have

mX · T1X ≤ T1TX ·mX · T1X ≤ T T1X ·mX · T1X ≤ mX · T1X .

by lax naturality of m.

1.9.4 Proposition. Let T be a lax extension to V-Rel of a monad T = (T,m, e) on Set.The following are equivalent:

(i) T is associative;

(ii) T : V-Rel −→ V-Rel preserves composition and m : T −→ T T is natural;

(iii) t (s r) = (t s) r for all V-relations t : TZ −→7 W , s : TY −→7 Z and right unitaryV-relations r : TX −→7 Y .

Proof. For (i) =⇒ (ii), consider V-relations r : X −→7 Y and s : Y −→7 Z. We first prove thatan associative lax extension preserves composition. Since all of r], s], and T1Z are unitary(Exercise 1.N), one has

T1Z (s] r]) = (T1Z s]) r] .

This identity is equivalent to T (s · r) = T s · T r: indeed,

T1Z (s] r]) = T (s] · T (r]) ·mX) ·mX = T (s] · T r) ·mX = T (s · r)

by using Lemma 1.9.2 twice, and

(T1Z s]) r] = T (s]) ·mY · T (r]) ·mX = T s · T r


by Lemma 1.9.2 again. To see that m is natural, we compute

T1Y (T1Y r]) = T1Y · T (T1Y · T (r]) ·mX) ·mX = T1Y · T T r ·mX = T T r ·mXand

(T1Y T1Y ) r] = T1Y · T T1Y ·mY · T (r]) ·mX = mY · T rusing Lemmata 1.9.2 and 1.9.3. Since Kleisli convolution is associative on unitary relations,we obtain T T r ·mX = mY · T r.For (ii) =⇒ (iii), we use right unitariness of r to write

t (s r) = t · T (s · T (r · T1X) ·mX) ·mX= t · T (s · T r · T T1X ·mX) ·mX= t · T (s · T r ·mX · T1X) ·mX= t · T s · T T r · T (mX) · T T1X ·mX= t · T s · T T r · (TmX) · T T1X ·mX= t · T s · T T r ·mTX ·mX · T1X= t · T s ·mY · T r ·mX · T1X= t · T s ·mY · T (r · T1X) ·mX = (t s) r .

(iii) =⇒ (i) is immediate by definition of an associative lax extension.

1.9.5 Remark. In the case where the lax extension T to V -Rel satisfies T (r) = (T r) forall V-relations r, the naturality condition for m in Proposition 1.9.4(ii) can equivalentlybe expressed as naturality of m : T T −→ T (see Exercise 1.J).

1.9.6 Examples.(1) The identity extension I to V-Rel of the identity monad is associative. Indeed, the

Kleisli convolution of V-relations is their usual composition, which is associative(Exercise 1.C).

(2) The lax extensions P, P of the powerset monad P (Example 1.5.2(2)) to Rel are bothassociative. Let us verify (ii) of Proposition 1.9.4 for P (the verifications for P aresimilar). Since A (P r) B is defined as A ⊆ r(B), the equivalence

A ⊆ (s · r)(C) ⇐⇒ ∃B ⊆ Y (A ⊆ r(B) & B ⊆ s(C))

(for all A ⊆ X, C ⊆ Y , and relations r : X −→7 Y , s : Y −→7 X) shows that the laxextension P preserves relational composition. As the monad multiplication ⋃ yields alax natural transformation ⋃ in Rel, one only needs to verify that ⋃Y ·P r ⊆ P P r ·⋃X :suppose that for A ⊆ X and B ⊆ PY , one has A ⊆ r(⋃B); then the subsetA = x | x ∈ A ⊆ PX is such that ⋃A = A and for all A′ ∈ A, there is B ∈ Bsuch that A′ ⊆ r(B).


(3) The largest lax extension T> to V-Rel of a monad T on Set is associative.

Let us exhibit now a non-associative lax extension. To this end, we consider the five-elementframe C depicted by

>

w

OO

u

==

v

aa

⊥`` >>

(considered as a quantale with ⊗ = ∧). The identity monad I = (I = 1Set, 1, 1) on Set maybe extended non-identically to C-Rel by

Ir(x, y) =

> if w ≤ r(x, y)r(x, y) otherwise,

for all x ∈ X, y ∈ Y , and C-relations r : X −→7 Y .1.9.7 Proposition. C is a commutative, integral and lean quantale, and I = (I , m = 1, e =1) is a flat lax extension of the identity monad I to C-Rel which makes m : I −→ I I anatural transformation. But I fails to preserve Kleisli convolution, so I is not associative.

Proof. The claims about C are immediate. One also easily sees that

r ≤ Ir = I Ir and If = f , I(f ) = f

for all r : X −→7 Y in C-Rel, and f : X −→ Y in Set. Hence, all claims but the last willfollow from

Is · Ir ≤ I(s · r)(with s : Y −→7 Z). To see this, suppose first that for x ∈ X and z ∈ Z, we haveIs · Ir(x, z) = >. Then necessarily Ir(x, y) = > = Is(y, z) for some y ∈ Y , that is,w ≤ r(x, y) and w ≤ s(y, z). In this case, w ≤ (s · r)(x, z) and I(s · r)(x, z) = > follows.Suppose now that

(Is · Ir)(x, z) = ∨y∈Y Ir(x, y) ∧ Is(y, z) ≤ w < > ,

and consider any y ∈ Y . We may assume Ir(x, y) < w (since Ir(x, y) and Is(y, z) cannotboth have value w), so that Ir(x, y) = r(x, y). But then, regardless of whether Is(y, z) = >(in which case w ≤ s(y, z)) or Is(y, z) < > (in which case Is(y, z) = s(y, z)), we obtain

Ir(x, y) ∧ Is(y, z) = r(x, y) ∧ s(y, z) .

Consequently,

(Is · Ir)(x, z) = ∨y∈Y Ir(x, y) ∧ Is(y, z) = (s · r)(x, z) < > ,


and (Is · Ir)(x, z) ≤ I(s · r)(x, z) follows.For the last claim, consider X = 0, 1, 2, 3 and r : X −→7 X with

r(0, 1) = > , r(1, 3) = u , r(0, 2) = v , r(2, 3) = > ,

while r(x, y) = ⊥ for all other pairs (x, y). Then Ir = r, and therefore,

(Ir · Ir)(0, 3) = (r · r)(0, 3) = ∨y∈X r(0, y) ∧ r(y, 3) = u ∨ v = w ,

which, however, implies (Ir · Ir)(0, 3) < > = I(r · r)(0, 3).

1.10 The Barr extension. Finding a lax extension of a functor T : Set −→ Set to V-Relrequires some effort in general, but the problem is much simpler for V = 2. Recall that2-Rel ∼= Rel. Given a relation r : X × Y −→ 2, we will denote its representation as a subsetof X × Y by R (compare with II.1.2). With π1 : R −→ X and π2 : R −→ Y the respectiveprojections, r is represented as a span, that is, as a diagram of the form

Rπ1

~~

π2

X Y

and we haver = π2 · π1

in Rel.

1.10.1 Definition. The Barr extension of a functor T : Set −→ Set to Rel is defined by

Tr := Tπ2 · (Tπ1) .

Elementwise, for x ∈ TX and y ∈ TY the Barr extension is given by

x Tr y ⇐⇒ ∃w ∈ TR (Tπ1(w) = x & Tπ2(w) = y ) .

1.10.2 Remarks.

(1) The Barr extension T preserves the order on hom-sets. Indeed, if s ≤ r, then we mayassume S ⊆ R with S the domain of the span representing s; hence, with i : S → Rdenoting the inclusion map,

Ts = Tπ2 · Ti · (Ti) · (Tπ1) ≤ Tπ2 · (Tπ1) = Tr .


(2) One easily verifies that

T (r) = (Tr) and Tf = Tf

for all relations r : X −→7 Y and Set-maps f : X −→ Y . Moreover, given Set-mapsf : A −→ X and g : Y −→ B, one has by definition

T (g · r) = Tg · Tr and T (r · f ) = Tr · (Tf) .

(3) In the definition of Tr the pair (π1, π2) can be replaced by any other mono-sourcerepresenting r, or even by any other source (p, q) with r = q · p if T sends surjectionsto surjections. (Recall that, in the presence of the Axiom of Choice, every Set-functorpreserves surjections, since every epimorphism in Set splits if and only if the Axiomof Choice holds, see Exercise II.2.C.)Given any other factorization r = q · p via maps p : P −→ X and q : P −→ Y , theequation r = q · p says precisely that the canonical map P −→ X × Y has image Rand therefore defines a surjection l : P −→ R. Moreover, this map l is a bijection if(p, q) forms a mono-source: l is monic by a trivial cancellation rule for mono-sources(see II.5.3 in the dual situation, and Exercise II.5.A).In general, for a factorization r = q · p, one has

Tq · (Tp) = T (π2 · l) · (T (π1 · l)) = Tπ2 · T l · (T l) · (Tπ1) ≤ Tπ2 · (Tπ1)

with equality holding if T l · (T l) = 1TX , that is, if T l is surjective (Proposition 1.2.2).

1.10.3 Examples.

(1) The Barr extension 1Set of the identity functor 1Set on Set is simply the identity functor1Rel on Rel.

(2) For the filter functor F : Set −→ Set, the Barr extension F is obtained as follows. First,notice that for filters a ∈ FX, b ∈ FY , and a relation r : X −→7 Y ,

a (Fr) b ⇐⇒ ∃c ∈ FR (π1[c] = a & π2[c] = b ) .

If such a filter c exists, then for all A ∈ a, one has: C := π−11 (A) ∈ c, and the set

r(A) := y ∈ Y | ∃x ∈ A (x r y)

must be in b , as it contains π2(C) and π2(C) ∈ π2[c] = b . Similarly, one observesthat r(B) ∈ a for all B ∈ b . Conversely, if r(A) ∈ b and r(B) ∈ a for all A ∈ a andB ∈ b , the sets CA,B = π−1

1 (A) ∩ π−12 (B) (with A running through a, and B through

b ) form a filter base for c ∈ FR such that π1[c] = a and π2[c] = b .


Therefore, the Barr extension of the filter functor is given by

a (Fr) b ⇐⇒ r[a] ⊆ b & r[b ] ⊆ a

for all a ∈ FX and b ∈ FY , and relations r : X −→7 Y , where r[a] is the filter generatedby the filter base r(A) | A ∈ a (this notation coincides with the image-filter notationfor maps of II.1.12).

(3) In the previous example, if both a and b are ultrafilters, then

r[a] ⊆ b ⇐⇒ r[b ] ⊆ a .

Indeed, for an ultrafilter b ∈ βY and A′ ⊆ X, one has

A′ ∈ b ⇐⇒ ∀B ∈ b (A′ ∩B 6= ∅) .

Hence, r[a] ⊆ b means that for all A ∈ a and B ∈ b , one has r(A) ∩B 6= ∅, that is,A ∩ r(B) 6= ∅, and one obtains r[b ] ⊆ a (the other implication also follows). TheBarr extension of the ultrafilter functor β is therefore described by

a (βr) b ⇐⇒ r[a] ⊆ b ⇐⇒ r[b ] ⊆ a ,

for all a ∈ βX, b ∈ βY and relations r : X −→7 Y , or equivalently by

a (βr) b ⇐⇒ ∀A ∈ a, B ∈ b ∃x ∈ A, y ∈ B (x r y) .

(4) The equivalent descriptions of the Barr extension of the ultrafilter monad lead todistinct extensions when we consider the filter monad instead. Similarly to theextensions P , P of the powerset functor (Example 1.4.2(2)), one obtains two laxextensions of the filter functor, neither of which is the Barr extension of F . First, bysetting

a (F r) b ⇐⇒ a ⊇ r[b ]⇐⇒ ∀B ∈ b ∃A ∈ a ∀x ∈ A ∃y ∈ B (x r y)

for all relations r : X −→7 Y , and filters a ∈ FX, b ∈ FY , one obtains a non-flatlax extension whose lax algebras, similarly to the Barr extension of β, provide aconvergence description of the category of topological spaces (see Theorem 2.2.5 andCorollary IV.1.5.4). By contrast, the lax algebras with respect to the lax extensiongiven by

a (F r) b ⇐⇒ b ⊇ r[a]⇐⇒ ∀A ∈ a ∃B ∈ b ∀y ∈ B ∃x ∈ A (x r y)

and their lax homomorphisms form a category isomorphic to the category of closurespaces (Exercise 1.O).


We still have to address the question of whether the Barr extension is actually a lax extensionof the functor T or, even better, of the monad T. To this end the functor needs to satisfyan important additional condition that we now proceed to describe. Reassuringly, all thefunctors presented above satisfy this condition and therefore their Barr extensions are laxextensions (see Examples 1.12.3).

1.11 The Beck–Chevalley condition. Consider Set-maps f : X −→ Z and g : Y −→ Z,and let

Pp2//

p1

Y

g

Xf// Z

be a pullback diagram. The inequality p2 ·p1 ≤ g ·f holds by commutativity of the diagram,while the pullback property forces equality: p2 · p1 = g · f . More generally, a commutativediagram

Wh2 //

h1

Y

g

Xf// Z

(1.11.i)

is a Beck–Chevalley square, or simply a BC-square, if the maps involved satisfy h2 ·h1 = g ·f ,or equivalently, if h1 · h2 = f · g, that is, if

Wh2 // Y

X

_h1

OO

f// Z

_ g

OO

or equivalentlyW

h1

Yh2

oo

g

X Zfoo

commutes in Rel. In fact, we may trade here Rel = 2-Rel with any non-trivial quantale V :

1.11.1 Lemma. The following conditions are equivalent for the commutative square (1.11.i):

(i) h2 · h1 = g · f in Rel;

(ii) h2 · h1 = g · f in V-Rel, for every quantale V;

(iii) h2 · h1 = g · f in V-Rel, for some non-trivial quantale V;

(iv) (1.11.i) is a weak pullback diagram in Set, that is, the canonical map c : W −→ X×Z Yis surjective.

Proof. The unique quantale homomorphism ι : 2 −→ V (with ⊥ 7−→ ⊥, > 7−→ k) inducesa faithful functor ι : Rel −→ V-Rel which sends r : X −→7 Y to ιr : X −→7 Y (with(ιr)(x, y) = ι(r(x, y))); moreover, ι(f) = f and ι(f ) = f for all maps f = f. Hence,


ι facilitates the proof that (i),(ii),(iii) are equivalent. For (i) ⇐⇒ (iv) we just note thatc(w) = (h1(w), h2(w)) for all w ∈ W , and

(x (g · f) y ⇐⇒ f(x) = g(y)) , (x (h2 · h1) y ⇐⇒ ∃w ∈ W (h1(w) = x & h2(w) = y))

for all x ∈ X, y ∈ Y .

1.11.2 Definitions.

(1) A Set-functor T satisfies the Beck–Chevalley condition, or BC for short, if it sendsBC-squares to BC-squares:

h2 · h1 = g · f =⇒ Th2 · (Th1) = (Tg) · Tf

for all maps f, g, h1, h2 with h1 · f = g · h2.

(2) A natural transformation α : S −→ T between Set-functors S and T satisfies theBeck–Chevalley condition, or BC for short, if its naturality diagrams

SXαX //

Sf

TX

Tf

SYαY // TY

are BC-squares for all maps f : X −→ Y , that is, if αX · (Sf) = (Tf) · αY , orequivalently, if Sf · αX = αY · Tf .

Note that by Lemma 1.11.1, it does not matter whether we read the equational conditionsappearing in this definition in Rel or V-Rel (for a non-trivial V). Furthermore, we can easilyprove the following characterization:

1.11.3 Proposition. The following statements are equivalent for a Set-functor T :

(i) T satisfies BC;

(ii) T preserves weak pullback diagrams;

(iii) T transforms pullbacks into weak pullbacks and preserves the surjectivity of maps.

Proof. (i)⇐⇒ (ii) follows from Lemma 1.11.1. The first assertion of (iii) follows triviallyfrom (ii), and the second from (i) and Proposition 1.2.2: f : X −→ Y is surjective preciselywhen f · f = 1Y , that is, if and only if

Xf//

f

Y

1Y

Y1Y // Y


is a BC-square. Finally, for (iii) =⇒ (i), with the BC-square (1.11.i) and the canonicalsurjection c : W −→ X ×Z Y , the canonical map

TWTc // T (X ×Z Y ) t // TX ×TZ TY

is by hypothesis the composite of two surjectives, hence surjective.

1.11.4 Remarks.

(1) Assuming the Axiom of Choice, so that epimorphisms in Set split and are preservedby functors, we may add

(iv) T transforms pullbacks into weak pullbacks.c©

to the list of equivalent statements in Proposition 1.11.3.

(2) This list may be extended further by the following statement that is equivalent to (iv):

(v) for all Set-maps f : X −→ Z, g : Y −→ Z, the monotone map

(−) · t : Rel(Q,W ) −→ Rel(TP,W )

is fully faithful for all sets W ; here, P = X ×Z Y , Q = TX ×TZ TY , andt : TP −→ Q is the comparison map.

In fact, for (iv) =⇒ (v), one considers r, s : Q −→7 W with s · t ≤ r · t and concludess = s · t · t ≤ r · t · t = r since t is surjective. For (v) =⇒ (iv), one observes that t,like any map, satisfies t = t · t · t, in particular 1Y · t ≤ t · t · t. By hypothesis oneconcludes 1Y ≤ t · t, so that t must be surjective.

We can now return to our primary purpose of introducing BC.

1.11.5 Theorem. For a functor T : Set −→ Set, the following assertions are equivalent:

(i) the functor T satisfies BC;

(ii) the Barr extension T is a flat lax extension of T to Rel and a functor T : Rel −→ Rel;

(iii) there is some functor T : Rel −→ Rel which is a lax extension of T to Rel.

Moreover, any functor T : Rel −→ Rel as in (iii) is uniquely determined, that is, T = T .

Proof. (i) =⇒ (ii): To see that T is a flat lax extension, the only issue lies in verifyingTs · Tr = T (s · r) for relations r : X −→7 Y and s : Y −→7 Z with respective factorizations


r = π2 ·π1 and s = ρ2 ·ρ1. As the pullback (p1, p2) of R π2 // Y Sρ1oo yields a mono-source

that moreover forms a factorization p2 · p1 of the relation ρ1 · π2,

Pp1

p2

Rπ1

~~

π2

Sρ1

ρ2

X Y Z

one has T (ρ1 · π2) = Tp2 · (Tp1) (see Remark 1.10.2(3)), and Tp2 · (Tp1) = (Tρ1) · Tπ2since T satisfies BC. Consequently, with Remark 1.10.2(2) one obtains

T (s)·T (r) = Tρ2 ·(Tρ1) ·Tπ2 ·(Tπ1) = Tρ2 ·T (ρ1 ·π2)·(Tπ1) = T (ρ2 ·ρ1 ·π2 ·π1) = T (s·r) .

(ii) =⇒ (iii): This is trivial.(iii) =⇒ (i): Let h2 · h1 = g · f as in 1.11.2(1). Since functoriality makes T also flat oneobtains with Corollary 1.4.4

Th2 · (Th1) = T (h2 · h1) = T (g · f) = (Tg) · Tf .

For the same reasons one has T r = Tπ2 · (Tπ1) = Tr, for r = π2 · π1.

1.12 The Barr extension of a monad. Theorem 1.11.5 proves that if T satisfies theBeck–Chevalley condition, then the Barr extension T is a lax extension of T to Rel. It doesnot require much more effort to show that under the same assumption, the Barr extensionyields a lax extension of the monad T = (T,m, e).Consider first a natural transformation α : S −→ T between functors S, T : Set −→ Setprovided with their lax extensions S, T . Then, for a relation r : X −→7 Y with r = π2 · π1,we have

αY · Sr = αY · Sπ2 · (Sπ1) = Tπ2 · αR · (Sπ1) ≤ Tπ2 · (Tπ1) · αX = Tr · αX ,

that is, α : S −→ T is oplax. From the same computation, we observe that α : S −→ T is anatural transformation if all naturality diagrams

SXαX //

Sf

TX

Tf

SY αY// TY

form BC-squares.


Therefore, if T belongs to a monad T = (T,m, e), then m and e become oplax naturaltransformations in Rel:

m : TT −→ T and e : 1Rel −→ T .

An issue remains with the domain of the multiplication, which should be T T rather thanTT . Hence, in order to obtain a lax extension of the monad T to Rel, we show that theidentities 1TTX are the components of an oplax natural transformation T T −→ TT . Itfollows from Remark 1.10.2(3) and the equality Tr = Tπ2 · (Tπ1) that

TTr = TTπ2 · (TTπ1) ≤ T (Tr)

for any relation r = π2 · π1, with equality holding if T preserves surjections. Thus, the Barrextension T = (T ,m, e) is a lax extension of the Set-monad T = (T,m, e) to Rel providedthat T satisfies the Beck–Chevalley condition.

1.12.1 Theorem.c© For a monad T = (T,m, e) on Set, the following assertions are equiva-lent.

(i) the functor T satisfies BC;

(ii) the Barr extension yields a flat lax extension T = (T ,m, e) of T to Rel.

Proof. The implication (i) =⇒ (ii) follows from the previous discussion since one knowsfrom Theorem 1.11.5 that if T satisfies BC then T is a flat lax extension of T . The converseimplication is also an immediate consequence of the same resultc© , since a monad is a flat laxextension exactly when its underlying functor is one.

1.12.2 Corollary. Suppose that T = (T,m, e) is a monad on Set such that T and m satisfyBC. Then T is an associative lax extension of T to Rel.

Proof. The proof of (i) =⇒ (ii) in the previous Theorem yields that Ts · Tr = T (s · r)if T satisfies the BC condition; the discussion preceding Theorem 1.12.1 shows that if Tpreserves epimorphisms (as do functors that satisfy BC, see 1.11) and naturality diagramsof m are BC-squares, then m : T T −→ T is a natural transformation. The fact thatthe Barr extension is flat allows us to conclude that the Barr extension is associative byProposition 1.9.4.

1.12.3 Examples.

(1) The identity functor 1Set on Set obviously satisfies BC; it is also immediate that theBarr extension 1Rel is a lax extension.


(2) The filter functor F : Set −→ Set satisfies BC. Indeed, suppose that

Wh2 //

h1

Y

g

Xf// Z

is a BC-square. Since the square commutes, one immediately obtains the inequalityFh2 · (Fh1) ≤ (Fg) · Ff . For the other direction, we must show that for all filtersa ∈ FX and b ∈ FY ,

f [a] = g[b ] =⇒ ∃c ∈ FW (h1[c] = a & h2[c] = b ) .

But the sets h−11 (A) ∩ h−1

2 (B) (for A ∈ a, and B ∈ b ) form a base for a filter csatisfying h1[c] = a and h2[c] = b ; indeed, g · f ≤ h2 · h1 means that for any pair(x, y) ∈ A × B with f(x) = g(y), there is an element w ∈ W satisfying h1(w) = xand h2(w) = y. Thus, the Barr extension F described in Example 1.10.3(2) is a laxextension of F to Rel.

(3) c©The ultrafilter functor satisfies BC for similar reasons. In this case, to see that(βg) · βf ≤ βh2 · (βh1), one obtains a filter c from ultrafilters a ∈ FX, b ∈ FY asabove. By Proposition II.1.13.2, there is an ultrafilter x on W with c ⊆ x , so thata ⊆ h1[x ] and b ⊆ h2[x ], and maximality of ultrafilters yields the required equalities.Thus, as in the filter case, the Barr extension β of Example 1.10.3(2) is a flat laxextension of β.We now show that the multiplication m of the ultrafilter monad satisfies BC c©. Forany map f : X −→ Y and all x ∈ βX and Y ∈ ββY with mY (Y ) = βf(x ), we mustfind X ∈ ββX with

ββf(X ) = Y and mX(X ) = x .

By hypothesis, f(A) ∩ B 6= ∅, for all A ∈ x and B ∈ Y . One easily verifiesf(A) = βf(A), and from βf(A) ∩ B 6= ∅ one obtains A ∩ (βf)−1(B) 6= ∅.Therefore,

A | A ∈ x ∪ (βf)−1(B) | B ∈ Y is a filter base, and any ultrafilter X containing it has the desired property. Conse-quently, by Corollary 1.12.2, is associative.As in 1.9 one can now consider the category (, 2)-URel of sets and unitary (, 2)-relations. (The same statement holds for the filter monad with its Barr extension, butwe will not consider this particular instance any further.)

The unit of neither the filter nor the ultrafilter monad satisfies BC (see Exercise 1.Q). Inthe case of the ultrafilter monad, there is a general reason for this claim, as follows.


1.12.4 Proposition. Any monad T = (T,m, e) with T1 ∼= 1 and e satisfying BC must beisomorphic to the identity monad.

Proof. Since e satisfies BC, the diagram

TXT !X //

_eX

T1_ e1

X!X // 1

commutes in Rel for every set X. Expressed elementwise (with 1 = ?) this reads as

∀x ∈ TX (T !X(x ) = e1(?) ⇐⇒ ∃x ∈ X (eX(x) = x ) ) ,

with T !X(x ) = e1(?) holding since T1 ∼= 1. Hence, eX must be surjective, or even bijectiveif T is non-trivial (see Exercise II.3.A). But neither of the two trivial monads on Set has aunit satisfying BC (Exercise 1.Q).

1.13 A double-categorical presentation of lax extensions. The notion of a laxextension T : V-Rel −→ V-Rel of a functor T : Set −→ Set as given in 1.4.1 allows for a verynatural double-categorical interpretation. We briefly describe this presentation here withoutformally introducing double categories, their functors and natural transformations. To thisend, we consider diagrams

Xf//

_r

≤

Y_ s

U g// V

consisting of Set-maps f, g and V-relations r, s such that

g · r ≤ s · f

or, equivalently r ≤ g · s · f , that is: r(x, u) ≤ s(f(x), g(u)) for all x ∈ X, u ∈ U . Wecall these diagrams cells. The point is that such a cell may be considered as a morphismr −→ s horizontally as well as a morphism f −→ g vertically. With map compositionused horizontally and V-relational composition used vertically one obtains two intertwinedcategory structures whose main interaction is captured by the middle-interchange law:

α β

γ δ

for cells α, β, γ, δ that fit together as indicated above, one has

(δ · γ) (β · α) = (δ β) · (γ α)


(with vertical composition denoted by ). Cells and their compositions form a double categoryV-Rel . A double category functor T : V-Rel −→ V-Rel returns for every cell α a cell T α,preserves horizontal composition and identity morphisms strictly, and vertical compositionand identity morphisms laxly. Hence T is in fact given by a functor T : Set −→ Set and alax functor T : V-Rel −→ V-Rel such that

Xf//

_r

≤

Y_ s

U g// V

T7−→

TXTf//

_T r

≤

TY_T s

TUTg// TV

that is, g · r ≤ s · f implies Tg · T r ≤ T s · Tf .

1.13.1 Proposition. Double category functors T = (T, T ) : V-Rel −→ V-Rel are preciselythe lax extensions T : V-Rel −→ V-Rel of functors T : Set −→ Set.

Proof. It suffices to show that the cell preservation condition for T = (T, T ) is equivalentto the lax extension conditions Tf ≤ T f and (Tf) ≤ T (f ), given that T is a functor andT a lax functor. When one exploits the preservation conditions for the cells

X1X //

_1X

≤

Y_ f

Xf// X

andX

f//

_1X

≤

Y_ f

X 1X// X

one obtainsTf = Tf · T1X ≤ Tf · T1X ≤ T f · T1X = T f

and(Tf) = 1TX · (Tf) ≤ T1X · (Tf) ≤ T (f ) · Tf · (Tf) ≤ T (f ) ,

respectively. Conversely, by Corollary 1.4.4 a lax extension satisfies the preservationcondition: expressing g ·r ≤ s·f equivalently by r ≤ g ·s·f , one obtains T r ≤ (Tg) ·T s·Tf ,or Tg · T r ≤ T s · Tf .

One can now proceed to consider an appropriate monad structure on T , by suitable naturaltransformations m : T T −→ T and e : 1 −→ T : we require these transformations to be givenby horizontal natural transformations m : TT −→ T and e : 1 −→ T (in the ordinary sense)that are compatible with the vertical structure, so that there are cells

TTXmX //

_T T r

≤

TX

_T r

TTY mY// TY

andX

eX //

_r

≤

TX

_T r

Y eY// TY


for every V-relation r : X −→7 Y . But the existence requirement for these cells gives preciselythe condition that m : T T −→ T and e : 1 −→ T be op-lax (see 1.5.1). We therefore have:

1.13.2 Corollary. Monads of the double category V-Rel are precisely lax extensions toV-Rel of monads on Set.

Exercises

1.A The trivial and integral quantales. Show that a quantale V is trivial (that is, |V| = 1)if and only if ⊥ = k, where k denotes the neutral element of V . Furthermore, V is integral(that is, k = >) if and only if the terminal object of V-Cat is a generator.

1.B Lean but not integral. For a monoid M , the powerset PM has a quantale structure asin Exercise II.1.M. Then PM is lean, but integral only if M is trivial.

1.C Associativity of V-relational composition. Verify that

t · (s · r) = (t · s) · r

holds for all V-relations r : X −→7 Y , s : Y −→7 Z and t : Z −→7 A.

1.D A V-powerset monad. Given a quantale V, the V-powerset functor PV sends a set Xto its V-powerset VX , and a map f : X −→ Y to PVf : VX −→ VY , where

PVf(φ)(y) := ∨x∈f−1(y) φ(x) ,

for all φ ∈ VY , y ∈ Y . The multiplication µ : PVPV −→ PV and unit δ : 1Set −→ PV of theV-powerset monad PV are defined respectively by

µX(Φ)(y) := ∨φ∈VX Φ(φ)⊗ φ(y) and δX(x)(y) :=

k if x = y

⊥ otherwise,

for all x, y ∈ X, Φ ∈ VVX . The extension operation (−)PV of the corresponding Kleislitriple (see II.3.7) is given for any f : X −→ PVY by

fPV (φ)(y) = ∨x∈X φ(x)⊗ f(x)(y)

for all φ ∈ VX , y ∈ Y . The 2-powerset monad is simply the powerset monad P, and theKleisli category returns V-Rel:

SetPV = V-Rel .

1.E Extensions and liftings in V-Rel. For r : X −→7 Y and s : X −→7 Z, show that theextension s r in the quantaloid V-Rel (see II.4.8) may be described as

(s r)(y, z) =∧

x∈Xr(x, y) s(x, z) .


Likewise, for t : Z −→7 Y , the lifting r t is described by

(r t)(z, x) =∧

y∈Yt(z, y) r(x, y) .

1.F Symmetric and separated metric spaces. The category Metsym is bicoreflective in Met,and Metsep is strongly epireflective in Met (see Example 1.3.1(2)). The notion of symmetryand separation can be introduced for V-categories and the previous statements generalizedto V-Cat, for arbitrary V in lieu of P+.

1.G Limits and colimits in V-Cat.

(1) Products and coproducts of (Xi, ai)i∈I in V-Cat are formed by endowing the sets∏i∈I Xi and ∐i∈I Xi with the structures

p((xi)i∈I , (yi)i∈I) = ∧i∈I ai(xi, yi) and s((x, i), (y, j)) =

ai(x, y) if i = j

⊥ otherwise,

respectively.

(2) Show that the forgetful functor O : V-Cat −→ Set is topological. Conclude that V-Catis small-complete and small-cocomplete, with all limits and colimits in V-Cat preservedby O.

(3) For a surjective map of sets f : X −→ Y and z, w ∈ Y , a tuple (x0, y0, x1, y1, . . . , xn, yn)(with 0 ≤ n) is f -admissible for z, w if

z = f(x0), f(y0) = f(x1), f(y1) = f(x2), . . . , f(yn−1) = f(xn), f(yn) = w .

For f : (X, a) −→ (Y, b) in V-Cat prove that f is O-final if and only if

b(z, w) =∨

(x0, y0, . . . , xn, yn) isf -admissible for z, w

a(x0, y0)⊗ . . .⊗ a(xn, yn)

for all z, w ∈ Y .

(4) Describe equalizers and coequalizers in V-Cat and apply the description to Met.

1.H Yoneda Functor, Yoneda Lemma, initial density of V in V-Cat. Let V be a commutativequantale, considered as a V-category (V ,), and for a V-category X = (X, a), set X =[Xop,V ] (see Proposition 1.3.3).

(1) Show that y : X −→ X with

y(x) = a(−, x) : Xop −→ V

defines a V-functor.


(2) Prove a(y(x), φ) = φ(x) for all x ∈ X, φ ∈ X, where a denotes the V-categorystructure of X.

(3) Conclude that y is O-initial with respect to the forgetful functor O : V-Cat −→ Set(see Exercise 1.G).

(4) With evx denoting the evaluation V-functor at x ∈ X, show that the source

( X y// X

evx // V )x∈X

is O-initial and conclude that V is O-initially dense in V-Cat.

1.I Lax distributive laws and lax monad extensions. A lax distributive law of the V-powerset monad PV (Exercise 1.D) over the monad T on Set is a natural transformationλX : TPVX −→ PVTX such that the diagrams

TPVPV

Tµ

λPV //

≥

PVTPVPVλ // PVPVT

µT

TPV λ // PVT

TTPV

mPV

OO

Tλ //

≥

TPVTλT // PVTT

PVm

OO

TTδ

δT

≥

TPV λ // PVT

PV

ePV

]]

PVe

AA

≥

commute up to “≥” as indicated. There is a bijective correspondence between lax extensionsT of T to V-Rel and lax distributive laws λ of PV over T satisfying

r ≤ r′ =⇒ λY · Tr ≤ λY · Tr′

for all V-relations r, r′ : X −→7 Y presented as maps r, r′ : X −→ PVY : a lax extensionT : V-Rel −→ V-Rel yields a lax distributive law λ = (T1PVX : TPVX −→ PVTX)X∈ob Set,and a lax distributive law λ : TPV −→ PVT determines a lax extension T that sends aV-relation r : X −→ PVY to the V-relation λY · Tr : TX −→ PVTY .

1.J The dual lax extension. Let V be a commutative quantale. If T is a lax extension toV-Rel of a Set-functor T , then for any V-relation r : X −→7 Y , its dual

T r := (T (r))

is also a lax extension to V-Rel of T , and one has (T ) = T . If T = (T ,m, e) is a laxextension of a monad T, then so is T = (T ,m, e). In this case, T is associative if and onlyif T preserves composition of V-relations and m : T T −→ T is a natural transformation.For the lax extensions of the powerset and filter monads coming from Examples 1.4.2(2)and 1.10.3(4), one has

P = P and F = F .


The Barr extension of a functor T to Rel is always self-dual:

T = T .

1.K Checking (T,V)-functoriality. Prove that the following equivalences hold:

f · a ≤ b · Tf ⇐⇒ a ≤ f · b · Tf ⇐⇒ f · a · (Tf) ≤ b

for all Set-maps f : X −→ Y , and V-relations a : TX −→7 X, b : TY −→7 Y .

1.L Associativity of the Kleisli convolution. Consider a lax extension T to V-Rel of theunderlying functor of T = (T,m, e), and (T,V)-relations r : X −7 Y , s : Y −7 Z,t : Z −7 W .

(1) If mY · T r ≤ T T r ·mTX , in particular if m : T −→ T T is a natural transformation,then (t s) r ≤ t (s r).

(2) If T r · mX · T1X ≤ T r · mX and T preserves the composition of V-relations, thent (s r) ≤ (t s) r.

1.M Kleisli convolution for (T,V)-functors. For a monad T with a lax extension T, onedefines for a map f : X −→ Y the unitary V-relation f ] = (eY · f) · T1Y : TY −→7 X. Thecondition

a · (Tf) ≤ f · b ,for unitary V-relations a : TX −→7 X and b : TY −→7 Y , is then equivalent to

a f ] ≤ f ] b .

Briefly put, (T,V)-functoriality of a map f : X −→ Y between (T,V)-categories (X, a) and(Y, b) can be expressed in terms of the Kleisli convolution.

1.N Unitary (T,V)-relations. Let (T ,m, e) be a lax extension of a monad T = (T,m, e)on Set. Show that T r : X −7 TY and r] = eY · T r are unitary (T,V)-relations for everyV-relation r : X −→7 Y . Moreover, if (T,V)-relations r : X −7 Y and s : Y −7 Z arerespectively right and left unitary, then s r is unitary.

1.O Closure spaces via the filter monad. A lax extension of the filter monad F to Rel isobtained via

a F r b ⇐⇒ b ⊇ r[a]for all a ∈ FX, b ∈ FY , and relations r : X −→7 Y (Example 1.10.3(4)). Verify that fora map c : PX −→ PX and relation r : FX −→7 X, the relation rc : FX −→7 X and mapcr : PX −→ PX given by

a rc x ⇐⇒ x ∈ ⋂A∈a c(A) and x ∈ cr(A) ⇐⇒ A r x


(where A denotes the principal filter over A) determine an isomorphism between thecategories of (F, 2)-categories and of closure spaces:

(F, 2, F)-Cat ∼= Cls .

1.P Functors on Set preserving monomorphisms.

(1) The following statements are equivalent for a functor T : Set −→ Set:

(a) T preserves monomorphisms;(b) T preserves the monomorphism ∅ −→ T∅;(c) T preserves the pullback

∅ //

∅

∅ // T∅ ;

(d) T preserves the pullback of (c) weakly, that is, the canonical map T∅ −→ T∅×TT∅T∅ is surjective.

(2) Each of the following conditions implies (a)–(d) of (1).

(a) T∅ = ∅;(b) T preserves the disjointness of some binary coproduct, that is, TX×T (X+Y )TY = ∅

for some X, Y ;(c) T preserves some binary coproduct;(d) T preserves kernel pairs;(e) T preserves kernel pairs weakly;(f) T satisfies BC;(g) T is the functor of a monad T = (T,m, e).

(3) Show that there is a functor T : Set −→ Set, with T∅ = ?+? and TX = X+? forX 6= ∅, which does not preserve monomorphisms. But there are natural transformationsm : TT −→ T and e : 1Set −→ T with

m ·mT = m · Tm and m · eT = 1 .

1.Q Units of monads and BC.

(1) The units of the two trivial monads on Set (see Exercise II.3.A) do not satisfy BC.

(2) The units of the powerset and the filter monad do not satisfy BC. (Hint. Considerthe map !X : X −→ 1 for |X| ≥ 2.)


(3) The list monad (see Example II.3.1.1(2)) is cartesian, that is, its functor L preservespullbacks and every naturality square of the unit and of the multiplication is a pullback.The functor L also preserves surjectivity of maps. In particular, L and its monadmultiplication satisfy BC.


2 Fundamental examples

In this Section, we present some of the motivating examples of lax algebras that willaccompany us throughout this book. Further examples appear in Exercises 2.B and 2.D.

2.1 Ordered sets, metric spaces and probabilistic metric spaces. In Example 1.3.1(1),we saw that 2-categories with 2-functors (that is, (I, 2)-categories with (I, 2)-functors, whereI is identically extended to Rel, see 1.6) were equivalenty described as ordered sets withmonotone maps:

(I, 2)-Cat = 2-Cat = Ord .

Similarly, P+-categories and P+-functors are the metric spaces with non-expansive maps ofExample 1.3.1(2):

(I,P+)-Cat = P+-Cat = Met .

The quantale isomorphism between P+ and the unit interval with its multiplication:

([0,∞]op, + , 0) ∼= ([0, 1], · , 1)

(see Exercise 2.A) allows for a “probabilistic” interpretation of metric spaces. Say that amap φ : [0,∞] −→ [0, 1] is a distance distribution if

φ(v) = ∨w<v φ(w)

for all v ∈ [0,∞]; the convolution of two distance distribution maps φ, ψ : [0,∞] −→ [0, 1]yields a distance distribution map φ⊗ ψ : [0,∞] −→ [0, 1] via

(φ⊗ ψ)(u) = ∨v+w≤u φ(v) · ψ(w)

for all u ∈ [0,∞]; with κ : [0,∞] −→ [0, 1] defined by κ(0) = 0 and κ(u) = 1 if 0 < u, theset D of all distribution maps ordered pointwise forms a quantale

D = (D,⊗, κ) .

A probabilistic metric is then a map a : X ×X −→ D such that

a(x, x) = 1 and a(x, y)⊗ a(y, z) ≤ a(x, z)

for all x, y, z ∈ X. Here, the value p = a(x, y)(u) can be loosely interpreted as the“probability” that a given randomized metric a : X×X −→ [0,∞] satisfies a(x, y) < u. A setX with a probabilistic metric a : X ×X −→ D forms a probabilistic metric space (X, a), anda map f : X −→ Y between probabilistic metric spaces (X, a) and (X, b) is probabilisticallynon-expansive if

a(x, y) ≤ b(f(x), f(y))

2. FUNDAMENTAL EXAMPLES 187

for all x, y ∈ X. Probabilistic metric spaces with probabilistically non-expansive maps arethe objects and morphisms of the category ProbMet, and one observes

(I,D)-Cat = D-Cat = ProbMet .

In Example 3.5.2(2) below we exhibit full embeddings

Ord //Met // ProbMet

which make explicit how the structures discussed here generalize each other.

2.2 Topological spaces. Our paradigmatic example of a (T,V)-category comes from[Barr, 1970], in which topological spaces are presented as so-called relational algebras forthe ultrafilter monad; in our context this result reads as

(, 2)-Cat ∼= Top . c©

Recall from Examples II.3.1.1 that stands for the ultrafilter monad, and the requiredlax extension β of β : Set −→ Set to Rel ∼= 2-Rel is described in Example 1.10.3(3). Inthe following, we will freely use the notations introduced in II.1.12 and II.1.13. Thereare many ways to prove the isomorphism mentioned above, depending in particular onthe choice of the standard presentation of topological spaces: open or closed sets, interioror closure operations, or neighborhood systems. Here we work with closure operations,while in Chapter IV, with a more developed theory at our disposal, we will present anotherapproach. The idea of the isomorphism between (, 2)-categories and topological spacesis that a relation r : βX −→7 X represents convergence and specifies which ultrafiltersconverge to which points of X. We can then associate with r a finitely additive closureoperation c : PX −→ PX, and conversely, show that every finitely additive closure operationc determines a convergence relation r : βX −→7 X.In II.1.6, closure spaces were introduced as pairs (X, c) consisting of a set X and a closureoperation c : PX −→ PX on the powerset PX of X. We observed that such a closureoperation is a monotone map (or equivalently, an element of Ord(PX,PX)) carrying amonoid structure with respect to the compositional structure, that is,

c · c(A) ⊆ c(A) , A ⊆ c(A)

for all A ⊆ X. Furthermore, c : PX −→ PX defines a topology on X if and only if c isfinitely additive:

c(A ∪B) = c(A) ∪ c(B) , c(∅) = ∅

for all A,B ⊆ X (see Exercises II.1.F and II.1.G). To any relation r : βX −→7 X, we canassociate a finitely additive map clos(r) : PX −→ PX given by

clos(r)(A) = y ∈ X | ∃x ∈ βX (A ∈ x & x r y) .


Conversely, given a map c : PX −→ PX, we define a relation conv(c) : βX −→7 X by setting

x conv(c) y ⇐⇒ ∀A ∈ PX (A ∈ x =⇒ y ∈ c(A)) .

Note that we have the identities

conv(c)(A) = c(A) and clos(r)(A) = r(A) , (2.2.i)

for all A ⊆ X, where A = x ∈ βX | A ∈ x , and

r(A) = x ∈ X | ∃x ∈ A (x r x) .

2.2.1 Lemma.c© If Set(PX,PX) is equipped with the pointwise order, then the maps

clos : Rel(βX,X) −→ Set(PX,PX) and conv : Set(PX,PX) −→ Rel(βX,X)

form an adjunction clos a conv. The fixpoints of clos · conv are the maps c : PX −→ PXthat are finitely additive.

Proof. Monotonicity of clos and conv follows immediately from the definitions. One alsohas

1 ≤ conv · clos and clos · conv ≤ 1 .Indeed, for x ∈ βX and y ∈ X,

x r y =⇒ ∀A ∈ PX (A ∈ x =⇒ x r y) =⇒ x conv(clos(r)) y

and x conv(c) y implies that y ∈ c(A) for all A ∈ x ; thus, for any A ∈ PX, the setclos(conv(c))(A) is given by

y ∈ X | ∃x ∈ βX (A ∈ x & ∀B ∈ PX (B ∈ x =⇒ y ∈ c(B))) ⊆ c(A) . (2.2.ii)

In particular, for c : PX −→ PX one necessarily has clos · conv(c)(∅) = ∅ (since noultrafilter x ∈ βX contains ∅), and

clos · conv(c)(A ∪B) = clos · conv(c)(A) ∪ clos · conv(c)(B) ,

as A ∪ B ∈ x implies that either A ∈ x or B ∈ x (Lemma II.1.13.1), and the converseholds because x is an up-set. Thus, the fixpoints of clos · conv must be finitely additive.Consider now a map c : PX −→ PX that is finitely additive (and therefore monotone) andA ∈ PX with y ∈ c(A). The set

j := J ∈ PX | y /∈ c(J)

is an ideal on X. By Corollary II.1.13.5c© (with the principal filter a = A), one obtains theexistence of an ultrafilter x ∈ βX with x ⊇ A and x ∩ j = ∅; in other words, A ∈ xand y ∈ c(B) for all B ∈ x , so one can conclude that c(A) = clos · conv(c)(A) by (2.2.ii)above.


A crucial observation is that both maps clos and conv are homomorphisms of the monoids(, 2)-URel(X,X) and SLat(PX,PX) whose operations are given by Kleisli convolution andmap composition, respectively:

2.2.2 Proposition. c©The maps conv and clos satisfy

clos(s r) = clos(s) · clos(r) , conv(d · c) = conv(d) conv(c) ,clos(eX) = 1X , conv(1PX) = eX ,

for all (, 2)-relations r, s : X −7 X and finitely additive maps c, d : PX −→ PX.

Proof. For x ∈ βX and x ∈ X, if x conv(1PX) x, then x ∈ A for all A ∈ x ; thus,conv(1PX) = eX . Since 1PX is finitely additive, it is a fixpoint of clos · conv, so the previousequality yields 1PX = clos · conv(1PX) = clos(eX).Consider now finitely additive maps c, d : PX −→ PX, and let x ∈ βX, z ∈ X. Set also

a := ↑PXc(A) | A ∈ x ,

which is a filter since c is monotone. Then (2.2.i), together with Corollary II.1.13.3 c©, tells us

a ⊆ y ⇐⇒ ∃X ∈ ββX (mX(X ) = x & X β(conv(c)) y )

for all y ∈ βX. Assume first that x conv(d · c) z. Then a is disjoint from the ideal

j = B ⊆ X | z /∈ d(B) .

Applying Corollary II.1.13.5 c©, we see that there is an ultrafilter y ∈ βX containing aand disjoint from j . Hence y conv(d) z, and there exists X ∈ ββX with X β(conv(c))y and mX(X ) = x . We conclude that x (conv(d) conv(c)) z. Assume now thatx (conv(d) conv(c)) z, so there is a y ∈ βX such that a ⊆ y and y conv(d) z. From thiswe obtain z ∈ d · c(A) for every A ∈ x , that is, x conv(d · c) z.Finally, let r, s : X −7 X be (, 2)-relations, and consider A ⊆ X, z ∈ X. If z ∈clos(s r)(A), then we have X ∈ ββX and y ∈ βX with

A ∈ X , X βr y , y s z ;

and it follows that z ∈ clos(s)(r(A)) = clos(s)(clos(r)(A)). Thus, clos(s r) ≤ clos(s) ·clos(r). For the other inequality, we use that conv(d · c) = conv(d) conv(c) for fixpointsc, d of clos · conv, and in particular for c = clos(r), d = clos(s), and c · d:

clos(s)·clos(r) = clos · conv(clos(s)·clos(r)) = clos(conv(clos(s))conv(clos(r))) ≤ clos(sr)

because conv · clos ≤ 1 (by Lemma 2.2.1).

2.2.3 Lemma. c©Every (, 2)-relation r : X −7 X satisfies conv · clos(r) = eX r.


Proof. On one hand,conv(clos(r)) = conv(1PX · clos(r)) = eX conv(clos(r)) ≥ eX r.

On the other hand, consider x ∈ βX, y ∈ X, and suppose that x conv(clos(r)) y.Then for all A ∈ x , we have y ∈ clos(r)(A) = r(A), which implies that there is anX ∈ ββX with mX(X ) = x and X (βr) eX(y). Hence, x (eX r) y, which shows thatconv(clos(r)) ≤ eX r.

As a consequence, we obtain r = conv · clos(r) for every left unitary (, 2)-relation r : X −7X, and in particular for every reflexive and transitive (, 2)-relation.2.2.4 Proposition.c© The fixpoints of the adjunction clos a conv are on one hand the finitelyadditive maps c : PX −→ PX, and on the other hand, the left unitary relations r : βX −→7 X.

Proof. This follows from Lemmata 2.2.1 and 2.2.3.

2.2.5 Theorem.c© There is an isomorphism(, 2)-Cat ∼= Top

that commutes with the underlying-set functors.

Proof. By Proposition 2.2.4, clos and conv define a one-to-one correspondence betweenfinitely additive maps c : PX −→ PX and left unitary (, 2)-relations a : X −7 X. If c is atopological closure operation, so that 1PX ≤ c and c · c = c, then a := conv(c) satisfies

eX = conv(1PX) ≤ conv(c) = a anda a = conv(c) conv(c) = conv(c · c) = conv(c) = a

by Proposition 2.2.2, that is, (X, a : βX −→7 X) is a (, 2)-category. Likewise, from eX ≤ aand a a = a one gets 1PX ≤ c and c · c = c for c := clos(a), and therefore clos and convactually define a bijective correspondence between (, 2)-categorical structures and topo-logical closure operations on X. An easy verification shows that a (, 2)-functor preservesthe corresponding closure operation—and is therefore continuous—while a continuous mappreserves the corresponding (, 2)-categorical structure—and is therefore a (, 2)-functor.

Spelled out, the previous Theorem states that a topological space (X,OX) can be equiv-alently described as a pair (X, a), with a : βX −→7 X a relation representing convergencewhich, when we denote both a and βa by −→, satisfies

X −→ y & y −→ z =⇒ ∑X −→ z and x −→ x ,

for all z ∈ X, y ∈ βX, and X ∈ ββX; here X −→ y ⇐⇒ X ⊇ a[y ], and ∑ is theKowalsky sum restricted to ultrafilters (see II.1.12 and II.3.1.1(5)). In this context, thecontinuous maps f : (X, a) −→ (X, b) are exactly the convergence-preserving maps, that is,the maps f : X −→ Y such that

x −→ y =⇒ f [x ] −→ f(y)for all y ∈ X, and x ∈ βX.


2.3 Compact Hausdorff spaces. Because the Barr extension of to Rel is flat, it ispossible to exploit Theorem 2.2.5 to obtain an elegant description of the category of-algebras (that is, of Eilenberg–Moore algebras associated to , see II.3.2).Let us recall that a topological space X is compact if every open cover of X has a finitesubcover, that is, for every open cover A there exists a finite subset F ⊆ A with ⋃F = X.In the following results we freely exploit that a topological space X can equivalently bedescribed via a set OX of open sets, a closure operation c : PX −→ PX, or a convergencerelation a : βX −→7 X (Exercises II.1.F, II.1.G and Theorem 2.2.5). In fact, we can avoidthe closure operation by translating the definition of the convergence relation given in 2.2as

x −→ x ⇐⇒ ∀A ∈ OX (x ∈ A =⇒ A ∈ x ) (2.3.i)

(this easily follows from Lemma II.1.13.1 and OX = (c(B)) | B ∈ PX).

2.3.1 Proposition. c©The following statements are equivalent for a topological space X:

(i) X is compact;

(ii) every ultrafilter on X converges, that is, 1βX ≤ a · a.

Proof. Assume first that X is compact. Let x ∈ βX and set

A = A ⊆ X | A open, A /∈ x .

The set A cannot cover X since otherwise there would exist A1, . . . , An ∈ A with A1 ∪. . . ∪ An = X ∈ x , and therefore Ak ∈ x for some k ∈ 1, . . . , n by Lemma II.1.13.1,contradicting the definition of A. Thus, there exists x ∈ X such that every open set A withx ∈ A belongs to x . This implies that x −→ x by (2.3.i).Assume now that any ultrafilter x ∈ βX converges. Let A be a set of open subsets of Xwith the property that no finite subset of A covers X. Then

f = ↑PX(⋃F) | F ⊆ A,F finite

is a proper filter on X, and therefore contained in an ultrafilter x (Proposition 1.13.2) thatconverges to some x ∈ X by hypothesis. Hence x /∈ A for any A ∈ A (by (2.3.i) and thedefinition of x ⊇ f ), that is, A does not cover X.

2.3.2 Proposition. c©The following statements are equivalent for a topological space X:

(i) X is Hausdorff;

(ii) every ultrafilter on X has at most one convergence point, that is, a · a ≤ 1X .


Proof. Assume that X is Hausdorff and let x ∈ βX and x, y ∈ X with x −→ x andx −→ y. By (2.3.i), for all open subsets A ⊆ X and B ⊆ X with x ∈ A and y ∈ B one hasA ∩B 6= ∅, which is possible only if x = y.Assume now that every ultrafilter on X converges to at most one point, and let x, y ∈ X. Ifevery open neighborhood of x intersects every open neighborhood of y, then there exists anultrafilter x containing all neighborhoods of x and all neighborhoods of y, hence convergingto both x and y, which, by hypothesis, implies x = y.

2.3.3 Theorem.c© There is an isomorphism

Set ∼= CompHaus

that commutes with the respective forgetful functors to Set.

Proof. Combining both propositions above, we see that a topological space X is compactHausdorff if and only if its convergence relation a : βX −→7 X is actually a map a : βX −→ X,which therefore satisfies a · βa = a ·mX . (Since βa = βa because a is a map, the conditiona · βa ≤ a ·mX is an inclusion of graphs of functions with same domain, hence necessarilyan equality.) Furthermore, a continuous map f : X −→ Y between compact Hausdorffspaces with convergence structures a : βX −→ X and b : βY −→ Y respectively satisfiesf · a ≤ b · Tf , but since all relations involved are functions, this condition is actually anequality.

2.4 Approach spaces. Approach spaces provide a common framework for both topologi-cal and metric structures. More precisely, an approach space is a pair (X, δ) consisting of aset X and a function δ : X × PX −→ [0,∞] subject to the conditions:

(1) δ(x, x) = 0,

(2) δ(x,∅) =∞,

(3) δ(x,A ∪B) = minδ(x,A), δ(x,B),

(4) δ(x,A) ≤ δ(x,A(u)) + u,

for all x ∈ X, A,B ⊆ X, u ∈ [0,∞], and where

A(u) = A(u)δ = x ∈ X | δ(x,A) ≤ u .

A function δ : X×PX −→ [0,∞] satisfying the axioms (1)–(4) is called an approach distanceon X. A morphism f : (X, δ) −→ (Y, δ′) of approach spaces is given by a non-expansivemap f : X −→ Y , so that f satisfies

δ′(f(x), f(A)) ≤ δ(x,A)

for all x ∈ X and A ⊆ X. Approach spaces and non-expansive maps are the objects andmorphisms of the category App.


2.4.1 Examples.

(1) Every metric space (X, a) (in the sense of Example 1.3.1(2)) becomes an approachspace (X, δ) when one puts

δ(y, A) := infa(x, y) | x ∈ A .

In fact, this defines a full embedding

Met → App

that exhibits Met as a coreflective subcategory of App (this follows from Proposi-tion 3.4.2 together with Theorem 2.4.5 below).

(2) Every topological space (X, c) (as in Exercises II.1.F and II.1.G) becomes an approachspace (X, δ) when one sets

δ(y, A) :=

0 if y ∈ c(A),∞ otherwise.

In this way, one obtains a full embedding

Top → App

that exhibits Top as a coreflective subcategory of App (see 3.6 together with Theo-rems 2.2.5 and 2.4.5 below).

We now show that approach spaces can be seen as “numerified topological spaces”, that is,we prove the existence of an isomorphism

App ∼= (,P+)-Cat c©

for a suitable extension β : P+-Rel −→ P+-Rel of β : Set −→ Set (defined below). In order tokeep the proof similar to the one for topological spaces as presented in 2.2, we think ofmaps

δ : X × PX −→ [0,∞]as of morphisms “from X to X”. Given also γ : X ×PX −→ [0,∞], we define the compositeγ · δ : X × PX −→ [0,∞] by

γ · δ(z, A) := infγ(z, A(u)δ ) + u | u ∈ [0,∞] ,

for all A ⊆ X and z ∈ X. One easily verifies that this composition preserves the order inboth variables, and that the map εX : X × PX −→ [0,∞], sending (x,A) to 0 if x ∈ A andto ∞ otherwise, is neutral with respect to this composition. We say that δ : X ×PX −→ P+is finitely additive if

δ(y,∅) =∞ and δ(y, A ∪B) = minδ(y, A), δ(y,B) ,


for all y ∈ X and A,B ∈ PX. With this notation, δ : X × PX −→ [0,∞] is an approachdistance if and only if δ is finitely additive and

δ · δ ≥ δ , εX ≥ δ .

The former inequality is actually an equality thanks to the latter.We saw in 2.2 that β : Set −→ Set can be extended to a 2-functor β : Rel −→ Rel by settingx (βr) y ⇐⇒ r(x ) ⊆ y for every relation r : X −→7 Y . We now go one step further andextend the ultrafilter monad to a lax monad (β,m, e) on P+-Rel as follows. Let us recallfrom II.1.10 that ≤ always refers to the natural order of [0,∞]. For r : X × Y −→ P+ andv ∈ P+, we consider the relation rv : X −→7 Y defined by

x rv y ⇐⇒ r(x, y) ≤ v .

Given another P+-relation s : Y −→7 Z and u ∈ P+, one easily verifies su · rv ≤ (s · r)u+v. Wenow consider the Barr extension of β to P+-Rel:

βr := βP+r : βX × βY −→ P+

(x , y ) 7−→ infv ∈ P+ | x β(rv) y ,

which can be equivalently expressed by

βr(x , y ) = supA∈x ,B∈y infx∈A,y∈B r(x, y) ,

for all P+-relations r : X −→7 Y , x ∈ βX, and y ∈ βY (see Exercise 2.E). Furthermore,β(ru) ≤ (βr)u for any u ∈ P+; and (βr)u ≤ β(rv) whenever u < v in P+.

2.4.2 Lemma.c© Let r : X −→7 Y be a P+-relation, f a filter on X and y ∈ βY . Then thereexists an ultrafilter x ∈ βX such that f ⊆ x and

βr(x , y ) = supA∈f ,B∈y infx∈A,y∈B r(x, y)

Proof. Certainly, for every ultrafilter x ∈ βX with f ⊆ x we have

βr(x , y ) ≥ supA∈f ,B∈y infx∈A,y∈B r(x, y) .

Let us set v := supA∈f ,B∈y infx∈A,y∈B r(x, y). If v = ∞, then any x ⊇ f has the desiredproperty. If v <∞, we consider

j = A ⊆ X | supB∈y infx∈A,y∈B r(x, y) > v .

Of course, f ∩ j = ∅, and it is not hard to see that j is an ideal. By Corollary II.1.13.5,there exists an ultrafilter x ∈ βX with f ⊆ x and x ∩ j = ∅, and therefore βr(x , y ) ≤ v.

2.4.3 Proposition.c© The Barr extension = (β,m, e) is a flat associative lax extension toP+-Rel of the ultrafilter monad = (β,m, e). Moreover, β(r) = (βr) for every P+-relationr.


Proof. We show only

βs · βr = β(s · r) and mY · β βr = βr ·mX ,

for all P+-relations r : X −→7 Y and s : Y −→7 Z, as the other verifications are straightforward.In order to show the inequality βs · βr ≥ β(s · r), assume βs · βr(x , z) < u for x ∈ βX andz ∈ βZ. Therefore, there is some y ∈ βY satisfying

βr(x , y ) + βs(y , z) < u .

Consider u1, u2 ∈ P+ such that

βr(x , y ) < u1 , βs(y , z) < u2 , u1 + u2 = u .

Hence, we have x β(ru1) y and y β(su2) z, so that x β(su2 · ru1) z. Since su2 · ru1 ≤(s · r)u1+u2 , we conclude β(s · r)(x , z) ≤ u1 + u2 = u.To see βs · βr ≤ β(s · r), let x ∈ βX, z ∈ βZ and u ∈ P+ with

u > β(s · r)(x , z) = supA∈x ,C∈z infx∈A,z∈C s · r(x, z) .

Hence, for every A ∈ x and every C ∈ z, there exist x ∈ A, y ∈ Y and z ∈ C withr(x, y) + s(y, z) ≤ u, that is

BA,C := y ∈ Y | ∃x ∈ A, z ∈ Z : r(x, y) + s(y, z) ≤ u 6= ∅ .

Since BA∩A′,C∩C′ ⊆ BA,C ∩BA′,C′ , the set

BA,C | A ∈ x , C ∈ z

is a filter base; let y ∈ βY be any ultrafilter containing it c©. Then

βr(x , y ) = supA∈x ,B∈y infx∈A,y∈B ≤ u

because for any A ∈ x and B ∈ y , one has B∩BA,Z 6= ∅. Similarly, βs(y , z) ≤ u. Considerε > 0 and set u0 = βr(x , y ); if u0 = 0, we are done, so we can assume u0 > 0. Thus, thereis some A ∈ x with ru0−ε[A] /∈ y , and therefore its complement

B0 := y ∈ Y | ∀x ∈ A (r(x, y) > u0 − ε)

belongs to y . We show that βs(y , z) ≤ (u − u0) + ε. To this end, let B ∈ y and C ∈ z.Then B ∩B0 ∩BA,C 6= ∅, which implies that there are x ∈ A, y ∈ B and z ∈ C with

r(x, y) + s(y, z) ≤ u and r(x, y) > u0 − ε ,

therefore s(y, z) ≤ (u− u0) + ε. Consequently,

βs(y , z) ≤ supB∈y ,C∈z infy∈B,z∈C s(y, z) ≤ (u− u0) + ε .


To verify mY · β βr ≥ βr ·mX , let u, u′ ∈ P+ and X ∈ ββX, Y ∈ ββY be such that

β βr(X ,Y ) < u < u′ .

Hence, X β β(ru) Y and therefore X β β(ru′) Y . This implies mX(X ) β(ru′) mY (Y ), thatis, βr(mX(X ),mY (Y )) ≤ u′.Finally, we show mY · β βr ≤ βr ·mX . Let X ∈ ββX and y ∈ βY , and assume

βr(mX(X ), y ) < u′′ < u′ < u .

Since the Barr extension of β to Rel is associative, there is some Y ∈ ββY with X (β βru′′) Y ,so that one has X β(βr)u′ Y and X (β βr)u Y , that is, β βr(X ,Y ) ≤ u.Finally, since β commutes with the involution on P+-Rel, m : β −→ β β is a natural trans-formation too (see also Exercise 1.J). Proposition 1.9.4 then yields that is associative.

Every P+-relation r : βX ×X −→ P+ defines a finitely additive function

clos(r) : X × PX −→ [0,∞](y, A) 7−→ infr(x , y) | x ∈ βA ,

and every function δ : X × PX −→ [0,∞] yields a P+-relation

conv(δ) : βX ×X −→ P+

(x , y) 7−→ supδ(y, A) | A ∈ x .

As in 2.2, the following proposition contains the key facts about clos and conv.

2.4.4 Proposition.c© The operations conv and clos satisfy

clos(s r) = clos(s) · clos(r) , conv(γ · δ) = conv(γ) conv(δ) ,clos(eX) = εX , conv(εX) = eX ,

for all finitely additive functions δ, γ : X × PX −→ [0,∞], and (,P+)-relations r, s : X −7X.

Proof. We show only clos(sr) = clos(s) · clos(r), as the verification of the other statementsare either very similar or straightforward.Assume first that clos(s) · clos(r)(z, A) < v, with z ∈ Z, A ⊆ X and v ∈ [0,∞]. Then, forsome u ∈ [0,∞],

clos(s)(z, A(u)clos(r)) + u < v .

Consider u′ ∈ P+ such that

clos(s)(z, A(u)clos(r)) < u′ and u′ + u = v .


Hence, there exists y ∈ βX with A(u)clos(r) ∈ y and r(y , z) < u′. For every B ∈ y , there is an

element y ∈ B satisfying

inf y∈A r(y , y) = clos(r)(y, A) ≤ u .

Therefore, we obtainsupB∈y inf y∈A infy∈B r(y , y) ≤ u ,

and Lemma 2.4.2 guarantees the existence of some X ∈ ββX with A = x ∈ βX | A ∈x ∈ X and βr(X , y ) ≤ u. Since

A ∈ mX(X ) and s r(mX(X ), z) ≤ u′ + u = v ,

we deduce clos(s r)(z, A) ≤ v.Assume now clos(s r)(z, A) < v. Hence, for some x ∈ βA, we have s r(x , z) < v, andthere exist X ∈ ββX, y ∈ βX with

mX(X ) = x and βr(X , y ) + s(y , z) < v .

Let u ∈ [0,∞] with βr(X , y ) < u and u+ s(y , z) = v. We have

A(u)clos(r) ⊇ ru(A) ∈ y ,

and therefore clos(s)(z, A(u)clos(r)) + u ≤ v.

From the identities

clos(conv(δ))(y, A) = infx ∈βA conv(δ)(x , y) = infx ∈βA supB∈x δ(y,B) ,

we obtain δ(x,A) ≤ clos(conv(δ))(x,A), for all δ : X × PX −→ [0,∞], A ⊆ X and y ∈ X.Moreover, equality holds if δ(y, A) =∞. If δ(y, A) <∞, we can consider

j = B ⊆ X | δ(y,B) > δ(y, A) ;

when δ is finitely additive, j is an ideal with A /∈ j , and there exists an ultrafilter x ∈ βXwith A ∈ x and x ∩ j = ∅. Therefore, δ = clos(conv(δ)) if and only if δ is finitely additive.On the other hand,

conv(clos(r))(x , y) = supA∈x clos(r)(y, A) = supA∈x inf y∈βA r(y , y) ,

so that conv(clos(r)) = eX r (see Exercise 2.F), for every P+-relation r : βX −→7 X. Hence,r = conv(clos(r)) if and only if r is unitary.

2.4.5 Theorem. c©There is an isomorphism (,P+)-Cat ∼= App that commutes with theunderlying-set functors.


Proof. To every approach space (X, δ) we associate the (,P+)-category (X, conv(δ)), andto every (,P+)-category (X, r) corresponds the approach space (X, clos(r)). Then non-expansive maps preserve the corresponding (,P+)-categorical structure, and (,P+)-functorsbecome non-expansive. All said, we obtain functors

App −→ (,P+)-Cat and (,P+)-Cat −→ App

that are inverse to one another.

Thus, as in the case of topological spaces, we obtain an alternate description of approachspaces. Here, the relational arrow for a convergence relation −→ is replaced by a numerified“degree of convergence”, so that an approach space can be seen as a pair (X, a), witha : βX ×X −→ [0,∞] a map satisfying

βa(X , y ) + a(y , z) ≥ a(∑X , z) and a(x, x) = 0 ,

for all z ∈ X, y ∈ βX, and X ∈ ββX, and where

βa(X , y ) = supA∈X ,B∈y infx ∈A,y∈B a(x , y) .

With this description, the non-expansive maps f : (X, a) −→ (Y, b) between approach spacesare those satisfying

a(x , y) ≥ b(f [x ], f(y))for all y ∈ X, and x ∈ βX, that is, the maps that improve the “degree of convergence”.The embedding Top → App of 2.4.1(2) now describes topological spaces as those approachspaces whose measure of convergence a : βX × X −→ [0,∞] takes its values in 0,∞.Hence, ultrafilter convergence in X is defined by

x −→ y ⇐⇒ a(x , y) = 0

for all x ∈ βX, y ∈ X.

2.5 Closure spaces. Example 1.6.4(2) shows that when the powerset monad P is equippedwith the lax extension P : Rel −→ Rel

A (P r) B ⇐⇒ B ⊆ r(A) , (2.5.i)

(for all A ∈ PX, B ∈ PY , and relations r : X −→7 Y ), (P, 2)-category structures a : PX −→7X are in one-to-one correspondence with a closure operation c : PX −→ PX via

x ∈ c(A) ⇐⇒ A a x

for all A ∈ PX, x ∈ X. Under this correspondence, a (P, 2)-functor f : (X, a) −→ (X, b)is equivalently described as a continuous map f : (X, cX) −→ (Y, cY ), and one obtains anisomorphism

(P, 2)-Cat ∼= Cls


that commutes with the underlying-set functors. The extension formula (2.5.i) can also beused with the finite-powerset monad Pfin = (Pfin,

⋃, −) whose functor Pfin : Set −→ Set

sends a set X to the set PfinX of its finite subsets; the multiplication and unit of themonad are just the appropriate restrictions of those of P. By the same procedure as inthe powerset case, one can identify a (Pfin, 2)-category with a finitary closure space (alsocalled an algebraic closure space), that is, a closure space (X, c) whose closure operationc : PX −→ PX is finitary:

c(A) = ⋃B∈PfinA c(B) .

Denoting by Clsfin the full subcategory of Cls whose objects are finitary closure spaces, oneobtains an isomorphism

(Pfin, 2)-Cat ∼= Clsfin .


Exercises

2.A The probabilistic and metric quantales. The unit interval [0, 1] with its natural orderand multiplication yields a quantale

([0, 1], · , 1)

that is isomorphic to the quantale P+ = ([0,∞]op, + , 0).

2.B Ultrametric spaces. An ultrametric is a map a : X ×X −→ [0,∞] satisfying

maxa(x, y), a(y, z) ≥ a(x, z) and 0 = a(x, x)

for all x, y, z ∈ X, and an ultrametric space is a pair (X, a) composed of a set X and anultrametric a : X ×X −→ [0,∞]; a non-expansive map f : (X, a) −→ (X, b) is, as in 1.3.1(2),a map f : X −→ Y such that

a(x, y) ≥ b(f(x), f(y)) .With the quantale Pmax = ([0,∞]op,max, 0) (see II.1.10.1(3)), the category UltraMet ofultrametric spaces with non-expansive maps is the category of Pmax-categories and Pmax-functors:

(I,Pmax)-Cat = Pmax-Cat = UltraMet .

2.C The underlying order via ultrafilter convergence. c©Use Exercise II.1.F and the correspon-dence between convergence of ultrafilters and closure operations of Theorem 2.4.5 to showthat the underlying order of a (, 2)-category (X, a) is given by

x ≤ y ⇐⇒ x −→ y

for all x, y ∈ X.


2.D Bitopological spaces.c© When V = 22 is the diamond lattice of Exercise II.1.H, the Barrextension to 22-Rel of the ultrafilter functor β is defined by

βr(x , y ) =∨w ∈ 22 | x β(rw) y =

∧A∈x ,B∈y

∨x∈A,y∈B

r(x, y) ,

for all 22-relations r : X −→7 Y , x ∈ βX, and y ∈ βY . Verify that this yields a flat laxextension of the ultrafilter monad from Set to 22-Rel.A bitopological space is a triple (X,O1,O2), with (X,O1) and (X,O2) topological spaces,and a bicontinuous map f : (X,O1,O2) −→ (Y,O′1,O′2) between bitopological spaces is amap f : X −→ Y that is continuous as f : (X,O1) −→ (Y,O′1) and f : (X,O2) −→ (Y,O′2).Show that the category of (, 22)-categories obtained from the previous Barr extension isisomorphic to the category BiTop of bitopological spaces with bicontinuous maps:

(, 22)-Cat ∼= BiTop .c©

2.E An alternative description of β : P+-Rel −→ P+-Rel. The Barr extension to P+-Rel ofthe ultrafilter functor (see Proposition 2.4.3) can be equivalently expressed as

βr(x , y ) = supA∈x ,B∈y infx∈A,y∈B r(x, y) ,

for every P+-relation r : X −→7 Y , x ∈ βX and y ∈ βY .

2.F Unitary (,P+)-relations.c© Show

eX r(x , y) = supA∈x inf y∈A r(y , y) ,

for every (,P+)-relation r : X −7 X (recall from 2.2 that A = y ∈ βX | A ∈ y).

2.G Metric closure spaces. A metric closure space is a pair (X, c) composed of a set X anda metric closure operation c, that is, a map c : X × PX −→ [0,∞] such that

(1) c(x,A) = 0,

(2) A ⊆ B =⇒ c(x,B) ≤ c(x,A),

(3) c(x,A) ≤ c(x,A(u)) + u,

for all x ∈ X, A,B ∈ PX, u ∈ [0,∞], and where A(u) = x ∈ X | c(x,A) ≤ u. Amorphism f : (X, c) −→ (Y, c′) of metric closure spaces is a non-expansive map, that is, amap f : X −→ Y that satisfies

c′(f(x), f(A)) ≤ c(x,A)

for all x ∈ X and A ⊆ X. Metric closure spaces with non-expansive maps form the categoryMetCls. By equipping the powerset monad P with the lax extension given by

P r : PX × PY −→ P+ , (A,B) 7−→ supx∈A infy∈B r(x, y) ,


(for all x ∈ X, y ∈ Y , and V-relations r : X −→7 Y ), one obtains an isomorphism

(P,P+)-Cat ∼= MetCls .

2.H An alternative description of approach spaces. For a map δ : X × PX −→ [0,∞]satisfying conditions (1)–(3) of 2.4, condition (4) is equivalent to

supx∈X |δ(x,A)− δ(x,B)| ≤ minsupy∈A δ(y,B), supz∈B δ(z, A)

for all A,B ⊆ X.


3 Categories of lax algebras

In this subsection we prove topologicity of the underlying-set functor (T,V)-Cat −→ Set anddescribe how (T,V)-Cat varies functorially under appropriate changes of the parameters Tand V , where V is a quantale and T a monad on Set with a lax extension T to V-Rel.

3.1 Initial structures. Our first goal is to show that the forgetful functorO : (T,V)-Cat −→Set is topological (see II.5.8). It is in fact easy to give an explicit description of the O-initiallifting of sources, as follows.

3.1.1 Proposition. The O-initial lifting of a source (fi : X −→ Yi)i∈I , where (Yi, bi) is afamily of (T,V)-categories, is provided by the structure a := ∧

i∈I fi · bi · Tfi on the set X

or, in pointwise notation, by

a(x , y) =∧

i∈Ibi(Tfi(x ), fi(y))

for all x ∈ TX and y ∈ X.

Proof. Let us first verify that the V-relation a : TX −→7 X is reflexive and transitive. Sinceeach bi is reflexive, one has

1X ≤ f i · fi ≤ f i · bi · eYi · fi = f i · bi · Tfi · eX ,

so that 1X ≤ a · eX by taking the infimum on all i ∈ I. One also observes

a · T a ≤ ∧i∈I(f i · bi · Tfi) · ∧j∈J T (f j · bj · Tfj) (T monotone on hom-sets)≤ ∧i∈I,j∈J f i · bi · Tfi · (Tfj) · T bj · TTfj (Corollary 1.4.4)≤ ∧i∈I f i · bi · Tfi · (Tfi) · T bi · TTfi≤ ∧i∈I f i · bi · T bi · TTfi (Tfi · (Tfi) ≤ 1TYi)≤ ∧i∈I f i · bi ·mYi · TTfi (bi transitive)= a ·mX (m natural).

Thus, (X, a) is a (T,V)-category, and every fi : X −→ Yi is a (T,V)-functor because, bydefinition of a, one has

a(x , y) ≤ bi(Tfi(x ), fi(y))for all x ∈ TX and y ∈ X. To prove that the source (fi : (X, a) −→ (Yi, bi))i∈I is O-initial,consider a source (hi : (Z, c) −→ (Yi, bi)) with a map g : Z −→ X satisfiying hi = fi · g for alli ∈ I. One then has

g · c ≤∧

i∈If i · fi · g · c ≤

∧i∈I

f i · bi · Tfi · Tg = a · Tg ,

as desired.

3. CATEGORIES OF LAX ALGEBRAS 203

3.1.2 Examples. Rewriting the formula of Proposition 3.1.1 for the main example cate-gories considered previously, we obtain:

(1) x ≤ y ⇐⇒ ∀i ∈ I (fi(x) ≤ fi(y)) in Ord = (I, 2)-Cat.

(2) a(x, y) = supi∈I bi(fi(x), fi(y)) in Met = (I,P+)-Cat.

(3) x −→ y ⇐⇒ ∀i ∈ I(fi[x ] −→ fi(y)) in Top ∼= (, 2)-Cat.

(4) a(x , y) = supi∈I bi(fi[x ], fi(y)) in App ∼= (,P+)-Cat.

3.1.3 Theorem. For a quantale V and a lax extension T of a Set-functor T to V-Rel,the forgetful functor O : (T,V)-Cat −→ Set is topological. It therefore admits initial andfinal liftings, is transportable, has both a fully faithful left and a fully faithful right adjoint,and makes (T,V)-Cat a small-complete, small-cocomplete, well-powered and cowell-poweredcategory with a generator and a cogenerator.

Proof. Proposition 3.1.1 shows that every source (fi : X −→ O(Yi, bi)) admits an O-initial lifting, so that O is topological. Theorem II.5.9.1 takes care of the final liftings,transportability, and the existence of adjoints. Corollary II.5.8.4 and its dual prove thestatements on completeness and cocompleteness, since Set is both small-complete andsmall-cocomplete. Well- and cowell-poweredness of (T,V)-Cat are also consequences oftopologicity, see Exercise II.5.F. To obtain a generator in (T,V)-Cat, one can apply the leftadjoint of O to a generator of Set, and proceed dually for a cogenerator.

3.1.4 Remark. The proof of Proposition 3.1.1 makes it easy to describe explicitly thelimit (L, a) of a diagram D : J −→ (T,V)-Cat: just form the limit L of OD in Set and thenuse the formula for a as given in 3.1.1, where the fi (i ∈ ob J) are the limit projections inSet. There is unfortunately no easy general formula for the construction of O-final liftings,and consequently for colimits in (T,V)-Cat. Under additional hypotheses coproducts maybe described easily; see 4.3.

3.2 Discrete and indiscrete lax algebras. The topological functor O : (T,V)-Cat −→Set has both a left and right adjoint (Theorem 3.1.3) which we now proceed to describeexplicitly. We first consider the functor (−)d : Set −→ (T,V)-Cat that is the identity onmaps, and sends a set X to Xd = (X, 1]X), where the discrete structure 1]X on X is describedby

1]X = eX · T1X(see 1.7). The reflexivity condition for a (T,V)-category structure a is equivalent to 1]X ≤ a

by (1.8.i), so one obtains a (T,V)-functor εX : (X, 1]X) −→ (X, a) whose underlying mapis the identity on X. Thus, the natural transformations 1 : 1Set −→ O(−)d = 1Set andε : (−)dO −→ 1(T,V)-Cat trivially satisfy the triangular identities of an adjunction (see II.2.5),and one concludes that (−)d is left adjoint to O:

(−)d 1ε

O : (T,V)-Cat // Set .


As is the case for every topological functor (see Theorem II.5.9.1), the left adjoint (−)dembeds Set as a full coreflective category of (T,V)-Cat.The right adjoint (−)i : Set −→ (T,V)-Cat of O provides a set X with the O-initial structureof the empty source with domain X (see II.5.8). Hence, (−)i sends a set X to the (T,V)-category Xi = (X,>X), where >X : TX −→7 X is the indiscrete structure on X givenby

>X(x , y) = >for all x ∈ TX, y ∈ X. This describes a full reflective embedding of Set into (T,V)-Cat.As the discrete and indiscrete structures described above are determined by adjunctions,they correspond to the respective structures described for Ord = 2-Cat and Top ∼= (, 2)-Catin Example II.2.5.1(3).

3.3 Induced orders. Given a lax extension T to V-Rel of a monad T on Set, the structureof a (T,V)-category is a reflexive and transitive V-relation. This terminology not onlyextends the usual concept used for ordinary relations, but it also suggests that the structureinduces a natural order on the underlying set. Indeed, since a (T,V)-category structurea : TX −→7 X is left unitary, one has

eX · T a · eTX ≤ eX · T a ·mX = eX a = a .

The inequality in the other direction is just the expression of oplaxness of the unit e :1V-Rel −→ T , so we have

a = eX · T a · eTX (3.3.i)for any (T,V)-category structure a : TX −→7 X. This identity is used to prove the followingresult.

3.3.1 Proposition. Let T be a lax extension of a monad T on Set to V-Rel. If a : TX −→7 Xis a (T,V)-category structure, then the relation

x ≤ y ⇐⇒ k ≤ a(eX(x), y)

(for all x, y ∈ X) defines an order on X, called the underlying order induced by a (orsometimes simply the induced order). The structure a is then monotone in its secondvariable with respect to this order:

x ≤ y =⇒ a(x , x) ≤ a(x , y)

for all x, y ∈ X, x ∈ TX.

Proof. For the given relation ≤ on X, one immediately has x ≤ x since k ≤ a(eX(x), x) byreflexivity of a. By transitivity of a and the identity (3.3.i), if x ≤ y and y ≤ z, then

k ≤ a(eX(x), y)⊗ a(eX(y), z)) = T a(eTX(eX(x)), eX(y))⊗ a(eX(y), z) ≤ a(eX(x), z) ,


that is, x ≤ z, so the relation ≤ is also transitive. Finally, if x ≤ y then transitivity of aalso yields

a(x , x) = T a(eTX(x ), eX(x)) ≤ T a(eTX(x ), eX(x))⊗ a(eX(x), y) ≤ a(x , y) ,

so that a is monotone with respect to this order.

3.3.2 Corollary. The order induced by a (T,V)-category structure yields a functor

(T,V)-Cat −→ Ord

that makes (T,V)-Cat an ordered category. Hence, as in Ord, hom-sets in (T,V)-Cat areordered pointwise:

f ≤ g ⇐⇒ ∀x ∈ X (f(x) ≤ g(x))for f, g : (X, a) −→ (Y, b).

Proof. The order given by Proposition 3.3.1 makes every (T,V)-functor f : (X, a) −→ (Y, b)monotone since, for x, y ∈ X, x ≤ y implies

k ≤ a(eX(x), y) ≤ b(Tf · eX(x), f(y)) = b(eY · f(x), f(y)) ,

that is, f(x) ≤ f(y). Since the hom-sets in the ordered category Ord are ordered pointwise,the order on the hom-sets in (T,V)-Cat induced by (T,V)-Cat −→ Ord is also pointwise.

3.3.3 Corollary. The following conditions are equivalent for (T,V)-functors f, g : (X, a) −→(Y, b):

(i) f ≤ g;

(ii) ∀y ∈ TY, x ∈ X (b(y , f(x)) ≤ b(y , g(x));

(iii) ∀x ∈ TX, x ∈ X (a(x , x) ≤ b(Tf(x ), g(x)).

Proof. (i) =⇒ (ii) follows from 3.3.1.For (ii) =⇒ (iii), observe that a(x , x) ≤ b(Tf(x ), f(x)) ≤ b(Tf(x ), g(x)).Finally, k ≤ a(eX(x), x) ≤ b(Tf(eX(x)), g(x)) = b(eY (f(x)), g(x)) proves (iii) =⇒ (i).

3.3.4 Remark. If T is associative, then the following condition is also equivalent to (i)–(iii)of Corollary 3.3.3:

(iv) ∀x ∈ TX, y ∈ Y (b(Tg(x ), y) ≤ b(Tf(x ), y)).

A proof using Kleisli convolution is indicated in Exercise 3.E.


3.3.5 Examples.(1) For the identity lax extension of I to 2-Rel, we have 2-Cat = Ord (Example 1.3.1(1)),

and the underlying order on an ordered set (X, a) induced by a returns the originalorder on X. For the identity lax extension of I to P+-Rel, we had P+-Cat = Met(Example 1.3.1(2)), and the underlying order induced on a metric space (X, a) is givenby

x ≤ y ⇐⇒ a(x, y) = 0for all x, y ∈ X.

(2) For the Barr extension of the ultrafilter monad to 2-Rel, Theorem 2.2.5 states that(, 2)-Cat ∼= Top. Here, the underlying order on (X, a) is given by (when we write aas −→)

x ≤ y ⇐⇒ x −→ y .

By Exercise 2.C, this is precisely the underlying order of topological spaces II.1.9.The underlying order on an approach space (X, a) in the guise of a (,P+)-category(see Theorem 2.4.5) is defined by

x ≤ y ⇐⇒ a(x, y) = 0for all x, y ∈ X. In terms of the set-point distance δ : X × PX −→ P+, this means

x ≤ y ⇐⇒ δ(y, x) = 0 .

For a (T,V)-category (X, a) we wish to define an order on the set TX such that—as onewould expect of a hom-functor—a becomes order-reversing in its first variable. The strategyshould be to provide TX with a “natural” (T,V)-structure and then to consider the orderinduced via Proposition 3.3.1. Unfortunately, the free T-algebra structure mX on TX willgenerally not provide a (T,V)-category structure (unless T is flat, see Proposition 1.6.5) butthere is a (T,V)-category structure mX ≥ mX on TX satisfying our requirements, namelymX := T1X ·mX , as we show next.3.3.6 Proposition. There is a functor T : Set −→ (T,V)-Cat sending X to (TX, mX) thatmakes

(T,V)-CatO

##

Set

T;;

T // Setcommute. The order on TX induced by mX is given by

x ≤ y ⇐⇒ k ≤ T1X(x , y ) ,for all x , y ∈ TX. Any (T,V)-category structure a on X reverses this order in its firstvariable:

x ≤ y =⇒ a(y , z) ≤ a(x , z)for all x , y ∈ TX, z ∈ X, so that a : TX −→7 X becomes a module of ordered sets.


Proof. Reflexivity and transitivity of mX are easily verified:

1TX ≤ T1X = T1X ·mX · eTX = mX · eTX ,

and

mX · T mX = T1X ·mX · T T1X · TmX (Corollary 1.4.4)≤ T1X ·mX · TmX (m oplax)= mX ·mTX .

Also, for any map f : X −→ Y , Tf : TX −→ T Y is a (T,V)-functor:

Tf · mX = Tf · T (eX) ·mX ·mX (Lemma 1.7.2)≤ T (f · eX) ·mX ·mX

≤ T (eY · Tf) ·mX ·mX

= T (eY ) · TTf ·mX ·mX (Corollary 1.4.4)≤ T (eY ) ·mY · Tf ·mX

= T1Y ·mY · TTf = mX · TTf (Lemma 1.7.2).

By Proposition 3.3.1, the order induced by mX on TX is given by the V-relation T1X ·mX · eTX = T1X , which gives the description in the claim. Finally, if a is a (T,V)-categorystructure, then it is right unitary so that k ≤ T1X(x , y ) yields

a(y , z) ≤ T1X(x , y )⊗ a(y , z) ≤ a · T1X(x , z) = a(x , z) ,

which means that x ≤ y implies a(y , z) ≤ a(x , z), as claimed.

We will see in 5.4 below that T factors through the left adjoint of O, see Exercise 5.K. Also,alternative orders on TX will be considered (see in particular Examples 5.3.7).

3.4 Algebraic functors. Consider lax extensions S, T to V-Rel of monads S = (S, n, d)and T = (T,m, e) on Set. A morphism of lax extensions α : (S, S) −→ (T, T ) is a naturaltransformation α : S −→ T that becomes an oplax transformation S −→ T , so that

αY · Sr ≤ T r · αX

for all V-relations r : X −→7 Y . A monad morphism α : S −→ T which is also a morphismof lax extensions α : S −→ T is denoted by α : (S, S) −→ (T, T ). Any such naturaltransformation α induces a functor

Aα : (T,V)-Cat −→ (S,V)-Cat ,


sending (X, a) to (X, a · αX), and mapping morphisms identically. Indeed, one has 1X ≤a · eX = a · αX · dX , and

a · αX · S(a · αX) = a · αX · Sa · SαX≤ a · T a · αTX · SαX≤ a ·mX · αTX · SαX= a · αX · nX .

Moreover, a (T,V)-functor f : (X, a) −→ (Y, b) is an (S,V)-functor f : (X, a · αX) −→(Y, b · αY ):

f · a · αX ≤ b · Tf · αX = b · αY · Sf .The functor Aα is called the algebraic functor associated with α.Given a lax extension T of T = (T,m, e) to V-Rel, the unit e : 1Set −→ T immediately yieldsan algebraic functor. Indeed, oplaxness of e means precisely that there is a morphism oflax extensions e : (1Set, 1V-Rel) −→ (T, T ). As e is also a monad morphism e : I −→ T, oneobtains a functor

Ae : (T,V)-Cat −→ V-Catthat sends (X, a) to its underlying V-category (X, a ·eX) and commutes with the underlying-set functors (recall from 1.6 that we write V-Cat rather than (I,V)-Cat). This functor has aleft adjoint that we now proceed to describe. In 1.9, we defined the unitary (T,V)-relation

r] = eY · T r

associated to a V-relation r : X −→7 Y .3.4.1 Lemma. The (−)] transformation defined above satisfies

r ≤ r′ =⇒ r] ≤ r′] and s] r] ≤ (s · r)]

for all V-relations r, r′ : X −→7 Y and s : Y −→7 Z. Moreover, if T is associative, thenthe second inequality is actually an equality, and (−)] defines a 2-functor (−)] : V-Rel −→(T,V)-URel.

Proof. The first implication is straightforward, since composition and T both preserve theorder on hom-sets. For the second expression, we write

s] r] = eZ · T s · T (eY · T r) ·mX= eZ · T s · (TeY ) · T T r ·mX≤ eZ · T s · (TeY ) ·mY · T r= eZ · T s · T r≤ eZ · T (s · r)= (s · r)] .


If T is associative, then T preserves composition and m is a natural transformation(Proposition 1.9.4), so the inequalities in the previous displayed formulas become equalities.Since (1X)] = 1]X is the identity in (T,V)-URel, the (−)] transformation defines a functorV-Rel −→ (T,V)-URel.

3.4.2 Proposition. The algebraic functor Ae has a left adjoint

A : V-Cat −→ (T,V)-Cat

which associates to a V-category (X, r) the (T,V)-category (X, r]).

Proof. Let (X, r) be a V-category. Since 1X ≤ r and r · r ≤ r, we deduce from Lemma 3.4.1

1]X ≤ r] and r] r] ≤ (r · r)] ≤ r] ,

so that r] is both reflexive and transitive if r is. For a V-functor f : (X, r) −→ (Y, s), wealso have

f · r] = f · eX · T r ≤ eX · T (f · r) ≤ eX · T (s · f) = eX · T s · Tf = s] · Tf ,

which yields (T,V)-functoriality. This functor is left adjoint to Ae: for a V-category (X, r),

r ≤ eX · T r · eX = r] · eX ,

and for a (T,V)-category (X, a),

(a · eX)] = eX · T (a · eX) = eX · T a · TeX ≤ eX · T a ·mX = a .

3.4.3 Examples.

(1) For V = 2 and T = , we have 2-Cat = Ord and (, 2)-Cat = Top, and the functorAe : Top −→ Ord sends a topological space (X, a) to the ordered set (X, a · eX) whoseorder is the underlying order of the former; that is, Ae is the forgetful functor to Ordof Corollary 3.3.2. The left adjoint Ord → Top provides an ordered set (X,≤) withthe Alexandroff topology, that is, the topology whose open sets are generated by thedown-sets ↓x, for x ∈ X (Example II.5.10.5).

(2) For V = P+ and T = , we have P+-Cat = Met and (,P+)-Cat = App, so Ae : App −→Met sends an approach space (X, a) to the metric space (X, a · eX). The left adjointMet → App sends a metric space (X, r) to the approach space whose structure isgiven by

r](x , y) = supA∈x infx∈A r(x, y)for all x ∈ βX and y ∈ X. This coreflective embedding has been described in termsof point-set distance in Example 2.4.1(1).


3.5 Change-of-base functors. The algebraic functors deal with monads, that is, withthe first variable in (T,V)-Cat. The change-of-base functors deal with the second. Considerlax extensions T and T of the monad T = (T,m, e) to V-Rel and W-Rel, respectively.Let ϕ : V −→ W be a lax homomorphism of quantales (see 1.10), so that ϕ is order preservingand

ϕ(u)⊗ ϕ(v) ≤ ϕ(u⊗ v) , l ≤ ϕ(k)for all u, v ∈ V, and where k, l are the units of V, W, respectively. Then ϕ induces a laxfunctor

ϕ : V-Rel −→W-Relwhich leaves objects unchanged and sends r : X × Y −→ V to ϕr : X × Y −→ W. Clearly,for any Set-map f , we have

f ≤ ϕf and f ≤ ϕ(f )

where f and f are considered as W-relations when appearing on the left of the inequalitysign, and as V-relations on the right. Furthermore, we assume that ϕ is compatible withthe respective lax extensions T and T of T to V-Rel and W-Rel, that is, T (ϕr) ≤ ϕ(T r) forall V-relations r:

V-Rel T //

ϕ

≤

V-Relϕ

W-RelT

//W-Rel .

Under these conditions, ϕ induces a functor

Bϕ : (T,V)-Cat −→ (T,W)-Cat ,

called the change-of-base functor associated to ϕ, sending (X, a) to (X,ϕa) and leavingmaps unchanged. Indeed, we observe that eX ≤ ϕ(eX) ≤ ϕa holds, as well as

ϕa ϕa = ϕa · T (ϕa) ·mX≤ ϕa · ϕ(T a) · ϕ(mX)≤ ϕ(a · T a ·mX)= ϕa .

Moreover, given a (T,V)-functor f : (X, a) −→ (Y, b), one can adapt Corollary 1.4.4 to ϕ toobtain the last equality in

f · ϕa ≤ ϕf · ϕa ≤ ϕ(f · a) ≤ ϕ(b · Tf) = ϕb · Tf .

3.5.1 Proposition. Adjunctions of maps become adjunctions of the corresponding change-of-base functors. More precisely, suppose that ϕ : V −→ W and ψ : W −→ V are lax


homomorphisms of quantales that are compatible with the respective lax extensions of T toV-Rel and W-Rel. Then one has:

ϕ a ψ =⇒ Bϕ a Bψ .

Proof. The composite BψBϕ sends a (T,V)-category (X, a) to (X,ψϕa). If a is seen as amap a : TX ×X −→ V , then ψϕa is the map ψ · ϕ · a with a ≤ ψ · ϕ · a because 1V ≤ ψ · ϕ;therefore, the identity 1X : (X, a) −→ (X,ψϕa) is a (T,V)-functor. Dually, the identity1X : (X,ϕψb) −→ (X, b) is a (T,W)-functor for a (T,W)-category (X, b). These maps thenyield the respective components of the unit and counit of an adjunction Bϕ a Bψ since thetriangular identities are trivially satisfied.

3.5.2 Examples.

(1) The quantale homomorphism ι : 2 −→ V always has a right adjoint p : V −→ 2 that is alax homomorphism of quantales (with p(v) = > if k ≤ v and p(v) = ⊥ otherwise, seeExercise II.1.I). In fact, p is a quantale homomorphism if and only if

k ≤ u⊗ v =⇒ k ≤ u and k ≤ v

for all u, v ∈ V. As a monotone map, ι has a left adjoint o : V −→ 2 if and only ifk = >, given then by o(v) = > if and only if ⊥ < v; furthermore, o is a quantalehomomorphism if and only if

u⊗ v = ⊥ =⇒ u = ⊥ or v = ⊥

for all u, v ∈ V. These maps are all obviously compatible with the identical laxextensions of the identity monad I to Rel and V-Rel, and Proposition 3.5.1 yieldsadjunctions

Ord ⊥⊥

// V-Cat .Bp

oo

Booo

In this diagram, Bι : 2-Cat −→ V-Cat is the embedding Ord → V-Cat and Bp is theforgetful functor V-Cat −→ Ord from Corollary 3.3.2 (in the T = I case).

(2) By (1) there is in particular a full reflective and coreflective embedding Ord → Metwhich provides an ordered set (X,≤) with the metric d given by d(x, y) = 0 if x ≤ yand d(x, y) =∞ otherwise. There is also a full embedding

Bδ : Met // ProbMet

(see 2.1) which is induced by the Dirac morphism

δ : P+ = ([0,∞]op,+, 0) −→ D = (D,⊗, κ) , w 7−→ δw ,


with

δw(v) =

0 if v ≤ w,1 if v > w,

for all v, w ∈ [0,∞]. (That δ is indeed a quantale homomorphism is easy to verify.)Indeed, Bδ sends the metric space (X, d) to the probabilistic metric space (X, a) with

a(x, y)(v) =

0 if d(x, y) ≥ v,1 if d(x, y) < v

(x, y ∈ X, v ∈ [0,∞]). For another metric space (Y, c) with Bδ(Y, c) = (Y, b), a mapf : (X, a) −→ (Y, b) is probabilistically non-expansive if and only if

∀x, y ∈ X, v ∈ [0,∞] (a(x, y)(v) ≤ b(f(x), f(y))(v)) ;

this means that d(x, y) < v always implies c(f(x), f(y)) < v, or equivalently thatf : (X, d) −→ (Y, c) is non-expansive. Consequently, Bδ is indeed full.The Dirac morphism δ has a left adjoint

ω : D −→ P+ , φ 7−→ supv ∈ [0,∞] | φ(v) ≤ 0 ,

that is a quantale homomorphism. As a monotone map, δ also has a right adjoint

ρ : D −→ P+ , φ 7−→ infv ∈ [0,∞] | 1 ≤ φ(v)

that, although it satisfies ρ(κ) = 0 and ρ(φ⊗ ψ) = φ+ ψ, does not preserve arbitrarysuprema, and is therefore only a lax homomorphism of quantales. Proposition 3.5.1then yields adjunctions

Met ⊥⊥

// ProbMet .Bρ

oo

Bωoo

3.6 Fundamental adjunctions.

Set, Ord, V-Cat, and (T,V)-Cat. In the general setting of (T,V)-categories (as in 1.6),the composite of the algebraic functor Ae : (T,V)-Cat −→ V-Cat (Proposition 3.4.2) withthe underlying-order functor Bp : V-Cat −→ Ord (see Example 3.5.2(1)) is precisely theinduced-order functor of Corollary 3.3.2. This functor has a left adjoint, since both Bp

and Ae have left adjoints. We may further compose this left adjoint with the discrete-order functor Set −→ Ord which then gives the following decomposition of the adjunction(−)d a O : (T,V)-Cat −→ Set described in 3.2:

Set // Ord⊥

Ooo

Bι // V-Cat⊥Bpoo

A // (T,V)-Cat .⊥Aeoo


Ord and Met. In the case where V = P+, Example 3.5.2(1) yields the embedding Bι :Ord → Met, together with its right and left adjoints Bp and Bo, respectively. The functorBι has been described in Example 3.5.2(2), and its adjoints provide a metric space (X, a)with the orders given by

Bo : x ≤ y ⇐⇒ a(x, y) <∞ and Bp : x ≤ y ⇐⇒ a(x, y) = 0 .

Top and App. The homomorphism ι is compatible with the lax extensions of the ultrafiltermonad to Rel and P+-Rel, and the induced change-of-base functor is the embedding Top →App described at the end of 2.4.The lax homomorphism p : P+ −→ 2 is also compatible with the lax extensions and providesthe embedding with a right adjoint Bp : App −→ Top. This adjoint sends an approach space(X, a) to a topological space in which an ultrafilter x converges to a point x precisely whena(x , x) = 0.Unfortunately, the lax homomorphism o : P+ −→ 2 is not compatible with the ultrafilterlax extensions. Nevertheless, given an approach space (X, a), one can still consider thepair (X, oa) that has an ultrafilter x converging to x precisely when a(x , x) < ∞. Thisstructure satisfies the reflexivity but not the transitivity condition for topologies defined viaconvergence. In other words, (X, oa) is just a pseudotopological space (see Exercise 3.D).But we may apply the left adjoint of the full reflective embedding Top → PsTop to (X, oa)to obtain a topological space and thereby a left adjoint L : App −→ Top to the embeddingTop → App.

Ord, Met, Top, and App. The following diagram relates the functors described in thepreceding paragraphs with the adjunction of Proposition 3.4.2. This diagram commuteswith respect to both the solid and the dotted arrows (but not the dashed arrows); moreover,the two full embeddings Ord → App described by it coincide.

Top

a

⊥⊥

Ae

// AppLoo

a Ae

Bpoo

Ord ⊥⊥

?

A

OO

//Met . ?

A

OO

Bpoo

Booo

Exercises

3.A The initial lax extension. The pair (1Set, 1V-Rel) is the initial object in the metacategoryV-LXT of lax extensions of Set-functors to V-Rel and morphisms of lax extensions.


3.B Functoriality of the lax-extension-to-lax-algebras transformation. The correspondence

(α : (S, S) −→ (T, T )) 7−→ (Aα : (T,V)-Cat −→ (S,V)-Cat)

of 3.4 defines a functor (V-LXT)op −→ CAT from the dual of V-LXT (see Exercise 3.A) tothe metacategory CAT of categories and functors.

3.C Functoriality of the change-of-base transformation. Given a monad T on Set, thecorrespondence

(ϕ : (V , T) −→ (W , T)) 7−→ (Bϕ : (T,V)-Cat −→ (T,W)-Cat)

of 3.5 defines a 2-functor QNT(T) −→ CAT from the metacategory QNT(T) of quantaleswith lax extension of T and compatible lax homomorphisms to the metacategory CAT ofcategories and functors. As a consequence, this functor preserves adjoint pairs (see alsoProposition 3.5.1).

3.D Pseudotopological spaces.c© The category PsTop of pseudotopological spaces is defined asfollows: its objects are sets equipped with a reflexive relation a : βX −→7 X representingconvergence of ultrafilters to points, and its morphisms are the convergence-preservingmaps. Theorem 2.2.5 shows in particular that any topological space can be regarded as apseudotopological space. In fact, one has a full reflective embedding

Top → PsTop .

Hint. Given a reflexive relation a : βX −→7 X, define A ⊆ X to be open precisely when

∀x ∈ X, y ∈ βX (y a x & x ∈ A =⇒ A ∈ y ) .

3.E Order and Kleisli convolution. For a (T,V)-functor f : (X, a) −→ (X, b), define(T,V)-relations f∗ : X −7 Y and f ∗ : Y −7 X by

f∗ := b · Tf and f ∗ := f · b .

Then:

(1) f∗ f ∗ ≤ b.

(2) a ≤ f ∗ f∗ if m satisfies BC.

(3) f ≤ g in (T,V)-Cat if and only if f ∗ ≤ g∗ in V-Rel.

(4) Condition 3.3.4(iv) holds if and only if g∗ ≤ f∗ in V-Rel.

(5) If T is associative, thenf ∗ ≤ g∗ ⇐⇒ g∗ ≤ f∗ ,

and condition 3.3.4(iv) is equivalent to 3.3.3(i)–(iii).Hint. Use Kleisli convolution and statements (1) and (2).


3.F Adjunctions in (T,V)-Cat. Use Exercise 3.E(5) to prove that if T is associative, thenf : (X, a) −→ (Y, b) is left adjoint to g : (Y, b) −→ (X, a) in the ordered category (T,V)-Catif and only if

a(x , g(y)) = b(Tf(x ), y)for all x ∈ TX, y ∈ Y .

3.G Tensored V-categories. A V-category (X, a) is called tensored if for all x ∈ X andu ∈ V there exists z ∈ X such that

∀y ∈ X ( a(z, y) = (u a(x, y)) ) . (3.6.i)

(1) Show that (X, a) is tensored if and only if, for all x ∈ X, the V-functor

a(x,−) : X −→ V

has a left adjoint in V-Cat. Conclude that any element z satisfying (3.6.i) is, up toorder-equivalence in X, uniquely determined by x and u; one writes z = x⊗ u.

(2) Show that the V-category V is tensored and so is X = [Xop,V ] (see Exercise 1.H), forevery V-category X.

3.H Change-of-base needs lax homomorphisms. For quantales V and W , let ϕ : V −→ W bemonotone. The following assertions are equivalent:

(i) For every V-category (X, a), the pair (X,ϕa) forms a W-category;

(ii) (V , ϕh) is a W-category, with h = (−) (−);

(iii) ϕ is a lax homomorphism of quantales.

3.I Characterizing change-of-base functors. Let V and W be quantales. We write (V , h) =(V ,) (see Exercise 3.H).

(1) For a lax homomorphism ϕ : V −→ W show that the change-of-base functor Bϕ :V-Cat −→W-Cat is a 2-functor that preserves initial morphisms with respect to theunderlying-Set functors.

(2) Let F : V-Cat −→ W-Cat be a 2-functor preserving underlying sets and initial mor-phisms. Writing (X, a) for F (X, a), show:

(a) h(k,−) : V −→ W is monotone.(b) For every tensored V-category (X, a),

∀u ∈ V ∀x, y ∈ X (a(x⊗ u, y) = h(u, a(x, y))) ;

in particular,∀x, y ∈ X (a(x, y) = h(k, a(x, y))) . (3.6.ii)


(c) Formula (3.6.ii) holds for every V-category, thanks to initiality of the Yonedafunctor (see Exercise 1.H(3)).

(d) F = Bϕ, for a unique lax homomorphism ϕ : V −→ W of quantales.

3.J Many structures on 1. For the topological functor O : V-Cat −→ Set, the complete latticeO−11 (see Theorem II.5.9.1) of V-category structures on a singleton set is order-isomorphicto v ∈ V | k ≤ v, v ⊗ v ≤ v, which is closed under infima in V but not under suprema(unless V is trivial).

3.K Lax extensions of the same monad. Let T and T be lax extensions to V-Rel of themonad T = (T,m, e) on Set with T r ≤ T r for all V-relations r. Then there is a full andfaithful algebraic functor (T,V , T)-Cat −→ (T,V , T)-Cat.

3.L Restricting a lax extension from V-Rel to Rel. Let T come with a lax extension T toV-Rel, and assume

k = > and (u⊗ v = ⊥ =⇒ u = ⊥ or v = ⊥)

(for all u, v ∈ V), so that the left adjoint o to ι : 2 −→ V becomes a quantale homomorphism.For r : X −→7 Y in Rel = 2-Rel, define T r := oT (ιr) and assume that

T (ιr) = ι(T r)

holds for all relations r. Then T is a lax extension to Rel of T, and ι induces the change-of-base functor

Bι : (T, 2, T)-Cat −→ (T,V , T)-Cat .

In particular, for V = P+ and T = the Barr extension of to P+-Rel, T is the Barrextension of to Rel.

4. EMBEDDING LAX ALGEBRAS INTO A QUASITOPOS 217

4 Embedding lax algebras into a quasitopos

In Theorem 3.1.3 we showed that the forgetful functor O : (T,V)-Cat −→ Set is topological,giving an explicit description of O-initial structures and consequently of limits (see Proposi-tion 3.1.1 and Remark 3.1.4). In this subsection, we introduce a supercategory of (T,V)-Catwhich remains topological over Set but, unlike (T,V)-Cat, also allows for an easy descriptionof colimits. In addition, this supercategory turns out to be a quasitopos—a term that wewill explain in 4.8 below—and therefore cartesian closed. For (T,V) = (, 2) this quasitoposis the category PsTop of pseudotopological spaces. We also discuss in general terms the roleof the intermediate category PrTop of pretopological spaces, which still enjoys an importantproperty of quasitopoi (existence of a partial-map classifier), although it fails to be cartesianclosed.We continue to work with a quantale V which has a ⊗-neutral element k, and with a monadT = (T,m, e) on Set which comes with a lax extension T to V-Rel (see 1.5).

4.1 (T,V)-graphs. A (T,V)-graph (X, a) is a set X equipped with a reflexive (T,V)-relation a : X −7 X (see 1.6), that is, a V-relation a : TX −→7 X with eX ≤ a. A morphismf : (X, a) −→ (X, b) of (T,V)-graphs is defined as for (T,V)-categories (f · a ≤ b · Tf) andis therefore also called a (T,V)-functor in the more general context of the category

(T,V)-Gph

of (T,V)-graphs. There is a string of full subcategories

(T,V)-Cat → (T,V)-UGph → (T,V)-RGph → (T,V)-Gph ,

with (T,V)-RGph the category of right unitary (T,V)-graphs (X, a) that must satisfya eX ≤ a, and with (T,V)-UGph the category of unitary (T,V)-graphs (X, a) for which ais also left unitary: eX a ≤ a. Recall from Proposition 1.7.3 that right-unitariness of a(T,V)-relation a is equivalently expressed by a · T1X ≤ a, so it comes for free when T isflat. For a map f : X −→ Y between (T,V)-graphs (X, a), (Y, b) one therefore has

b · Tf = b · T1X · Tf = b · T f

when (Y, b) is right unitary, so that the condition f · a ≤ b · T f then suffices to make f a(T,V)-functor.Before presenting examples, let us state these definitions elementwise:

4.1.1 Lemma. A set X with a V-relation a : TX −→7 X is a (T,V)-graph if

k ≤ a(eX(x), x)

for all X ∈ X. It is right unitary if

T1X(x , y )⊗ a(y , y) ≤ a(x , y)


for all y ∈ X, x , y ∈ TX, and unitary if, in addition,

T a(X , eX(y)) ≤ a(mX(X ), y)

for all X ∈ TTX, y ∈ X.

Proof. By definition,

(a · T1X)(x , y) = ∨y∈TX T1X(x , y )⊗ a(y , y)

for all x ∈ TX, y ∈ X. Furthermore, eX a ≤ a means eX · T a ≤ a ·mX which, whenexpressed elementwise, gives the stated condition for (X, a) being unitary.

Note also that with 1]X = eX · T1X , (X, a) is a left unitary (T,V)-graph if and only if

1]X ≤ a = 1]X a ,

and that it becomes unitary precisely under the additional condition

a = a 1]X

(see Proposition 1.7.3). With f ] = f · 1]Y , (T,V)-functors are equivalently described by

a f ] ≤ f ] b

(see 1.8).

4.1.2 Remark. The definition of a (T,V)-graph does not depend on the monad multiplica-tion m of T = (T,m, e) or the lax extension T of the functor T . Right-unitariness dependson T but not on m, whereas unitariness depends on both m and T .

4.1.3 Examples.

(1) (I,V)-graphs, with I identically extended to V-Rel, are simply sets X with a reflexiveV-relation a : X −→7 X. For V = 2 one obtains sets with a reflexive relation, and forV = P+ sets with a function a : X ×X −→ [0,∞] which is 0 on the diagonal of X ×X.(I,V)-graphs are automatically unitary.

(2) (, 2)-graphs, where is equipped with its Barr extension, are sets X with a relationa : βX −→7 X that is only required to satisfy x −→ x for all x ∈ X. (Here wewrite x −→ x instead of x a x.) These are precisely the pseudotopological spaces ofExercise 3.D. Indeed, since the Barr extension is flat, being right unitary comes forfree, and the only remaining condition (x −→ x for all x ∈ X) is characteristic forpseudotopologicity; hence,

(, 2)-Gph = (, 2)-RGph = PsTop .


Unitary (, 2)-graphs must satisfy the additional condition

(X −→ y) =⇒ (∑X −→ y) (4.1.i)

where ∑X is the Kowalsky sum of X ∈ ββX, and where X −→ y amounts to

∀A ∈ X ∃x ∈ A (x −→ y) .

Hence, (4.1.i) means equivalently (see Example 1.10.3(3)),

(x ∈ βX | x −→ y ∈ X ) =⇒ (∑X −→ y) (4.1.ii)

for all X ∈ ββX, y ∈ X. From Proposition 2.2.4 we obtain immediately that unitary(, 2)-graphs on a set X correspond bijectively to maps c : PX −→ PX with

A ⊆ c(A) , c(∅) = ∅ , c(A ∪B) = c(A) ∪ c(B) ,

for all A,B ⊆ X. A set X equipped with such a map c is called a pretopological space.A morphism f : X −→ Y in the category PrTop must satisfy f(cX(A)) ⊆ cY (f(A)) forall A ⊆ X, and it is now easy to see that there is an isomorphism

(, 2)-UGph ∼= PrTop c©

which leaves underlying sets unchanged.

(3) For the filter monad F extended by

a (F r) b ⇐⇒ a ⊇ r[b ]

(for all relations r : X −→7 Y , a ∈ FX, b ∈ FY ) as in Example 1.10.3(4), a rightunitary (F, 2)-graph (X,−→) must satisfy

x −→ x and (a ⊇ b & b −→ y =⇒ a −→ y) , (4.1.iii)

and (X,−→) is unitary if

(a ∈ FX | a −→ y ∈ A) =⇒ (∑A −→ y) (4.1.iv)

for all A ∈ FFX, x, y ∈ X. The category (F, 2)-RGph contains the full subcategory(F, 2)-RGphPs whose objects (X,−→) satisfy (4.1.iii) and the condition

(Ps) if a is a filter such that for every proper filter b ⊇ a there exists a proper filterc ⊇ b with c −→ y, then a −→ y

for all a ∈ FX, y ∈ X. It is not difficult to see that (F, 2)-RGphPs is isomorphic c©toPsTop (Exercise 4.I). Moreover, the full subcategory (F, 2)-UGph of (F, 2)-RGphPs is


isomorphic to PrTop (Exercise 4.L), so using (2) above, one obtains the followingdiagram of full embeddings and isomorphisms:

(F, 2)-UGph∼= // PrTop

∼= c©

c©// PsTop

=

∼=c©

// (F, 2)-RGphPs _

(, 2)-UGph // (, 2)-RGph=

// (F, 2)-RGph _

(, 2)-Gph // (F, 2)-Gph .

The horizontal arrows on the right are obtained by extension of the convergencerelation: for a ∈ FX and y ∈ X,

(a −→ y) ⇐⇒ ∀x ∈ βX ( x ⊇ a =⇒ x −→ y) .

(4) We writePsApp := (,P+)-Gph = (,P+)-RGph

(with the Barr extension of to P+-Rel of 2.4) and calls its objects pseudo-approachspaces. PsApp contains the full subcategory

PrApp := (,P+)-UGph

of pre-approach spaces whose objects may be described equivalentlyc© as sets equippedwith a finitely additive distance function δ : X×PX −→ [0,∞] satisfying δ(x, x) = 0for all x ∈ X. The needed isomorphism is established as in Theorem 2.4.5, withProposition 2.4.4 providing the key ingredient to the proof. (See also Exercise 4.O.)

4.1.4 Proposition. The forgetful functor O : (T,V)-Gph −→ Set is topological. O-initialliftings of sources may be formed as for (T,V)-categories (see Proposition 3.1.1), while asink (fi : (Xi, ai) −→ (Y, b))i∈I in (T,V)-Gph is O-final precisely when

b = eY ∨∨i∈I

fi · ai · (Tfi) . (4.1.v)

For an epi-sink (fi)i∈I , this formula simplifies to b = ∨i∈I fi · ai · (Tfi).

Proof. For a family (Xi, ai) of (T,V)-graphs and Set-maps fi : Xi −→ Y (i ∈ I), define bby (4.1.v). Trivially, b is reflexive, and every (fi : (Xi, ai) −→ (Y, b) has trivially become a(T,V)-functor. For O-finality of (fi)i∈I , consider a (T,V)-graph (Z, c) and a map h : Y −→ Zsuch that

h · fi · ai ≤ c · T (h · fi)for all i ∈ I. Then

h · fi · ai · (Tfi) ≤ c · Th · Tfi · (Tfi) ≤ c · Th ,


and therefore fi · ai · (Tfi) ≤ h · c · Th by adjunction (for all i ∈ I). SinceeY ≤ eY · (Th) · Th = h · eZ · Th ≤ h · c · Th ,

one has b ≤ h · c · Th, and h is a (T,V)-functor. This concludes the proof that O istopological, with O-final sinks characterized by (4.1.v). That O-initial sources may bedescribed as in Proposition 3.1.1 follows from a direct verification (see Exercise 4.A). Finally,if (fi)i∈I is epic, with Proposition 1.2.2 one obtains 1Y ≤

∨i∈I fi · f i . Consequently,

eY ≤∨i∈I fi · f i · eY = ∨

i∈I fi · eXi · (Tfi) ≤ ∨i∈I fi · ai · (Tfi) .

The category (T,V)-RGph is reflective in (T,V)-Gph, with the reflection morphisms givenby (X, a) −→ (X, a · T1X). Indeed, any (T,V)-functor f : (X, a) −→ (Y, b) with (Y, b) rightunitary is a (T,V)-functor f : (X, a · T1X) −→ (Y, b):

f · a · T1X ≤ b · Tf · T1X ≤ b · T f = b · T1Y · Tf = b · Tf .4.1.5 Corollary. The forgetful functor O : (T,V)-RGph −→ Set is topological. O-initialliftings of sources may be formed as for (T,V)-categories (see Proposition 3.1.1), while asink (fi : (Xi, ai) −→ (Y, b))i∈I in (T,V)-RGph is O-final precisely when

b = 1]Y ∨∨i∈I

fi · ai · T (f i ) . (4.1.vi)

For an epi-sink (fi)i∈I , this formula simplifies to b = ∨i∈I fi · ai · T (f i ).

Proof. We may apply Theorem II.5.10.3. O-final liftings in (T,V)-RGph are obtained byreflecting the O-final lifting in (T,V)-Gph. With b0 denoting the codomain structure of theO-final lifting in (T,V)-Gph, an easy computation in the quantaloid V-Rel shows

b = b0 · T1Y = (eY · T1Y ) ∨ ∨i∈I(fi · ai · (Tfi) · T1Y ) = 1]Y ∨∨i∈I fi · ai · T (f i ) .

4.1.6 Remarks.(1) Since (T,V)-Cat → (T,V)-Gph preserves initial sources, one may use the Taut Lift

Theorem II.5.11.1 to obtain its reflectivity; likewise for (T,V)-Cat → (T,V)-RGph.The reflector provides a (T,V)-graph (X, a) with the (T,V)-category structure∧

c : X −7 X | a ≤ c, c c ≤ c .In 4.2 below, using ordinal recursion we give an alternative description of the reflector(T,V)-RGph −→ (T,V)-Cat that turns out to be essential in 4.3 when describingcoproducts in (T,V)-Cat.

(2) A surjective morphism f : X −→ Y in (T,V)-RGph with X in (T,V)-Cat that isfinal with respect to O : (T,V)-RGph −→ Set will generally fail to make Y into a(T,V)-category and be final with respect to the forgetful functor (T,V)-Cat −→ Set,even when T = I and V = 2 (Exercise 1.G(3)). Nevertheless, the explicit descriptionof O-final sinks in (T,V)-RGph turns out to be useful for the computation of colimitsin (T,V)-Cat in special instances (see 4.2).


4.2 Reflecting (T,V)-RGph into (T,V)-Cat. For a right unitary (T,V)-graph (X, a) wedefine recursivelyc© (see II.1.14) an ascending chain of (T,V)-relations aν , for every ordinalnumber ν, as follows:

a0 := eX ,

aν+1 := a aν ,aλ := ∨

ν<λ aν (λ a limit ordinal).

Since a is right unitary, a1 = a eX = a, and reflexivity makes the chain ascending:aν ≤ eX aν ≤ a aν = aν+1. Consequently, this chain in the set V-Rel(TX,X) mustbecome stationary, that is,

aµ = aµ+1

for some ordinal number µ. Next we will show that under a mild additional assumptionon the lax extension of T, the pair (X, a∞) := (X, aµ) provides a reflection of (X, a) into(T,V)-Cat.

4.2.1 Proposition. For the lax extension T = (T ,m, e) of T, suppose that m : T −→ T Tis a natural transformation. Then (T,V)-Cat is reflective in (T,V)-RGph, with the reflectionmorphisms obtained by ordinal recursion as described above.

Proof. When m is a natural transformation, then (ts)r ≤ t(sr) for all (T,V)-relationsr : X −7 Y , s : Y −7 Z, t : Z −7 W (Exercise 1.L). In particular,

(a aν) aµ ≤ a (aν aµ)

for all ordinals ν, µ. Consequently, if aµ = aµ+1, one can easily show aν aµ ≤ aµ forall ν by ordinal recursion, and aµ aµ ≤ aµ follows. Since eX ≤ a ≤ aµ, one has that(X, aµ) is a (T,V)-category, and 1X : (X, a) −→ (X, aµ) is a (T,V)-functor. It remains tobe shown that every (T,V)-functor f : (X, a) −→ (Y, b) with (Y, b) a (T,V)-category givesa (T,V)-functor f : (X, aµ) −→ (Y, b), and for that is suffices to show that a ≤ f · b · Tfimplies aν ≤ f ·b ·Tf for all ordinals ν. We show the successor step of the ordinal recursion:

aν+1 = a aν ≤ (f · b · Tf) (f · b · Tf)≤ f · b · Tf · (Tf) · T b · TTf ·mX≤ f · b · T b ·mY · Tf= f · (b b) · Tf = f · b · Tf .

It follows from Proposition 4.2.1 that (T,V)-Cat is reflective in (T,V)-UGph, but there isno obvious simplification in this proof when applying the ordinal recursion to a unitary(T,V)-graph. However, under additional hypotheses on the lax extension of T, there is aneasy one-step construction for a reflector (T,V)-RGph −→ (T,V)-UGph, as follows.


4.2.2 Proposition. Suppose that the lax extension T = (T ,m, e) of T to V-Rel is associa-tive. Then

1X : (X, a) −→ (X, eX a)is a reflection of the right unitary (T,V)-graph (X, a) into (T,V)-UGph.

Proof. By Proposition 1.9.4, eX a is right unitary because a is:

(eX a) 1]X = eX (a 1]X) = eX a .

Similarly, eX a = 1]X a is also left unitary:

1]X (1]X a) = (1]X 1]X) a = 1]X a .

Furthermore, if f : (X, a) −→ (Y, b) is a (T,V)-functor with (Y, b) unitary, f : (X, eX a) −→(Y, b) is also a (T,V)-functor:

eX a ≤ eX (f · b · Tf)= eX · T (f · b · Tf) ·mX= eX · (Tf) · (T b) · TTf ·mX≤ f · eY · (T b) ·mY · Tf≤ f · (eY b) · Tf = f · b · Tf .

4.3 Coproducts of (T,V)-categories. In the topological category (T,V)-RGph over Set,the coproduct of a family (Xi, ai) of right unitary (T,V)-graphs is given by the disjointunion

ti : Xi −→ X = ∐i∈I Xi (i ∈ I)

endowed with the final structure a = ∨i∈I ti · ai · T (ti ) (see Corollary 4.1.5). Since

tj · ti =

1Xj if j = i,⊥ otherwise,

each tj is not just a (T,V)-functor but satisfies the equation

tj · a = ∨i∈I t

j · ti · ai · T (ti ) = aj · T (tj) .

In what follows we use the following terminology on which we will elaborate further inChapter V.

4.3.1 Definition. A morphism f : (X, a) −→ (Y, b) in (T,V)-RGph is nearly open if f · b ≤a · T (f ), and open if f · b ≤ a · (Tf).


Since every (T,V)-functor satisfies a · (Tf) ≤ f · b, near openness of f means equivalentlyf · b = a · T (f ), and openness f · b = a · (Tf). Clearly, openness implies near openness,and the converse statement holds if

T (f ) = T1X · (Tf) (4.3.i)

for all maps f : X −→ Y .4.3.2 Lemma. Let f : (X, a) −→ (Y, b) be a (T,V)-functor from a (T,V)-category to a rightunitary (T,V)-graph. Then near openness of f implies near openness of f : (X, a) −→ (Y, b∞)with b∞ = ∨

ν bν (as in 4.2), provided that T preserves V-relational composition.

Proof. Proceeding to show f ·b∞ ≤ a · T (f ) by ordinal recursion, we consider the successorstep and assume f · bν ≤ a · T (f ). If T preserves composition of V-relations, we thenobtain

f · bν+1 = f · (b bν)= f · b · T bν ·mY≤ a · T (f ) · T bν ·mY≤ a · T (f · bν) ·mY≤ a · T (a · T (f )) ·mY= a · T a · T T (f ) ·mY≤ a ·mX ·mX · T (f ) = a · T (f ) .

4.3.3 Theorem. If T is associative, then (T,V)-Cat is closed under coproducts in (T,V)-RGph.

Proof. For (T,V)-categories (Xi, ai), one forms the coproduct (ti : (Xi, ai) −→ (X, a))i∈Iin (T,V)-RGph, with a = ∨

i∈I ti · ai · T (ti ), and one obtains the coproduct in (T,V)-Catby applying the reflector (X, a) −→ (X, a∞). Hence, we must show a∞ = a. But, byLemma 4.3.2, every ti : (Xi, ai) −→ (X, a∞) is nearly open. Since (ti)i∈I is epic, nearopenness yields

a∞ = ∨i∈I ti · ti · a∞ = ∨

i∈I ti · ai · T (ti ) = a .

4.3.4 Remarks.

(1) Near openness of every ti : (Xi, ai) −→ (X, a) (i ∈ I) is characteristic for O-finality ofthe sink (ti)i∈I (see Exercise 4.B).

(2) Since T preserves injections (see Exercise 1.P), every open injection f : (X, a) −→ (Y, b)in (T,V)-Gph is O-initial. Indeed,

f · b · Tf = a · (Tf) · Tf = a

since Tf is injective. We refer to such (T,V)-functors as open embeddings.


(3) With T an associative lax extension to V-Rel satisfying (4.3.i), a coproduct (ti :(Xi, ai) −→ (X, a))i∈I in (T,V)-Cat satisfies

a =∨

i∈Iti · ai · (Tti) .

Hence, for all x ∈ TX, y ∈ X, one has

a(x , y) =

ai(x , y) if x ∈ TXi, y ∈ Xi for some i ∈ I,⊥ else;

here we write x ∈ TXi instead of x = Tti(x i) for some x i ∈ TXi.

(4) In Lemma 4.3.2, the hypotheses that T preserves V-relational composition may betraded for the assumption that T (r · g) = T r · (Tg) for all r : X −→7 Y andg : Z −→ X. In fact, in this case a nearly open map f : (X, a) −→ (Y, b) is open, sincea · T (f ) = a · T1X · (Tf) = a · (Tf).

4.3.5 Definition. A functor T : Set −→ Set is taut if it sends every BC-square

M

m

g// N

n

Xf// Y

(4.3.ii)

with n monic to a BC-square; that is, n · f = g ·m with n monic implies (Tn) · Tf =Tg · (Tm).

Clearly, a BC-square (4.3.ii) with n monic is actually a pullback diagram: since the canonicalmorphism c : M −→ X ×Y N factors through the monomorphism m, it is injective, but alsosurjective (by Lemma 1.11.1).Trivially, a functor satisfying BC is taut, and every taut functor preserves monomorphisms,since m is a monomorphism if and only if

M

1

1 //M

m

M m // X

is a BC-square (and then necessarily a pullback square).

4.3.6 Corollary. c©A Set-functor T is taut if and only if T preserves pullback squares (4.3.ii)with n monic.

Proof. This is immediate from the previous discussion.


In what follows, in addition to tautness of T , we also need that the underlying ordered setof the quantale V is cartesian closed, so that both ⊗ and ∧ distribute over suprema in V.The status of this assumption for our purposes is clarified by the following Lemma.4.3.7 Lemma. The following conditions are equivalent for a quantale V:

(i) V is cartesian closed;

(ii) the underlying ordered set of V is a frame;

(iii) the right Frobenius law(r ∧ s · f) · f = r · f ∧ s

holds in V-Rel for all f : X −→ Y , r : X −→7 Z, s : Y −→7 Z;

(iv) the left Frobenius lawf · (r ∧ f · s) = f · r ∧ s

holds in V-Rel for all f : X −→ Y , r : Z −→7 X, s : Z −→7 Y ;

Proof. Straightforward verifications.

4.3.8 Proposition. Let V be a cartesian closed quantale. If T is taut, then open embeddingsare stable under pullback in (T,V)-RGph, and if T satisfies BC, all open maps are stableunder pullback.

Proof. Consider a pullback diagram

(P, d) q//

p

(Y, b)g

(X, a) f// (Z, c)

in (T,V)-RGph with g open. Then d is the O-initial structure with respect to (p, q), that is,d = (p · a · Tp) ∧ (q · b · Tq), and g · c = b · (Tg). If T satisfies BC, with Lemma 4.3.7one obtains openness of p as follows:

p · a = p · (a ∧ a)≤ p · (a ∧ (f · c · Tf))≤ p · a ∧ (p · f · c · Tf)= p · a ∧ (q · g · c · Tf)= p · a ∧ (q · b · (Tg) · Tf)= p · a ∧ (q · b · Tq · (Tp)) (BC)=((p · a · Tp) ∧ (q · b · Tq)

)· (Tp) (V cartesian closed)

= d · (Tp) .

Tautness of T suffices in lieu of BC when g is injective.


One says that a coproduct (ti : Xi −→ X)i∈I in a category is universal (or stable underpullback) if for all morphisms f : Y −→ X and pullback diagrams

Yisi //

fi

Y

f

Xiti // X ,

the family (si : Yi −→ Y )i∈I is a coproduct.

4.3.9 Theorem. Suppose that V is cartesian closed and T is taut. Then coproducts in(T,V)-RGph are universal if T (f ) = T1X · (Tf) for all f : X −→ Y . If moreover T isassociative, coproducts are also universal in (T,V)-Cat.

Proof. Under the stated hypotheses, coproduct injections in (T,V)-RGph are open and stableunder pullback. This implies universality of coproducts by Remark 4.3.4(1). The additionalhypothesis makes sure that (T,V)-Cat is closed under coproducts in (T,V)-RGph.

A coproduct (ti : Xi −→ X)i∈I in a category is disjoint if for all i 6= j, the pullback of tiand tj is given by an initial object in the category. This is certainly true in Set and, sincethere is only one structure on ∅ in (T,V)-RGph and (T,V)-Cat, likewise in these categories.A category with coproducts (of small families of objects) and pullbacks is extensive if itscoproducts are universal and disjoint (see also Exercise 4.E and V.5.1).

4.3.10 Corollary. Suppose that V is cartesian closed, that T is taut, and that T is asso-ciative. If T (f ) = T1X · (Tf) for all f : X −→ Y , then (T,V)-RGph and (T,V)-Cat areextensive categories.

Proof. The result restates Theorem 4.3.9.

4.4 Interlude on partial products and local cartesian closedness. A finitely com-plete category C is locally cartesian closed if its comma categories C/Z are cartesian closedfor all Z ∈ ob C. By definition of cartesian closedness, this means that for all f : X −→ Zin C the functor

(−)×Z X : C/Z −→ C/Z , (Y, g) 7−→ (Y ×Z X, g · π1 = f · π2)

(with π1 and π2 the pullback projections) has a right adjoint. Since the domain functordomZ : C/Z −→ C always has a right adjoint, given by

W 7−→ (W × Z,W × Z −→ Z) ,

local cartesian closedness of C makes the composite functor

domZ((−)×Z X) : C/Z −→ C , (Y, g) 7−→ Y ×Z X


have a right adjoint.Let us spell out the condition for a C-object Y to admit a (domZ((−)×Z X))-couniversalarrrow. Such an arrow is given by an object (P, p : P −→ Z) in C/Z and a morphismε : P ×ZX −→ Y in C such that for any (Q, q : Q −→ Z) in C/Z and any g : Q×ZX −→ Y inC, there is a uniquely determined morphism h : Q −→ P with p ·h = q and ε · (h×Z 1X) = gin C:

Y P ×Z Xεoo

π2 //

π1

X

f

Q×Z X

gbb

h×Z1X99 44

Pp

// Z

Q

h

99

q

44

One calls P = P (Y, f) together with the projection p and the evaluation ε a partial productof Y over f . The category C has partial products if P (Y, f) exists for all (Y, f).Note that for f = 1X , P (Y, 1X) ∼= Y ×X is simply a product. If Z is a terminal object,the existence of P (Y, ! : X −→ Z) means precisely the existence of the exponential Y X in C,that is, the existence of a ((−)×X)-couniversal arrow for Y in C (see II.4.4). In particular,local cartesian closedness of C implies cartesian closedness of C.

4.4.1 Examples.

(1) The existence of P = P (Y, f) means, by definition, the existence of a natural bijectivecorrespondence

Q −→ P

Q −→ Z , Q×Z X −→ Y .Hence, when Q is a terminal object, we see that in C = Set the partial product Pmust be, up to isomorphism,

P = (j, z) | z ∈ Z, j : f−1z −→ Y ∼=∐z∈Z Y

f−1z ,

with p : (j, z) 7−→ z and ε the evaluation map. In fact, one can write P ×Z X as

P ×Z X = (j, x) | x ∈ X, j : f−1f(x) 7−→ Y ∼=∐x∈X Y

f−1f(x)

with ε : (j, x) 7−→ j(x). In this way one sees that Set has partial products.

(2) Ord is cartesian closed but fails to have partial products (see Exercise 4.F).

(3) Top fails to be cartesian closed (see Exercise 4.G).

4.4.2 Lemma. For a category A with equalizers, a functor F : A −→ C/X has a rightadjoint if domX F : A −→ C has a right adjoint.


Proof. Let κ : domX −→ ∆X be the natural transformation with κ(W,t) = t for every object(W, t) in C/X, and let J be right adjoint to domZ F . Then by adjunction, κF : domX −→ ∆Xcorresponds to a natural transformation σ : 1A −→ ∆JX. Given (W, t), one forms theequalizer

G(W, t) // JWJt //

σJW// JX

whose domain defines a right adjoint G of F . Indeed, for every A-object A, one obtains theequalizer diagram in Set given by the top row of

A(A,G(W, t)) // A(A, JW )∼=

A(A,Jt)//

A(A,σJW )// A(A, JX)

∼=

C(domX FA,W )C(domX FA,t)

//

φ// C(domX FA,X)

in which the map A(A, σJW ) has constant value σA by naturality of σ. Consequently, themap φ that makes the diagram commute (in the obvious sense) must send every C-morphismto κFA, which corresponds to σA by adjunction. The equalizer of the two lower maps inSet is

h : domX FA −→ W | t · h = κFA ∼= C/X(FA, (W, t)) ,

which is therefore isomorphic to A(A,G(W, t)), naturally in A.

4.4.3 Proposition. For a morphism f : X −→ Z of a finitely complete category C, thefollowing assertions are equivalent and characterize f as an exponentiable object of C/Z:

(i) f is exponentiable in C/Z, that is, (−)×Z X : C/Z −→ C/Z has a right adjoint;

(ii) for all C-objects Y , the partial product P (Y, f) exists in C;

(iii) the pullback functor

f ∗ : C/Z −→ C/X , (Y, g) 7−→ (Y ×Z X, π2 : Y ×Z X −→ X)

has a right adjoint.

Proof. (i) =⇒ (ii) follows from the definitions. Lemma 4.4.2 yields (ii) =⇒ (iii) bycommutativity of the diagram

C/XdomX

C/ZdomZ((−)×ZX)

//

f∗<<

C .


For (iii) =⇒ (i), consider the commutative diagram

C/Xf!

C/Z(−)×ZX

//

f∗@@

C/Z

with f! : (W, t) 7−→ (W, f · t). Actually, one easily shows f! a f ∗ (so that (−)×Z X is thefunctor part of a comonad on C/Z). Hence, if f ∗ has a right adjoint, then the compositefunctor f!f

∗ has a right adjoint.

4.4.4 Example. In C = Set, the right adjoint f∗ : C/X −→ C/Z of f ∗ sends (W, t) in C/Xto (P, p), with

P = (j, z) | z ∈ Z, j : f−1z −→ W, t · j = (f−1 → X)

and p : (j, z) 7−→ z. One may write

P ×Z X = (j, x) | x ∈ X, j : f−1f(x) −→ W, t · j = (f−1f(x) → X) ,

with evalutation map ε : (j, x) 7−→ j(x); this ε is a C/X-morphism since t · ε = π2. Whensetting (W, t) = (Y ×X, Y ×X −→ X), one retrieves the partial product P (Y, f) from thisconstruction.

4.5 Local cartesian closedness of (T,V)-Gph. Throughout this subsection we assumethat

• V is cartesian closed and integral;

• T satisfies the Beck–Chevalley condition.

(In fact, the tensor product of the quantale V and the multiplication m of the monad Tplay no role in this subsection: it suffices that V is a frame and T an endofunctor equippedwith a natural transformation e : 1Set −→ T . In particular, no lax extension of T to V-Rel isneeded.)Our goal is to construct partial products in (T,V)-Gph and thereby show local cartesianclosedness of this category. For that, it is useful to point out that the ordered set V(considered as a category) is locally cartesian closed. Indeed, for α, β, γ, υ ∈ V , one has theequivalence

υ ≤ γ ∧ (α −→ β)υ ≤ γ , υ ∧ α ≤ β ,

where α −→ β denotes the internal hom on V with its cartesian structure ∧. Hence, if α ≤ γ,the partial product P (β, α ≤ γ) exists in V and is given by γ ∧ (α −→ β).


Let us now consider f : (X, a) −→ (Z, c) and (Y, b) in (T,V)-Gph, and form the set

P = (j, z) | z ∈ Z, j : (Xz, az) −→ (Y, b) a (T,V)-functor ,

where Xz = f−1z and az is the restriction of a. We can write

W = P ×Z X = (j, x) | x ∈ X, j : (Xf(x), af(x)) −→ (Y, b)

and define p : P −→ X and ε : W −→ Y as at the level of sets (see Example 4.4.1(1); but notethat P is a subset of the partial product of Y over f in Set). The V-relation d : TP −→7 Pis defined by

d(p, (j, z)) = c(Tp(p), z) ∧∧

w ,x(a(Tπ2(w), x) −→ b(Tε(w), j(x))) (4.5.i)

(for p ∈ TP , (j, z) ∈ P ), with the meet ranging over all w ∈ TW with Tπ1(w) = p andx ∈ X with f(x) = z.

4.5.1 Theorem. The pair (P, d) is a partial product of (Y, b) over f : (X, a) −→ (Z, c) in(T,V)-Gph.

Proof. We show reflexivity of d, (T,V)-functoriality of p and ε, and verify the universalproperty of (P, d).• d is reflexive. For (j, z) ∈ P one has

k ≤ c(eZ(z), z) = c(eZ · p(j, z), z) = c(Tp(eP (j, z)), z) .

We now consider the pullback diagrams

Xz

!

s //W

π1

π2 // X

f

1 r // Pp// Z

in Set, with r the constant map to (j, z) and s the map (x′ 7−→ (j, x′)). Since for everyw ∈ TW with Tπ1(w) = eP (j, z) and every x ∈ X with f(x) = z one has

Tπ1(w) = eP · r·!(x) = Tr(e1(!(X))) ,

and since T satisfies BC, there is x ∈ TXz with T !(x ) = e1(!(x)) and Ts(x ) = w .Considering T (π2 · s) as a subset inclusion, one obtains by (T,V)-functoriality of j :(Xz, az) −→ (Y, b)

a(Tπ2(w), x) = a(Tπ2 · Ts(x ), x)= a(x , x)≤ b(Tj(x ), j(x))= b(T (ε · s)(x ), j(x)) = b(Tε(w), j(x)) .


Consequently, k ≤ a(Tπ2(w), x) −→ b(Tε(w), j(x)) which implies k ≤ d(eP (j, z), (j, z)).• p and ε are (T,V)-functors. Since d(p, (j, z)) ≤ c(Tp(p), z) by definition of d, p is amorphism in (T,V)-Gph. Denoting the structure of W by m, for all w ∈ TW and (j, x) ∈ W ,one has by definition of d

m(w , (j, x)) = d(Tπ1(w), (j, f(x))) ∧ a(Tπ2(w), x)≤ (a(Tπ2(w), x) −→ b(Tε(w), j(x))) ∧ a(Tπ2(w), x)≤ b(Tε(w), j(x)) = b(Tε(w), ε(j, x)) .

• (P, d) satisfies the universal property. For morphisms q : (Q, l) −→ (Z, c), g : (Q×ZX,n) −→(Y, b) in (T,V)-Gph, we must show that the map

h : Q −→ P

t 7−→ (jt, q(t))with

jt : Xq(t) −→ Y

x 7−→ g(t, x)is a well-defined (T,V)-functor. For t ∈ Q consider the pullback diagrams

Xq(t)

!

u // U

π1

π2 // X

f

1 t // Qq// Z

in Set, with U = Q×ZX and u the map (x 7−→ (t, x)). Since k = > one has t : (1,>) −→ (Q, l).Consequently, u : (Xq(t), aq(t)) −→ (U, n) and

j = g · u : (Xq(t), aq(t)) −→ (Y, b)are (T,V)-functors. Hence, (jt, q(t)) ∈ P . Finally, for q ∈ TQ, t ∈ Q, we first note

l(q , t) ≤ c(Tq(q), q(t)) = c(Tp(Th(q)), q(t)) .Exploiting BC for the left pullback diagram in

U

π1

h×1X //W

π1

π2 // X

f

Qh // P

p// Z

for every w ∈ TW with Tπ1(w) = Th(q) we obtain u ∈ TU with T π1(u) = q andT (h× 1X)(u) = w , hence T π2(u) = Tπ2(w). Now, for every x ∈ Xq(t) we have

l(q , t) ∧ a(Tπ2(w), x) = l(T π1(u), t) ∧ a(T π2(u), x)= n(u, (t, x))≤ b(Tg(u), g(t, x))= b(Tε(T (h× 1X)(u)), g(t, x)) = b(Tε(w), jt(x)) .

Consequently, l(q , t) ≤ a(Tπ2(w), x) −→ b(Tε(w), jt(x) and l(q , t) ≤ d(Th(q), h(t)) fol-lows.


4.5.2 Corollary. The category (T,V)-Gph is locally cartesian closed.

4.5.3 Examples.

(1) For T = I identically extended to V-Rel, every (T,V)-graph is unitary, Corollary 4.5.2gives that the category

V-Gph := (I,V)-Gph = (I,V)-UGph

is locally cartesian closed whenever V is cartesian closed with k = >. For V = 2, thisis the category

RRel = 2-Gph

of reflexive relations, that is, of sets endowed with a reflexive relation which, evenin the absence of transitivity, we write as ≤; morphisms are then “monotone” maps.Keeping the notation of 4.4, by Theorem 4.5.1 one can form the partial productP = P (Y, f) in RRel by endowing the set

P = (j, z) | z ∈ Z, j : Xz −→ Y monotone

with the relation

(j, z) ≤ (j′, z′) ⇐⇒ z ≤ z′ & ∀x ∈ Xz, x′ ∈ Xz′ (x ≤ x′ =⇒ j(x) ≤ j′(x′)) .

While p and ε are defined as for sets (see Example 4.4.1(1)), we note that O : RRel −→Set does not preserve partial products, not even exponentials.

(2) For general V (as in (1) above), Theorem 4.5.1 constructs partial products in V-Gphas follows. For f : (X, a) −→ (Z, c) and (Y, b) in V-Gph and

P = (j, z) | z ∈ Z, j : (Xz, az) −→ (Y, b) a V-functor ,

provide (P, d) = P (Y, f) with the structure

d((j, z), (j′, z′)) = c(z, z′) ∧ ∧x∈Xz ,x′∈Xz′ (a(x, x′) −→ b(j(x), j′(x′)) .

In particular, in the category

RNRel := Pmax-Gph

of reflexive numerical relations and non-expansive maps, the “metric” d on the partialproduct P of f over Y is given by

d((z, j), (z′, j′)) = maxc(z, z′), supx∈Xz ,x′∈Xz′ (a(x, x′) −→ b(j(x), j′(x′)

(see Example II.1.10.1(5)).


(3) Theorem 4.5.1 shows thatPsTop = (, 2)-Gph = (, 2)-RGph

is locally cartesian closed, with the ultrafilter convergence −→ onP = P (Y, f) = (j, z) | z ∈ Z, j : Xz −→ Y continuous

described as follows: p −→ (j, z) ∈ P if and only if• p(p) −→ z in Z (with p : P −→ Z the projection),• for all x ∈ Xz, w ∈ β(P ×Z X) with π1[w ] = p, π2[w ] −→ x in X impliesε[w ] −→ j(x) in Y (with ε : P ×Z X −→ Y the evaluation map). With ultrafilterstraded for filters, the same condition describes partial products in (F, 2)-Gph (seeExample 4.1.3(3)).

4.6 Local cartesian closedness of subcategories of (T,V)-Gph. Throughout thissubsection we assume that

• V is cartesian closed and integral;• T satisfies the Beck–Chevalley condition.

Let us first give sufficient conditions for cartesian and local cartesian closedness of reflectiveor coreflective full subcategories of a cartesian or locally cartesian closed category.4.6.1 Proposition. Let A be a full replete subcategory of a cartesian closed category Xwith finite products.

(1) If A is reflective, with the reflector preserving binary products, then A is cartesianclosed with exponentials (that is, internal hom-objects) formed as in X.

(2) If A is coreflective and closed under binary products in X, then A is cartesian closedwith exponentials in A obtained by coreflection of exponentials formed in X.

Proof.

(1) For A-objects A and B, let BA be the exponential formed in X with counit ε : BA×A −→B, and let ρ : BA −→ R(BA) be the reflection into A. Since R(BA ×A) ∼= R(BA)×Aby hypothesis, the morphism ρ × 1A : BA × A −→ R(BA) × A serves as a reflectioninto A. Hence, ε factors uniquely as ε = ε · (ρ× 1A). The mate s of ε that makes

A×BA ε // B

A×R(BA)

1A×s

OO

ε

99

commute must be inverse to ρ. Consequently, BA ∼= R(BA) lies in A and serves therein the same capacity as in X.


(2) The claim is straightforward.

4.6.2 Corollary. Let A be a full replete subcategory of a locally cartesian closed categoryX with pullbacks.

(1) If A is reflective, with the reflector preserving pullbacks, then A is locally cartesianclosed with partial products formed as in X.

(2) If A is coreflective and closed under pullbacks in X, then A is locally cartesian closedwith partial products in A obtained by coreflection of partial products formed in X.

Proof. Apply Proposition 4.6.1 to the reflective or coreflective subcategory A/B of X/B(for B ∈ ob A).

4.6.3 Corollary. If the functor R : (T,V)-Gph −→ (T,V)-RGph, sending (X, a) to (X, a ·T1X), preserves binary products, then (T,V)-RGph is cartesian closed with exponentialsformed as in (T,V)-Gph. Similarly, if R : (T,V)-Gph −→ (T,V)-RGph preserves pullbacks,then (T,V)-RGph is locally cartesian closed with partial products formed as in (T,V)-Gph.

Proof. By Corollary 4.1.5, R is the reflector of (T,V)-Gph onto (T,V)-RGph. Hence, Propo-sition 4.6.1 and Corollary 4.6.2 may be applied.

4.6.4 Example. With the filter monad F extended by F (as in Example 1.10.3.(4)), thereflector R : (F, 2)-Gph −→ (F, 2)-RGph extends the reflexive filter convergence relation −→on a set X to

a −→ x ⇐⇒ ∃b ∈ FX (a ⊇ b & b −→ x)

for all x ∈ X, a ∈ FX. It is not difficult to show that R preserves finite limits. Hence,(F, 2)-RGph is locally cartesian closed with partial products formed as in (F, 2)-Gph.

In addition to the two general hypotheses on T and V stated at the beginning of 4.6, wenow assume that

• T is associative;

• T (f ) = T1X · (Tf) for all f : X −→ Y (see (4.3.i)).

Under these hypotheses, not only coproduct sinks in (T,V)-RGph are universal (as provedin Theorem 4.3.9), but so are all O-final epi-sinks, where O : (T,V)-RGph −→ Set is theforgetful functor:

4.6.5 Proposition. O-final epi-sinks are stable under pullback in (T,V)-RGph.


Proof. For an O-final epi-sink (gi : (Yi, bi) −→ (Z, c))i∈I in (T,V)-RGph, the formula given inCorollary 4.1.5 simplifies under the additional hypotheses to c = ∨

i∈I gi · bi · (Tgi). Whenfor f : (X, a) −→ (Z, c) we form the pullbacks

(Pi, di)pi //

qi

(X, a)f

(Yi, bi)gi // (Z, c)

we have di = (pi ·a ·Tpi)∧ (qi · bi ·Tqi) for all i ∈ I and need to show a = ∨i∈I pi ·di · (Tpi):∨

i∈I pi · di · (Tpi) = ∨i∈I pi ·

((pi · a · Tpi) ∧ (qi · bi · Tqi)

)· (Tpi)

= ∨i∈I a ∧ (pi · qi · bi · Tqi · (Tpi)) (4.3.7)

= a ∧ ∨i∈I f · gi · bi · (Tgi) · Tf (BC)= a ∧ f · (∨i∈I gi · bi · (Tgi)) · Tf= a ∧ (f · c · Tf) = a .

4.6.6 Corollary. Colimits are stable under pullback in (T,V)-RGph.

Proof. Colimit sinks in (T,V)-RGph are those O-final epi-sinks which are colimits in Set.

For f : (X, a) −→ (Z, c) in (T,V)-RGph we now consider the commutative diagram

(T,V)-RGph/(Z, c)OZ

f∗// (T,V)-RGph/(X, a)

OX

domX // (T,V)-RGphO

Set/Z f∗// Set/X domX // Set .

(4.6.i)

Since O is topological, OX is also topological (and so is OZ), with OX-final structuresformed like O-final structures, that is, the upper row domain functor domX (see 4.4)transforms OX-final structures into O-final sinks (see Exercise 4.H). More importantly, byProposition 4.6.5, the upper row pullback functor f ∗ (see Proposition 4.4.3) sends OZ-finalepi-sinks to OX-final epi-sinks. In particular, f ∗ preserves all colimits. In fact, f ∗ has aright adjoint by virtue of the Special Adjoint Functor Theorem (see Exercise II.5.J).

4.6.7 Theorem. Suppose that T is associative and T (f ) = T1X ·(Tf) for all f : X −→ Y .Then the category (T,V)-RGph is locally cartesian closed and therefore has all partialproducts.

Proof. We must verify that the sufficient conditions of the dual of the Special AdjointFunctor Theorem are satisfied by the functor f ∗. But the topological category (T,V)-RGphover Set inherits the relevant properties from Set and passes them on to its comma categories.In fact, for every category C and C-object X, one easily verifies the following facts:

(1) if C is locally small, then C/X is locally small (because domX : C/X −→ C is faithful);


(2) if C is wellpowered, then C/X is wellpowered (because domX preserves and reflectsisomorphisms);

(3) if C is small-complete, then C/X is small-complete (see Exercise II.2.J);

(4) if C is locally small and G is generating in C, then the class of arrows ⋃G∈G C(G,X) isgenerating in C/X.

4.6.8 Remark. It is important to notice that in general f ∗ will not transform all OZ-finalsinks into OX-final sinks, but only those that are epic. Otherwise, we could have applied theTaut Lift Theorem (II.5.11) to (4.6.i), with the result that the lower-row right adjoints wouldhave been lifted along the vertical topological functors. But a partial product in (T,V)-Gphis generally not formed by endowing the partial product of the underlying Set-data witha (T,V)-graph structure. However, one can apply the Generalized Taut Lift Theorem(Exercise II.5.U) as an alternate method of proof of Theorem 4.6.7.

4.7 Interlude on subobject classifiers and partial-map classifiers. Throughoutthis subsection we consider a class M of morphisms in a category C with pullbacks suchthat:

• SplitMono C ⊆M ⊆ Mono C;

• M is closed under composition with isomorphisms;

• M is stable under pullback.

We adopt the notation of II.5.2, and write subX = subMX :=M/X for the full subcategoryof C/X whose objects lie in M, and assume throughout that the separated reflection(see II.1.3)

subX −→ sub'X := subX/ '

comes with a section in SET, so that it is a split epimorphism in SET. Since the pullbackfunctors f ∗ : C/Y −→ C/X restrict to f ∗ : subY −→ subX and satisfy (1X)∗ ∼= 1subX and(g · f)∗ ∼= f ∗g∗ (for f : X −→ Y , g : Y −→ Z in C), one has a functor

sub' : Cop −→ SET .

An M-subobject classifier is a C-object Ω that represents sub', so that C(−,Ω) ∼= sub'.Hence, Ω is characterized by the existence of maps

ψX : C(X,Ω) −→ subX

that are pseudo-natural in X (with respect to the order of subX) and are such that forevery subobject m : M −→ X there is a unique morphism f : X −→ Ω with ψX(f) ' m;


one calls f the characteristic morphism of m. Here is the hands-on characterization ofM-subobject classifiers:

4.7.1 Proposition. An object Ω is an M-subobject classifier in C if and only if there isan M-subobject t : 1 −→ Ω (with 1 a terminal object of C) such that, for any M-subobjectm : M −→ X there is a unique morphism f : X −→ Ω making

M //

m

1t

Xf// Ω

(4.7.i)

a pullback diagram.

Proof. For an M-subobject classifier Ω, one puts t := ψΩ(1Ω) : Z −→ Ω. Then, usingpseudo-naturality of ψ, to obtain the characteristic morphism f of m : M −→ X in M wecan chase 1Ω ∈ C(Ω,Ω) in two ways around the diagram

C(Ω,Ω) ψΩ //

C(f,Ω)

sub Ωf∗

C(X,Ω) ψX // Ω

yielding f ∗(t) ' m, so that (4.7.i) is a pullback diagram with 1 replaced by Z. We nowshow that Z must necessarily be terminal in C. Considering m = 1X for any object X, weobtain a pullback diagram

Xu //

1X

Z

t

Xf// Ω .

For any other v : X −→ Z,X v //

1X

Z

t

X t·v // Ωis a pullback diagram since t is a monomorphism. Hence, t · v is a characteristic morphismfor 1X , and we must have t · v = f = t · u, which implies u = v. The sufficiency condition isleft to the reader.

4.7.2 Remarks.

(1) The proof of Proposition 4.7.1 shows that a category C with anM-subobject classifiernecessarily has a terminal object and, in the presence of pullbacks, is therefore finitelycomplete. Furthermore, if C is locally small, C must be M-wellpowered.


(2) Since m in (4.7.i) is a pullback of the split monomorphism 1 −→ Ω, one necessarily hasM⊆ RegMono C if C has an M-subobject classifier; we speak of a regular-subobjectclassifier ifM = RegMono C, while subobject classifier refers to the caseM = Mono C.Hence, the existence of a subobject classifier in C forces Mono C = RegMono C.

4.7.3 Examples.

(1) Every two-element set 2 is a subobject classifier in Set, since 2 represents the powersetfunctor P of Set. Less trivially, every functor category SetDop has a subobject classifierfor a small category D (see [Johnstone, 1977], [Mac Lane and Moerdijk, 1994]).

(2) If U : A −→ X is topological with right adjoint I, and Ω is a regular-subobject classifierin X, then IΩ is a regular-subobject classifier in A, see Exercise 4.J. In particular, atwo-element set will assume that role both in Ord and in Top when provided with itsrespective indiscrete structure.

An M-partial map from X to Z in C consists of an M-subobject m : M −→ X and amorphism g : M −→ Z. The (possibly large) class

part(X,Z) = partM(X,Z)

of all M-partial maps from X to Z is ordered by

(m, g) ≤ (m′, g′) ⇐⇒ ∃h (m′ · h = m & g′ · h = g) .

For f : X −→ Y in C there is now a pullback functor

f ∗ : part(Y, Z) −→ part(X,Z) , (n, g) 7−→ (f ∗(n), g · fn) ,

(with n : N −→ Y and g : N −→ Z) defined by the pullback diagram

· fn//

f∗(n)

N

n

Xf// Y .

As for sub, when taking isomorphism classes of M-partial maps, part(−, Z) becomes afunctor

part'(−, Z) : Cop −→ SET .

Note that for a terminal object Z = 1, the functor part'(−, Z) is isomorphic to sub'.An M-partial-map classifier for Z is an object Z∗ that represents part'(−, Z), so thatC(−, Z∗) ∼= part'(−, Z). Hence, Z∗ is characterized by the existence of maps

φX : C(X,Z∗) −→ part(X,Z)


that are pseudo-natural in X and are such that for all (m, g) ∈ part(X,Z), there is a uniquemorphism f : X −→ Z∗ with φX(f) ' (m, g).4.7.4 Proposition. Let Z be an object in C. An object Z∗ is an M-partial-map classifierfor Z if and only if there is a morphism tZ : Z −→ Z∗ in M such that, for any M-partialmap (m, g) from X to Z, there is a unique morphism f : X −→ Z∗ making

Mg//

m

Z

tZ

Xf// Z∗

(4.7.ii)

a pullback diagram.

Proof. If Z∗ is an M-partial-map classifier for Z, one sets

(tZ : WZ −→ Z∗, sZ : WZ −→ Z) := φZ∗(1Z∗) .

Given (m, g) from X to Z, let f : X −→ Z∗ be such that φX ' (m, g). Chasing 1Z∗ aroundthe pseudo-naturality diagram

C(Z∗, Z∗) φZ∗ //

C(f,Z∗)

part(Z∗, Z)f∗

C(X,Z∗) φX // part(X,Z)

one obtains (m, g) ' (f ∗(tZ), sZ · ftZ ), as in the diagram

·ftZ //

f∗(tZ)

WZ

tZ

sZ // Z

Xf// Z∗ .

A comparison with (4.7.ii) shows that if sZ is an isomorphism, then the necessity part ofthe statement is proved. When we specialize (m, g) = (1Z , 1Z), one obtains sZ · ftZ forthe corresponding morphism f : Z −→ Z∗. Furthermore, chasing f ∈ C(Z,Z∗) both waysthrough

C(Z∗, Z∗) φZ //

C(sZ ,Z∗)

part(Z,Z)s∗Z

C(WZ , Z∗)

φWZ // part(WZ , Z)

we obtain φWZ(f · sZ) ' (1WZ

, sZ). Since φWZ(tZ) ' (1WZ

, sZ) trivially, we concludef · sZ ' tZ . Hence, the split epimorphism sZ is a monomorphism, and therefore anisomorphism.The proof of the sufficiency of the stated condition is left to the reader.


As for M-subobject classifiers, the term partial-map classifier refers to the case M =Mono C, and the term regular-partial-map classifier to the caseM = RegMono C. Actually,as we shall see below in 4.7.6, the general hypotheses on M leave no choice for M whenthere is an M-partial-map classifier.

4.7.5 Examples.

(1) A partial-map classifier for Z in Set is Z∗ = Z + 1. More generally, such classifiersexist in every functor category SetCop with C small (see [Johnstone, 1977], [Mac Laneand Moerdijk, 1994]).

(2) Unlike subobject classifiers, partial-map classifiers may not be lifted along topologicalfunctors. Indeed, the category Ord fails to have a regular-partial-map classifier forthe two-chain 2 = 0 < 1, see Exercise 4.K. A similar statement holds for Top.Top does have M-partial-map classifiers for all spaces when M is the class of openembeddings or the class of closed embeddings. However, for these choices of M thegeneral hypothesis that M contain all split monomorphisms fails, while all others aresatisfied.

4.7.6 Corollary. In a finitely complete category C with an M-partial-map classifier, Mis precisely the class of equalizers in C.

Proof. Setting g = m ∈M in (4.7.ii), one sees that m is an equalizer of tX and f for somef : X −→ X∗. Conversely, if m : M −→ X is the equalizer of some pair g, h : X −→ Y ofmorphisms, then m is a pullback of the split monomorphism 〈1X , h〉 along 〈1X , g〉 and musttherefore be a morphism in M:

M m //

m

X

〈1X ,h〉

X〈1X ,g〉

// X × Y .

Consequently, when C has anM-partial-map classifer and cokernel pairs of regular monomor-phisms, so that every regular monomorphism is an equalizer (Exercise II.2.D), thenM = RegMono C.

4.7.7 Theorem. For a finitely complete category C, the following assertions are equivalent:

(i) C has partial products and an M-subobject classifier;

(ii) C is locally cartesian closed and has an M-subobject classifier;

(iii) C is cartesian closed and has M-partial-map classifiers for all objects.


Proof. The equivalence of (i) and (ii) follows from Proposition 4.4.3.For (i) =⇒ (iii) one forms the partial product P = P (Z, t : 1 −→ Ω) and considers thediagram

Z P ×Ω 1εoo //

τ

1t

Z

1Z\\

〈h,!〉== 44

1Z

Pp

// Ω

Z

h

<<

f

44

where f is the characteristic morphism for 1Z . With the induced morphism h, the evaluationmorphism ε becomes a split epimorphism. In addition, the partial-product propertyguarantees h · ε = τ , which makes ε a monomorphism since τ is one. Hence ε is anisomorphism and we may assume ε = 1Z . Now it is easy to see that τ : Z −→ P classifiesM-partial maps to Z.For (iii) =⇒ (i) we prove that f ∗ : C/Z −→ C/X has a right adjoint (for every f : X −→ Zin C, see 4.4.3) by factoring f as

f = ( X 〈f,1X〉// Z ×X π1 // Z )

and showing that both π∗1 and 〈f, 1X〉∗ have right adjoints. Since π∗1 makes the diagram

C/Zπ∗1 //

domZ

C/Z ×XdomZ×X

C (−)×X// C

(with π∗1(W,u) = (W × X, u × 1X)) commute, and (−) × X is left adjoint by cartesianclosedness of C, the functor π∗1 is also left adjoint by 4.4.3.The split monomorphism m := 〈f, 1X〉 : X −→ Z ×X lies in M by our general hypothesison M, so the M-partial morphism (m, 1X) is represented by h : Z ×X −→ X∗. For (W,w)in C/X we construct the value m∗(w), to obtain an adjunction m∗ a m∗ : C/X −→ C/Z×X,as follows. For g : W ∗ −→ X∗ representing the M-partial morphism (tW , w), one obtainsm∗(w) as a pullback of g along h, and the counit εw : m∗(m∗(w)) −→ w by the pullbackproperty of W = X ×X∗ W ∗:

X ×Z×X Y

m∗(m∗(w))~~

εw //Ww

~~

tW

X

1X // X

tX

Ym∗(w)

//W ∗

g

Z ×X h // X∗


That εw assumes the alleged role as a counit is an easy diagrammatic exercise.

4.7.8 Examples. Set satisfies the equivalent conditions of Theorem 4.7.7. Ord is cartesianclosed and has a regular-subobject classifier, but not all regular-partial-map classifiers. Thecategory Set∗ of pointed sets and pointed maps has all partial-map classifiers but fails tobe cartesian closed (Exercise 4.K), where a pointed set (X, x0) is a set X with x0 ∈ X,and a pointed map f : (X, x0) −→ (Y, y0) is a map f : X −→ Y with f(x0) = y0. Top has aregular-subobject classifier but none of the other properties of 4.7.7, and Cat is cartesianclosed but not locally so, and does not have a regular-subobject classifier.

The following Theorem gives in particular a necessary and sufficient condition for a topo-logical category over Set to have regular-partial-map classifiers with underlying sets as inSet.

4.7.9 Theorem. Let U : A −→ X be a topological functor, with X finitely complete. Thefollowing assertions are equivalent:

(i) A has regular-partial-map classifiers and U preserves them;

(ii) X has regular-partial-map classifiers, and U -final sinks in A are stable under pullbackalong regular monomorphisms.

Proof. (ii) =⇒ (i): In order to construct a regular-partial-map classifier for B in A, onetakes a regular-partial-map classifier tZ : Z −→ Z∗ for Z = UB in X and considers the(possibly large) family of all regular partial maps

(mi : Mi −→ Ai, gi : Mi −→ B)i∈I (4.7.iii)

in A with codomain B. Since a morphism in A is a regular monomorphism if and only if itis U -initial and its U -image is a regular monomorphism in X (Exercise II.5.D), for everyi ∈ I one obtains a unique morphism hi : UAi −→ Z∗ in X which makes

UMi

Umi

Ugi // Z

tZ

UAihi // Z∗

(4.7.iv)

a pullback diagram. We let (fi : Ai −→ B∗)i∈I be a U -final lifting of (hi)i∈I . Since thefamily (4.7.iii) contains in particular (1B, 1B), there is j ∈ I with mj = gj = 1B and hj = tZ .We claim that sB := fj : B −→ B∗ is a regular-partial-map classifier. Since UsB = tZ is aregular mono, we must show that sB is U -initial. Let n : C −→ B∗ be a U -initial lifting oftZ : Z −→ UB∗. Then the diagrams (4.7.iv) with hi = Ufi show that there are morphismsgi : Mi −→ C with Ugi = Ugi, by U -initiality of n, and the U -initiality of every mi shows


that allMi

mi

gi // C

n

Aifi // B∗

(4.7.v)

are pullback diagrams in A. By hypothesis (ii) the sink (gi)i∈I is U -final, so that Ugi = Ugifor all i ∈ I yields C ≤ B in U−1Z, with B ≤ C holding trivially by U -initiality of n. Itfollows that sB is a regular monomorphism, and the diagrams (4.7.v) show that it has therequired universal property in A. Furthermore, UsB = tZ is a regular-partial-map classifierin X.(i) =⇒ (ii): For the existence of regular-partial-map classifiers in X, see Exercise 4.N. Nowconsider a family of pullback diagrams

Mi

mi

gi // N

n

Aifi // B

(i ∈ I) in A, with n a regular monomorphism and (fi)i∈I a U -final morphism, and considerk : UN −→ UC in X such that k · Ugi = Uhi with hi : Mi −→ C in A for all i ∈ I. Inorder to confirm U -finality of (gi)i∈I we must lift k to a morphism N −→ C in A. Butwith sC a regular-partial-map classifier for C in A, for every i ∈ I there is a uniquemorphism ti : Ai −→ C∗ in A exhibiting mi as a pullback of sC along ti which restricts to hi.Furthermore, since U preserves this classifier, there is a unique morphism j : UB −→ UC∗

in X exhibiting Un as pullback of UsC along j which restricts to k. The diagram

UC

UsC

UMiUgi //

Uhi11

Umi

UN

k

;;

Un

UAiUfi //

Uti --

UBj

##

UC∗

makes it obvious that j lifts to an A-morphism B −→ C∗ by U -finality of (fi)i∈I , and that kthen lifts to an A-morphism N −→ C by U -initiality of sC .

4.8 The quasitopos (T,V)-Gph. A category C is a quasitopos if

• C is finitely complete and finitely cocomplete;


• C is locally cartesian closed;• C has a regular-subobject classifier.

A quasitopos is a topos if it has a subobject classifier. Hence, a topos is a quasitopos inwhich all monomorphisms are regular. Set and, more generally, every functor categorySetDop (with D small) is a topos.As a topological category over Set, the category (T,V)-Gph is small-complete and small-cocomplete, and Theorem 4.5.1 gives sufficient conditions on T and V for (T,V)-Gph to belocally cartesian closed. The existence of a regular-subobject classifier is easily established,not only in (T,V)-Gph but also in its relevant subcategories.4.8.1 Proposition. The categories (T,V)-Gph, (T,V)-RGph, (T,V)-UGph and (T,V)-Catall have a regular-subobject classifier.Proof. Simply put the indiscrete structure on the subobject classifier 2 of Set (see Exer-cise 4.J).

As a consequence, (2,>) is a regular-subobject classifier in all four categories.4.8.2 Corollary. The category (T,V)-Gph is a quasitopos provided that V is cartesian closedand integral, and T satisfies BC. If, in addition, T is associative and T (f ) = T1X · (Tf)for all f : X −→ Y , then (T,V)-RGph is also a quasitopos.Proof. The statement follows immediately from the previous discussion, with Theorems 4.5.1and 4.6.7.

Theorem 4.7.9 applied to O : (T,V)-RGph −→ Set provides simpler conditions for (T,V)-RGphto have regular-partial-map classifiers.4.8.3 Proposition. For T taut, let T satisfy T t = Tt · T1Z for every injective functiont : Y −→ Z. Then (T,V)-RGph has regular-partial-map classifiers preserved by O.Proof. We must show that O-final sinks in (T,V)-RGph are hereditary, that is: stable underpullback along embeddings. Hence, consider a sink (fi : (Xi, ai) −→ (Y, b)) in (T,V)-RGphwith

b = 1]Y ∨∨i∈I

fi · ai · T (f i ) = eY · T1Y ∨∨

i∈Ifi · ai · (Tfi) · T1Y

(see Proposition 4.1.4), an embedding t : (Z, c) → (Y, b) with c = t · b · Tt, and letfi : (f−1

i (Z), di) −→ (Z, c) be the restriction of fi with di = si · ai · Tsi, si · f−1i (Z) → Xi.

From t · eY = eZ · (Tt) one obtains eZ = t · eY · Tt since Tt is injective. With thehypothesis on T and T ,

c = t · eY · T1Y · Tt ∨∨i∈I t

· fi · ai · (Tfi) · T1Y · Tt= t · eY · Tt · T1Z ∨

∨i∈I t

· fi · ai · (Tfi) · Tt · T1Z= eZ · T1Z ∨

∨i∈I fi · si · ai · Tsi · (T fi) · T1Z

= 1]Z ∨∨i∈I fi · di · T (f i ) ,


so that (fi)i∈I is O-final, as claimed.

4.8.4 Remark. Examining the proof of Theorem 4.7.9, one sees that the hypothesisT t = Tt · T1Z is used only for t : Z → Z∗ = Z ∪ ? with ? /∈ Z. Since T t = T1Z∗ · Ttalways holds, one can in fact show that it suffices that T satisfy

x ∈ TZ , ⊥ < T1Z∗(x , y ) =⇒ y ∈ TZ (4.8.i)for Proposition 4.8.3 to hold true.

4.8.5 Examples.(1) The categories RRel, RNRel, PsTop, PsApp are quasitopoi (see Examples 4.1.3 and 4.5.3),

but none of them is a topos (since monomorphisms need not be regular in these cate-gories).

(2) Condition (4.8.i) fails for the lax extension F of the monad F, but it is satisfied for thelax extension F of 1.10.3(4). Since (F, F, 2)-RGph is cartesian closed (Example 4.6.4),it is a quasitopos.

(3) PrTop ∼= (F, 2)-UGph (see 4.1.3(3)) has a regular-partial-map classifier which may beconstructed via Theorem 4.7.9 or Proposition 4.8.3, but PrTop is not cartesian closed(see [Herrlich et al., 1991]).

4.9 Final density of (T,V)-Cat in (T,V)-Gph. Let us finally show that (T,V)-Gph is“not too big” an extension of (T,V)-Cat, in the sense that every (T,V)-graph is the codomainof a small O-final sink with domains in (T,V)-Cat, where O : (T,V)-Gph −→ Set is theforgetful functor. Here we assume that the functor T of the monad T preserves disjointness,so that X ∩ Y = ∅ implies TX ∩ TY = ∅; more precisely,

∅ //

TX

TY // T (X + Y )

is a pullback diagram for all sets X and Y . This is certainly the case when T∅ = ∅ and Tis taut, or when T preserves binary coproducts.We commence by constructing a (T,V)-category structure c on the set

Z = X + 1depending on a given set X and elements x ∈ TX and v ∈ V . Writing 1 = ? and usingthe disjointness hypothesis we define c = cx ,v by

c(z, z) =

k if z = eZ(z),v if z = x and z = ?,⊥ otherwise,


for all z ∈ Z, z ∈ TZ. We observe that the V-relation c : TZ −→7 Z has finite fibres, that is,

c(z) = z ∈ TZ | ⊥ < c(z, z)

is finite for all z ∈ Z. We say that the lax natural transformation e : T −→7 1 is finitelystrict if

TX //Tr //

_eX

TY_ eY

X r // Y

commutes for every V-relation r with finite fibres (that is, for those r : X −→7 Y withr(y) = x ∈ X | ⊥ < r(x, y) finite for all y ∈ Y ).

4.9.1 Proposition. Suppose that T preserves disjointness, T is flat, and e is finitelystrict. Then

(X + 1, cx ,v)is a (T,V)-category for any set X and all x ∈ TX, v ∈ V.

Proof. As c is reflexive by definition, we must only show transitivity:

T c(Z, z)⊗ c(z, z) ≤ c(mZ(Z), z)

for all z ∈ Z = X + 1, z ∈ TZ, Z ∈ TTZ. With i : X → Z denoting the inclusion onetrivially has i · c = i · eZ , hence (Ti) · T c = (Ti) · (TeZ) since T is flat. This means inpointwise terms

T x(Z, z) =

k if Z = TeZ(z),⊥ otherwise,

whenever z ∈ TX. In proving transitivity of c, we can disregard cases where c(z, z) = ⊥ orT c(Z, z) = ⊥; hence, we are left to consider the following cases.Case 1: z ∈ X, z = eZ(z), Z = TeZ(z). Then all three terms appearing in the transitivitycondition are equal to k, and the inequality is actually an equality.Case 2: z = ?, z = x , Z = TeZ(z). Then, similarly to the previous case, one obtains

T c(Z, z)⊗ c(z, z) = k ⊗ v = v = c(mZ(Z), z) .

Case 3: z = ?, z = eZ(?). Since e is finitely strict, one has eZ · T c = c · eTZ , so that

T c(Z, z) =∨

w∈eTZ(Z) c(w , ?) .

As this value is assumed not to be ⊥, there is w ∈ TZ with ⊥ < c(w , ?) and eTZ(w) = Z,and for every such w one has c(mZ(Z), z) = c(w , z). Consequently,

T c(Z, z)⊗ c(z, z) = c(mZ(Z, z))⊗ k = c(mZ(Z), z) ,

which concludes the last case.


4.9.2 Theorem. Suppose that T preserves disjointness, T is flat, and e is finitely strict.Then (T,V)-Cat is finally dense in (T,V)-Gph.

Proof. For (X, a) a (T,V)-graph, x ∈ X and x ∈ TX, we claim that

fx ,x : (X + 1, cx ,a(x ,x)) −→ (X, a)

is a (T,V)-functor, with the map fx ,x induced by 1X : X −→ X and x : 1 −→ X. Indeed, forz ∈ Z = X + 1 and z ∈ TZ with ⊥ < c(z, z) (where c = cx ,a(x ,x)), one has z = eZ(z) orz = x , z = ? ∈ 1. In the former case,

c(z, z) = k ≤ a(eX(fx ,x(z)), fx ,x(z)) = a(Tfx ,x(z), fx ,x(z))

since a is reflexive, and in the latter case

c(z, z) = a(x , x) = a(Tfx ,x(z), fx ,x(z))

since fx ,x|X = 1X . Furthermore, this last equation also shows

a(x , x) = (fx ,x · cx ,a(x ,x)) · (Tfx ,x))(x , x) ,

so that the epi-sink (fx ,x)x ∈TX,x∈X must beO-final for the forgetful functorO : (T,V)-Gph −→Set (see Proposition 4.1.4).

Since the O-final sink just considered is small, we may state Theorem 4.9.2 more stronglyas follows.

4.9.3 Corollary. Suppose that T preserves disjointness, T is flat, and e is finitely strict.Then every (T,V)-graph is a quotient of a coproduct of (T,V)-categories. If, moreover, T isassociative, then every (T,V)-graph is a quotient of a (T,V)-category.

Proof. The small O-final epi-sink used in the proof of Theorem 4.9.2 factors through acoproduct of (T,V)-categories followed by a quotient map. That coproduct is again a(T,V)-category in case T is associative and flat by Theorem 4.3.3.

This Corollary can be applied to Examples 4.1.3 with the following consequence.

4.9.4 Corollary.c© Every pseudotopological space is a quotient of a topological space, andevery pseudo-approach space is a quotient of an approach space.

Exercises

4.A Initial sources in (T,V)-RGph and (T,V)-UGph. For the forgetful functorO : (T,V)-RGph −→Set, O-initial sources are characterized as in Proposition 3.1.1. Likewise for (T,V)-UGph −→Set.


4.B Sinks of nearly open morphisms. Let (fi : (Xi, ai) −→ (Y, b))i∈I be an epi-sink ofmorphisms in (T,V)-RGph, with all fi nearly open for all i ∈ I. Then (fi)i∈I is O-final,with O : (T,V)-RGph −→ Set.

4.C Taut functors.

(1) The Set-functor T with T∅ = ∅, and TX = 1 if X 6= ∅, preserves monomorphismsbut is not taut.

(2) The filter functor F is weakly terminal amongst all taut functors, that is: for everytaut functor T : Set −→ Set, there is a natural transformation α : T −→ F .

Hint. Since T preserves monomorphisms, one may assume TA ⊆ TX for A ⊆ X, and defineαX(x ) = A ⊆ X | x ∈ TA for x ∈ TX.

4.D Powerset graphs. With respect to both lax extensions of the powerset monad P and P(see Example 1.4.2(2)), show

(P, 2)-Gph = (P, 2)-RGph

and identify (P, 2)-UGph as a subcategory of Ord and Cls, respectively (see Examples 1.6.4).

4.E Extensive Categories. For a category C with small-indexed coproducts and pullbacks,the following statements are equivalent.

(i) C is extensive (that is, coproducts are universal and disjoint);

(ii) coproducts are universal, and for all small families (fi : Xi −→ Yi)i∈I , the diagrams

Xj//

fj

∐i∈I Xi∐

ifi

Yj //∐i∈I Yi

are pullback diagrams;

(iii) for all small families (Yi)i∈I of C-objects, the functor∏i∈I C/Yi −→ C/∐i∈I Yi , (fi : Xi −→ Yi)i∈I 7−→

∐i∈I fi

is an equivalence of categories.

4.F Ord is not locally cartesian closed. Ord is cartesian closed but not locally so: the partialproduct P = (0, 1, 0, 1 → 0, 1

2 , 1) does not exist in Ord.


4.G Top is not cartesian closed.

(1) Consider the topological spaces X, Y , W , Z whose non-empty open sets are illustratedby the diagram

X =0

1

q′

//0 a

b 1= Y

q

W = 0 1 // 0 12 1 = Z .

Conclude that quotient maps fail to be stable under pullback in Top, and that Topfails to be locally cartesian closed.

(2) Find a topological space X and a quotient map q : Y −→ Z such that q×1X : Y ×X −→Z ×X fails to be a quotient map. Conclude that X is not exponentiable.

Hint. Consider X = Q, the rational numbers, as a subspace of R.

4.H Slicing topological functors. Consider a topological functor U : A −→ X and an A-objectA. The induced functor UA : A/A −→ X/UA is also topological, with the domain functordomA : A/A −→ A sending UA-final sinks to U -final sinks.

4.I Pseudotopological spaces via filter convergence.c© With the help of the Axiom of Choice,condition (Ps) of Example 4.1.3(3) can equivalently be expressed as

(∀x ∈ βX ( x ⊇ a =⇒ x −→ y)) =⇒ (a −→ y)

for all a ∈ FX, y ∈ X. Use this to show that the categories PsTop and (F, 2)-RGphPs ofExample 4.1.3(3) are isomorphic. Describe the reflector PsTop −→ PrTop.

4.J Lifting regular-subobject classifiers. If I is right adjoint to the topological functorU : A −→ X, and if Ω is a regular-subobject classifier of X, then IΩ is a regular-subobjectclassifier of A (see Example 4.7.3(2) and Exercise II.5.D).

4.K Partial-map classifiers. The categories Ord and Top fail to have partial-map classifiers.The category Set∗ of pointed sets has partial-map classifiers but fails to be cartesian closed.

4.L Pretopological spaces via filter convergence. The multiplication and unit of the filtermonad F satisfy ∑ A = ⋂A and (X ⊇ A) =⇒ (∑X ⊇ ⋂A)

for all A ⊆ FX and X ∈ FFX. Hence, a (F, 2)-graph (X, a) is unitary if and only if it isright unitary and

ν(y) −→ y


holds for all y ∈ X, where ν(y) = ⋂a ∈ FX | a −→ y. Deduce that unitary graphs on aset X are in bijective correspondence with maps c : PX −→ PX satisfying

c(∅) = ∅ , A ⊆ c(A) , c(A ∪B) = c(A) ∪ c(B) ,

for all A,B ⊆ X. The categories PrTop and (F, 2)-UGph (see Example 4.1.3(3)) are thereforeisomorphic. Describe the reflector PrTop −→ Top.Hint. Consider the correspondence between maps ν : X −→ PPX and c : PX −→ PX givenby

c(A) = y ∈ X | B ∈ ν(y) =⇒ A∩B 6= ∅ and ν(y) =B ∈ PX | y ∈ c

(B)

,

where B = X \B denotes the complement of B in X.

4.M Partial-map classifiers in V-Cat. If k = > in V, then an object (Z, c) in V-Cat mayhave a regular-partial-map classifier only when Z is indiscrete, that is, when c(x, y) = >for all x, y ∈ Z. Hence, V-Cat is not a quasitopos, even when ⊗ = ∧ in V .

4.N Inheriting regular-partial-map classifiers. Let A be a finitely complete category withregular-partial-map classifiers sA : A −→ A∗.

(1) If B is a full, replete and reflective subcategory of A with monic reflection morphismsrA : A −→ RA and the reflector R preserving regular monomorphisms, then rB∗ · sB isa regular-partial-map classifier for B in B.

(2) If U : A −→ X is topological, then X has regular-partial-map classifiers.Hint. Recall Exercise II.5.D and apply (1) to the subcategory of indiscrete objects inA.

4.O Top versus App. There are full embeddings

Top // _

PrTop // _

PsTop _

App // PrApp // PsApp

all of which are reflective, and the vertical ones are also coreflective (via a(x , x) = 0 if andonly if x −→ x, as in 3.6). Describe the reflectors and coreflectors.


5 Representable lax algebras

In this subsection we show that the Set-monad T with its lax extension T to V-Rel maybe lifted, first to V-Cat and then to (T,V)-Cat. The respective Eilenberg–Moore categoriesover V-Cat and (T,V)-Cat are topological over the Eilenberg–Moore category SetT and,somewhat surprisingly, turn out to be isomorphic. These categories lead us to interestingclasses of so-called ordered and metric compact Hausdorff spaces, respectively.

5.1 The monad T on V-Cat. We continue to work with a quantale V that is assumedto be non-trivial as in 1.2. As shown in 3.3, every lax extension T = (T ,m, e) of a monadT = (T,m, e) from Set to V-Rel induces an order relation on TX, namely

x ≤ y ⇐⇒ k ≤ T1X(x , y ) , (5.1.i)

for x , y ∈ TX. In fact, the V-relation T1X : TX −→7 TX makes TX into a V-category(TX, T1X) since

1TX = T (1X) ≤ T1X and T1X · T1X ≤ T (1X · 1X) = T1X ,

and its underlying order is given by (5.1.i). More generally, the same computation showsthat, for every V-category (X, a0),

T (X, a0) = (TX, T a0) , (5.1.ii)

is a V-category. This construction is the object part of a functor

T : V-Cat −→ V-Cat

as Tf is a V-functor Tf : T (X, a0) −→ T (Y, b0), for every f : (X, a0) −→ (Y, b0). Furthermore,oplaxness of m and e with respect to T imply that

mX : TT (X, a0) −→ T (X, a0) and eX : (X, a0) −→ T (X, a0)

are V-functors, for every V-category (X, a0). Hence, the monad T = (T,m, e) on Set liftsto a monad on V-Cat, which we denote by T = (T,m, e) again.

5.1.1 Proposition. A lax extension T of a monad T on Set induces a 2-monad T on V-Catvia (5.1.ii), that is, a monad T = (T,m, e) with T a 2-functor.

Proof. We are left to show that T : V-Cat −→ V-Cat is a 2-functor. To this end, letf, g : (X, a0) −→ (Y, b0) be V-functors with f ≤ g, that is, 1X ≤ g · b0 · f . Then

1TX ≤ T1X ≤ T (g · b0 · f) = (Tg) · T b0 · Tf ,

that is Tf ≤ Tg.

5. REPRESENTABLE LAX ALGEBRAS 253

5.1.2 Examples.(1) The identity monad on Set extended to the identity monad on V-Rel yields the identity

monad on V-Cat.

(2) For the powerset monad P = (P, −,⋃) with its extensions P, P and P to Rel (seeExample 1.5.2(2) and 1.12), for an ordered set (X,≤) and A,B ⊆ X we find

A (P ≤) B ⇐⇒ A ⊆ ↓B ,

A (P ≤) B ⇐⇒ B ⊆ ↑A ,

A (P ≤) B ⇐⇒ A ⊆ ↓B & B ⊆ ↑A .

Less formally, A (P ≤) B whenever every element of A is covered by an element ofB, A (P ≤) B whenever every element of B covers an element of A, and A (P ≤) Bwhenever both conditions are satisfied. Note that in all three cases the order relationon PX need not be antisymmetric even if the order on X is so. For instance, forX = N with its natural order ≤, and E and O the sets of even and odd numbersrespectively, one has

O (P ≤) E and E (P ≤) O .

(3) For the list monad L = (L,m, e) and its Barr extension L to Rel, given by(x1, . . . , xn) Lr (y1, . . . , ym) ⇐⇒ n = m & x1 r y1 & . . . & xn r yn

for all (x1, . . . , xn) ∈ LX, (y1, . . . , ym) ∈ LY , and relations r : X −→7 Y (see also V.1.4),one has

L(X,≤) ∼=∐n∈N(X,≤)n

for every ordered set (X,≤). Under this identification Lf corresponds to ∐n∈N fn.

(4) With the lax extension P of P to P+-Rel given in Exercise 2.G, the powerset monadinduces a 2-monad P on P+-Cat ∼= Met. In particular, for a metric space (X, d), thesets (PX, Pd) become metric spaces with the (non-symmetric) Hausdorff metric

P d(A,B) = supx∈A infy∈B d(x, y) ,for all A,B ⊆ X.

(5) We now consider the ultrafilter monad on Set, together with the Barr-extension to Rel ∼= 2-Rel (see 1.10.3). Then, for an ordered set (X,≤), the order relation on βXis given by

x ≤ y ⇐⇒ ∀A ∈ x , B ∈ y ∃x ∈ A, y ∈ B (x ≤ y) ,for x , y ∈ βX.Finally, we consider the quantale P+ and the extension of the ultrafilter monad toP+-Rel described in 2.4. Here, for a metric space (X, d) and ultrafilters x , y ∈ βX,the distance between x and y is

supA∈x ,B∈y infx∈A,y∈B d(x, y) .


5.2 T-algebras in V-Cat. Let T be a monad lifted from Set to V-Cat as in 5.1. Followingthe notation of II.3.2, we denote the category of T-algebras and T-homomorphisms by

(V-Cat)T .

We also recall that there is an adjunction

(V-Cat)T >GT // V-Cat ,FToo

that (V-Cat)T is complete and that GT : (V-Cat)T −→ V-Cat preserves and creates limits.An object of (V-Cat)T can be described as a triple (X, a0, α) where (X, a0) is a V-categoryand α : TX −→ X is simultaneously a T-algebra structure on the set X and a V-functorα : T (X, a0) −→ (X, a0). For T-algebras (X, a0, α) and (Y, b0, β), a map f : X −→ Y isa T-homomorphism f : (X, a0, α) −→ (Y, b0, β) if and only if f : (X, a0) −→ (Y, b0) is aV-functor and f : (X,α) −→ (Y, β) is a morphism of T-algebras.

5.2.1 Examples.

(1) For the monad P on Ord = 2-Cat obtained from the extension P of P to Rel, anEilenberg–Moore algebra is an ordered set (X,≤) together with a monotone mapα : PX −→ X satisfying

α(α(A) | A ∈ A) = α(⋃A) and α(x) = x , (5.2.i)

for all x ∈ X and A ⊆ PX. We show that α(A) is a supremum of A ⊆ X in (X,≤).Clearly, x ≤ A in (PX,≤) for any x ∈ A, so that x ≤ α(A). Furthermore, for y ∈ Xwith x ≤ y for all x ∈ A, one has A ≤ y and therefore α(A) ≤ y. Hence, OrdPcan be equivalently described as the category of complete ordered sets with chosensuprema α : PX −→ X satisfying (5.2.i), and a morphism is a monotone map whichpreserves these chosen suprema. Of course, suprema are unique if X is separated,and the full subcategory of OrdP defined by the separated complete ordered sets isprecisely the category Sup (see II.2.1); moreover, Sup is a reflective subcategory ofOrdP.Dually, if we consider the extension P of P to Rel, the same reasoning applies withα : PX −→ X now representing chosen infima of an order relation on X. Hence, theEilenberg–Moore category OrdP has as objects complete ordered sets with choseninfima α : PX −→ X satisfying (5.2.i), and as morphisms those monotone maps whichpreserve the chosen infima. Furthermore, the category Inf of separated completeordered sets and inf-maps is a full reflective subcategory of OrdP.

(2) For the list monad L = (L,m, e) with its Barr extension to Rel (see Example 5.1.2(3)),the category OrdL is isomorphic to the category of ordered monoids, that is, of monoidsin the cartesian closed category Ord.


(3) An object in Ord (here we consider the Barr extension β of β to Rel) is a triple(X,≤, α) where (X,≤) is an ordered set and α : βX −→ X is the convergence relationof a compact Hausdorff topology on X (see 2.3); moreover, α : β(X,≤) −→ (X,≤) ismonotone. We write R ⊆ X×X for the graph of the order relation ≤ and π1 : R −→ X,π2 : R −→ X for the projection maps. Recall from 1.10 that, for x , y ∈ βX,

x (β(≤)) y ⇐⇒ ∃w ∈ βR (βπ1(w) = x & βπ2(w) = y ) .

Therefore, α : βX −→ X is monotone if and only if (α(βπ1(w)), α(βπ2(w))) ∈ R forevery ultrafilter w ∈ βR; that is, if and only if R is a closed subset of X ×X withrespect to the product topology.Hence, an Eilenberg–Moore algebra for the monad on Ord can be equivalently c©

described as a compact Hausdorff space X equipped with an order relation whosegraph is a closed subspace of X ×X; in other words, X is an “internal order” in Set.These spaces are known as ordered compact Hausdorff spaces. We write

OrdCompHaus

for the category of ordered compact Hausdorff spaces and morphisms. Every finiteordered set is an ordered compact Hausdorff space with the discrete topology. Inparticular, the ordered set 2 = 0 ≤ 1 can be seen as an ordered compact Hausdorffspace.

(4) c©Considering the extension β of β to P+-Rel of 2.4, one obtains a monad onMet = P+-Cat. The objects of Met are triples (X, d, α) where (X, d) is a metricspace and α is the convergence relation of a compact Hausdorff topology on X suchthat α : β(X, d) −→ (X, d) is non-expansive. We call these spaces metric compactHausdorff spaces, and denote the category of metric compact Hausdorff spaces andtheir morphisms by

MetCompHaus .The metric space [0,∞] with distance µ(u, v) = vu becomes a metric compact Haus-dorff space with ξ : β[0,∞] −→ [0,∞] sending u to supA∈u infu∈A u (see Exercise 5.J).

Besides sitting over V-Cat, the category (V-Cat)T also admits a forgetful functor

O : (V-Cat)T −→ SetT , (f : (X, a0, α) −→ (Y, b0, β)) 7−→ (f : (X,α) −→ (Y, β)) .

to SetT which can be seen as a lifting of the forgetful functor O : V-Cat −→ Set. Recall fromExample II.5.6.1(3) that every mono-source in SetT is initial with respect to the forgetfulfunctor GT : SetT −→ Set, and therefore a mono-source (fi : (X, a0, α) −→ (Xi, a0i, αi))i∈Iin (V-Cat)T is initial with respect to the forgetful functor (V-Cat)T −→ Set if and only if(fi : (X, a0) −→ (Xi, a0i))i∈I is initial with respect to the forgetful functor O : V-Cat −→ Set.

5.2.2 Proposition. The functor O : (V-Cat)T −→ SetT is topological.


Proof. For a family (fi : (X,α) −→ (Xi, αi))i∈I of morphisms of T-algebras where (Xi, a0i, αi)is in (V-Cat)T, let (X, a0) be the initial lift of (f : X −→ Xi)i∈I in V-Cat (see Theorem 3.1.3).Then (X, a0, α) is indeed an object of (V-Cat)T, since

α · T a0 = α · T (∧i∈I f i · a0i · fi)≤ ∧i∈I(α · T (f i · a0i · fi))= ∧

i∈I(α · (Tfi) · T a0i · Tfi)≤ ∧i∈I(f i · αi · T a0i · Tfi)≤ ∧i∈I(f i · a0i · fi · α)= (∧i∈I f i · a0i · fi) · α = a0 · α ,

with the penultimate equality holding by right adjointness of (−) · α (recall that α a α inV-Rel).

As a consequence, the functor O : (V-Cat)T −→ SetT has both a right adjoint and a leftadjoint given by the O-initial liftings of the empty family and the “all-family”, respectively(see Theorem II.5.9.1). In summary, in the commutative diagram

(V-Cat)T

O

GT // V-CatO

SetT GT // Setthe vertical arrows are topological and therefore have left and right adjoints; the horizontalarrows are monadic and therefore have left adjoints.For a flat extension T of T, the left adjoint (−)d : SetT −→ (V-Cat)T has a simple descriptionas (X,α)d = (X, 1X , α). If, moreover, V is integral so that k = > is the top elementof V, then (−)d : SetT −→ (V-Cat)T preserves limits and therefore has a left adjointπ0 : (V-Cat)T −→ SetT (see Exercise II.3.K and also Exercises 5.A and 5.B).

5.3 Comparison with lax algebras. The two structures a0 : X −→7 X and α : TX −→ X

of an object (X, a0, α) in (V-Cat)T can be combined to yield a single structure a = a0 · α :TX −→7 X turning X into a (T,V)-category (X, a). In fact, one has

a · eX = a0 · α · eX = a0 ≥ 1Xand

a · T a = a0 ·α · T (a0 ·α) = a0 ·α · T (a0) · T (α) ≤ a0 · a0 ·α · T (α) = a0 ·α ·mX = a ·mX .

Note also that the underlying V-category of (X, a) is (X, a0). Furthermore, every morphismf : (X, a0, α) −→ (Y, b0, β) in (V-Cat)T becomes a (T,V)-functor f : (X, a) −→ (Y, b) (whereb = β · b0). More generally, one has the following result.


5.3.1 Lemma. Let (X, a0, α), (Y, b0, β) be objects in (V-Cat)T with corresponding (T,V)-categories (X, a) and (Y, b), and let f : (X, a0) −→ (Y, b0) be a V-functor. Then f isa (T,V)-functor f : (X, a) −→ (Y, b) if and only if β · Tf ≤ f · α in V-Cat, that is,β · Tf(x ) ≤ f · α(x ) in the V-category (Y, b0), for all x ∈ TX.

Proof. First note that β·Tf ≤ f ·α in V-Cat is equivalent to b0·f ·α ≤ b0·β·Tf in V-Rel. Sincef ·a0 ≤ b0 ·f one obtains f ·a = f ·a0 ·α ≤ b0 ·f ·α ≤ b0 ·β ·Tf = b·Tf . Conversely, assumingf ·a ≤ b ·Tf , one derives f ·α ≤ b0 ·β ·Tf and therefore b0 ·f ·α ≤ b0 ·b0 ·β ·Tf ≤ b0 ·β ·Tf .

The previous constructions define a functor

K : (V-Cat)T −→ (T,V)-Cat , (X, a0, α) 7−→ (X, a0 · α) .

It is worth noting that K is a 2-functor.

5.3.2 Example. We apply the functor K to the Examples 5.2.1(3) and (4). The orderedcompact Hausdorff space 2 gives rise to the Sierpinski space 2 = 0, 1 having 0 asits only non-trivial open subset. The metric compact Hausdorff space ([0,∞], µ, ξ) (seeExample 5.2.1(4)) induces the approach space [0,∞] with convergence P+-relation λ :β[0,∞] −→7 [0,∞] defined by

λ(u, u) = u (supA∈u infv∈A v) .

5.3.3 Proposition. The functor K : (V-Cat)T −→ (T,V)-Cat sends initial sources withrespect to O : (V-Cat)T −→ SetT to initial sources with respect to O : (T,V)-Cat −→ Set.

Proof. Assume that (fi : (X, a0, α) −→ (Xi, a0i, αi))i∈I in (V-Cat)T is initial with respect toO : (V-Cat)T −→ SetT. Then

a = a0 · α = (∧i∈I f i · a0i · fi) · α= ∧

i∈I(f i · a0i · fi · α)= ∧

i∈I(f i · a0i · αi · Tfi) = ∧i∈I(f i · ai · Tfi) .

Consequently, applying the Taut Lift Theorem II.5.11.1 to the diagram

(V-Cat)T

O

K // (T,V)-CatO

SetTGT

// Set

we obtain the following corollary:

5.3.4 Corollary. The functor K : (V-Cat)T −→ (T,V)-Cat has a left adjoint M : (T,V)-Cat −→(V-Cat)T. In particular, K preserves limits.


Analyzing the proof of the Taut Lift Theorem II.5.11.1, we find that the underlying SetT-object of M(X, a) can be chosen as (TX,mX), for a (T,V)-category X = (X, a), and thenMf = Tf for a (T,V)-functor f . The unit X −→ KM(X) is given by eX : X −→ TX and,for every Y = (Y, b0, β) in (V-Cat)T, the counit MK(Y ) −→ Y is given by β : TY −→ Y . Wewill now give an explicit description of the V-category structure of M(X, a), under someconditions on the lax extension T .

5.3.5 Theorem. Assume that the lax extension T of T is associative. Then the left adjointof K : (V-Cat)T −→ (T,V)-Cat is given by

M : (T,V)-Cat −→ (V-Cat)T , (X, a) 7−→ (TX, T a ·mX ,mX) , f 7−→ Tf .

Proof. According to Theorem II.5.11.1 and Proposition 5.2.2, one hasM(X, a) = (TX, a,mX)where a is the initial V-category structure on TX with respect to the family of maps

TXTf−−→ TY

β−→ (Y, b0)

running over all (T,V)-functors f : (X, a) −→ (Y, b0 ·β) with (Y, b0, β) in (V-Cat)T. We showthat

a = T a ·mX .

Let (X, a) be a (T,V)-category. Firstly, (TX, T a ·mX) is a V-category since

T a ·mX ≥ T eX ·mX ≥ 1TX

and

T a ·mX · T a ·mX = T a · T T a ·mTX ·mX= T a · T T a · (TmX) ·mX≤ T a · T T a · T (mX) ·mX≤ T (a · T a ·mX) ·mX = T a ·mX .

Furthermore,

mX · T (T a ·mX) ≤ mX · T T a · TmX · T T1X≤ T a ·mTX · TmX · T T1X≤ T a ·mX ·mX · T T1X≤ T (a · T1X) ·mX ·mX = T a ·mX ·mX

since mX ·mTX = mX · TmX implies mTX · TmX ≤ mX ·mX ; therefore (TX, T a ·mX ,mX)is indeed an object of (V-Cat)T. The map eX : X −→ TX is actually a (T,V)-functor

eX : (X, a) −→ (TX, T a ·mX ·mX) ,


since T a · mX · mX · TeX = T a · mX ≥ T a · eTX ≥ eX · a; hence, the identity map1TX = mX · TeX on TX is a V-functor (TX, a) −→ (TX, T a ·mX), so that a ≤ T a ·mX .Secondly, for any (T,V)-functor f : (X, a) −→ (Y, b0 · β) with (Y, b0, β) in (V-Cat)T we findthat

β · Tf · T a ·mX ≤ β · T (f · a) ·mX≤ β · T b0 · Tβ · TTf ·mX≤ b0 · β · Tβ · TTf ·mX= b0 · β ·mY · TTf ·mX= b0 · β · Tf ·mX ·mX ≤ b0 · β · Tf .

Consequently, β · Tf : (TX, T a · mX) −→ (Y, b0) is a V-functor and we conclude thata ≥ T a ·mX .

5.3.6 Remark. Under the conditions of the theorem above, the unit eX : (X, a) −→KM(X, a) of M a K at (X, a) ∈ (T,V)-Cat is even O-initial (for the forgetful functorO : (T,V)-Cat −→ Set) since a is left unitary:

eX · T a ·mX ·mX · TeX = eX · T a ·mX = eX a = a .

The same computation shows that the (T,V)-structure a on X may be recovered froma = T a ·mX :

eX · a = eX · T a ·mX = a .

5.3.7 Examples.

(1) c©For the Barr extension β of the ultrafilter monad β to Rel and a topological space Xwith convergence relation a : βX −→7 X (see 2.2), the order (≤) = (βa ·mX) on βX isgiven by

x ≤ x ′ ⇐⇒ every closed set A ∈ x belongs to x ′

⇐⇒ every open set A ∈ x ′ belongs to x .

In fact, x ≤ x ′ if and only if there is some X ∈ ββX with mX(X ) = x and X (βa) x ′,that is,

∀A ∈ x (A ∈ X ) and ∀A ∈ X (a(A) ∈ x ′) ,

where A = a ∈ βX | A ∈ a (see Example 1.10.3(3)). Since a(A) = A is the closureof A, from x ≤ x ′ it follows that every closed set A ∈ x belongs to x ′. Conversely,this condition guarantees that the filter base

A | A ∈ x


is disjoint from the idealB ⊆ βX | a[B] /∈ x ′ ,

so that by Corollary II.1.13.5 there is an ultrafilter X ∈ ββX with mX(X ) = x andX (βa) x ′.

(2)c© Similarly, for the Barr extension β of β to P+-Rel and an approach space X withconvergence P+-relation a : βX −→7 X and corresponding approach distance δ : X −→7PX (see 2.4), the metric a = βa ·mX on βX can be written in terms of the approachdistance δ as

a(x , x ′) = infu ∈ [0,∞] | ∀A ∈ x (A(u) ∈ x ′) .To see this, let

v := a(x , x ′) = infβa(X , x ′) | X ∈ ββX (mX(X ) = x )

andw := infu ∈ [0,∞] | ∀A ∈ x (A(u) ∈ x ′) .

Sincev ≥ supA∈x

B∈x ′infa∈A

y∈Ba(a, y) ,

for every ε > 0, A ∈ x , and B ∈ x ′, there exist a ∈ A and y ∈ B with a(a, y) ≤ v+ ε;hence, δ(y, A) ≤ v + ε. Therefore, A(v+ε) ∩B 6= ∅, and we conclude that A(v+ε) ∈ x ′for every A ∈ x and ε > 0. This proves w ≤ v. For the reverse inequality, note thatA(w+ε) ∩B 6= ∅, for every ε > 0, A ∈ x and B ∈ x ′; this implies that

supA∈xB∈x ′

infa∈Ay∈B

a(a, y) ≤ w + ε .

Hence, by Lemma 2.4.2, there is some X ∈ ββX with

A | A ∈ x ⊆ X and βa(X , x ′) ≤ w + ε ,

so that v ≤ w.

5.4 The monad T on (T,V)-Cat. In order to be able to apply Theorem 5.3.5, throughoutthis subsection we assume that

• the lax extension T of T is associative.

The adjunction M a K of the previous subsection induces a monad on (T,V)-Cat, whosefunctor sends a (T,V)-category (X, a) to (TX, T a · mX · mX) and a (T,V)-functor f toTf . Since the multiplication and unit are given by m and e respectively, this monadconstitutes a lifting of the Set-monad T to (T,V)-Cat. We therefore denote this monadagain by T = (T,m, e).


The induced comparison functor is denoted by

K : (V-Cat)T −→ ((T,V)-Cat)T ;

here K(X, a0, α) = (X, a0 · α, α) and Kf = f .The algebraic functor Ae : (T,V)-Cat −→ V-Cat (defined by (X, a) 7−→ (X, a · eX), see 3.4)has a left adjoint

A : V-Cat −→ (T,V)-Cat , (X, a0) 7−→ (X, eX · T a0) ,

and composing the adjunction A a Ae with

((T,V)-Cat)T >GT // (T,V)-CatFToo

yields a new adjunction

((T,V)-Cat)T >GT0 // V-Cat .FT0

oo

A direct computation shows that F T0 a GT

0 induces the monad T on V-Cat, so we obtain thecomparison functor

Ae : ((T,V)-Cat)T −→ (V-Cat)T

that sends (X, a, α) to (X, a · eX , α). Clearly, AeK = 1, and that Ae and K are in factinverse to each other follows from a very pleasant property of the monad T on (T,V)-Cat,as we show next.

5.4.1 Theorem. Assume that the lax extension T of T is associative. Then the monad Ton (T,V)-Cat is of Kock–Zoberlein type.

Proof. Firstly, if f ≤ g for f, g : (X, a) −→ (Y, b) in (T,V)-Cat, then Tf ≤ Tg; indeed, itfollows from f · b ≤ g · b that

(Tf) · T b ·mY ·mY = T (f · b) ·mY ·mY ≤ T (g · b) ·mY ·mY = (Tg) · T b ·mY ·mY .

Secondly, we show that eTTX : TX −→ TTX is right adjoint to mX : TTX −→ TX, forevery (T,V)-category X = (X, a). Clearly, mX · eTX = 1TX . To see 1TTX ≤ eTX ·mX , werecall that the (T,V)-category structure on TX is given by T a ·mX ·mX , and the V-categorystructure on TTX by c = T a · TmX ·mTX , where a = T a ·mX . Using the fact that a ·mX

is unitary, we then compute:

mX · eTX · c = mX · eTX · T (a ·mX) ·mTX = mX · a ·mX ≥ mX ·mX ≥ 1TTX .

Hence, for all X ∈ TTX, we have c(X , eTX ·mX(X )) ≥ k; and therefore 1TTX ≤ eTX ·mX

in (T,V)-Cat.


For X = (X, a) in (T,V)-Cat, an Eilenberg–Moore structure α : TX −→ X in (T,V)-Catgives rise to an adjunction α a eX in (T,V)-Cat (see Proposition II.4.9.1). Since thealgebraic functor Ae : (T,V)-Cat −→ V-Cat is a 2-functor, the underlying V-functor

eX : Ae(X, a) = (X, a0) −→ AeT (X, a) = (TX, T a ·mX ·mX · eTX) = (TX, T a ·mX)

is right adjoint to α : (TX, T a ·mX) −→ (X, a0) as well. Hence, with Exercise 3.F appliedto V-Cat we obtain from the adjunction α a eX in V-Cat the equation

a0(α(x ), x) = T a ·mX(x , eX(x)) (5.4.i)

for all x ∈ TX and x ∈ X; this means a0 ·α = eX · T a ·mX = eX a = a. As a consequence,K and Ae are inverse to each other:

5.4.2 Corollary. If the lax extension T of T is associative, then ((T,V)-Cat)T ' (V-Cat)T.

The following diagram summarizes the situation exhibited so far:

((T,V)-Cat)TAe //'

GT

(V-Cat)T O //

GT

K

zz

⊥

K

oo SetT

GT

(T,V)-CatAe

//

M

::

V-CatO

// Set .

(5.4.ii)

One important consequence of Theorem 5.4.1 is that a (T,V)-category X admits up toequivalence at most one T-algebra structure α : TX −→ X, since necessarily α a eX in(T,V)-Cat (see Proposition II.4.9.1).

5.4.3 Definitions.

(1) A (T,V)-category X is representable if eX : X −→ TX has a left adjoint in (T,V)-Cat.Since T is of Kock–Zoberlein type, a (T,V)-functor α : TX −→ X is a left adjointof eX if and only if α · eX ' 1X . We hasten to remark that a representable (T,V)-category does not need to be a T-algebra since α a eX only implies α · eX ' 1X andα · Tα ' α ·mX . Of course, if X is separated, in the sense that its underlying order isseparated (see also V.2.1), then α is a T-algebra structure.

(2) A (T,V)-functor f : X −→ Y between representable (T,V)-categories X and Y , withleft adjoints α : TX −→ X and β : TY −→ Y respectively, is a pseudo-homomorphismwhenever

β · Tf ' f · α .

As before, if Y is separated, then we have equality above. We note that this conditiondoes not depend on the particular choice of the left adjoints α and β.


(3) The category of representable (T,V)-categories and pseudo-homomorphism will bedenoted as

(T,V)-RepCat .

(4) A (T,V)-category X is T-cocomplete if the V-functor eX : AeX −→ AeTX has a leftadjoint in V-Cat. While we will not elaborate on this notion in this book, here wenote that, by (5.4.i), X = (X, a) is T-cocomplete if and only if a can be written asa = a0 · α (with a0 = a · eX), for some map α : TX −→ X. Put differently, for everyx ∈ TX there must exist a tacitly chosen generic point x0 ∈ X so that

a(x , x) = a0(x0, x)

for all x ∈ X, and such a generic point is unique up to equivalence.

5.4.4 Proposition. Assume that the lax extension T of T is associative. Then a (T,V)-category X = (X, a) is representable if and only if X is T-cocomplete and a · T a = a ·mX .

Proof. Clearly, a representable (T,V)-category X = (X, a) is T-cocomplete and, witha = a0 · α,

a · T a = a0 · α · T a0 · Tα ≥ a0 · a0 · α · Tα = a0 · α ·mX = a ·mX .

Assume now that X = (X, a) is T-cocomplete and that a · T a = a · mX . Since T is ofKock–Zoberlein type, it is enough to verify that the map α : TX −→ X is a (T,V)-functor.In fact,

α · T a ·mX ·mX ≤ a · T a ·mX ·mX = a ·mX ·mX ·mX

≤ a ·mX = a · T a = a · T a0 · Tα = a · Tα .

5.5 Dualizing (T,V)-categories. In 1.3 we introduced the dual Xop of a V-categoryX = (X, a) as Xop = (X, a), for a commutative quantale V. This definition cannot beused directly for (T,V)-categories in general since a : X −→7 TX does not have the correcttype, and in this subsection we will discuss one possible way of dealing with this problem.Roughly speaking, we consider only those (T,V)-categories X = (X, a) where a = a0 · α,for some α : TX −→ X and where a0 = a · eX is the underlying V-category structure, thendualize just (X, a0) and combine the result with α; hence the structure of Xop is given bya0 · α. This defines, however, in general only a (T,V)-graph (see Proposition 5.5.3 below).We therefore consider the concept in this context.Throughout this subsection we assume that

• V is commutative.


A (T,V)-graph X = (X, a) is called dualizable whenever a0 = a · eX is transitive anda = a0 · α, for some map α : TX −→ X. For a dualizable (T,V)-graph X = (X, a), wewrite X0 to denote its underlying V-category X0 = (X, a0). We consider TX as a discreteV-category, so that α : TX −→ X0 is a V-functor. With this notation, a0 · α = α∗ (see 1.3)and, if α∗ = a = β∗, also α∗ = β∗ and therefore

a0 · α = (α∗) = (β∗) = a0 · β .

5.5.1 Lemma. Let X = (X, a) be a dualizable (T,V)-graph. Then (X, a0 ·α) is a dualizable(T,V)-graph as well, where the underlying V-category of (X, a0 · α) is (X0)op.

Proof. It suffices to show a0 = a0 ·α ·eX . From a = a0 ·α we infer a0 = a0 ·α ·eX = (α ·eX)∗,hence a0 = (α · eX)∗ and therefore a0 = a0 · α · eX .

The dual (T,V)-graph of a dualizable (T,V)-graph X = (X, a) is then defined as Xop =(X, a0 · α). This definition is independent of the choice of α : TX −→ X by the calculationgiven before Lemma 5.5.1.Every T-cocomplete (T,V)-category, seen as a (T,V)-graph, is dualizable. In particular,every V-category is dualizable and its dual in the sense above is just the usual dual. In 5.8we will see another important example of a dualizable (T,V)-graph.5.5.2 Lemma. For a (T,V)-functor f : (X, a) −→ (Y, b) between dualizable (T,V)-graphswith a = a0 · α and b = b0 · β, the map f : X −→ Y defines also a (T,V)-functorf op : Xop −→ Y op if and only if f · α ' β · Tf .

Proof. As for Lemma 5.3.1.

5.5.3 Proposition. Assume that the lax extension T of T is associative and let X = (X, a)be a T-cocomplete (T,V)-category. Then the following assertions are equivalent:

(i) the (T,V)-graph Xop is a (T,V)-category;

(ii) X satisfies a · T a = a ·mX ;

(iii) X is representable.

Proof. By Proposition 5.4.4, (iii) =⇒ (ii); and the implication (iii) =⇒ (i) can be shown asin 5.3. Assume now that Xop is a (T,V)-category. Since X is a (T,V)-category,

(α · Tα)∗ = a0 · α · Tα ≤ a0 · α · T a0 · Tα ≤ a0 · α ·mX = (α ·mX)∗;

similarly, since Xop is a (T,V)-category,

a0 · α · Tα ≤ a0 · α ·mX

and therefore (α ·Tα)∗ ≤ (α ·mX)∗. Consequently, (α ·Tα)∗ = (α ·mX)∗ by Exercise II.4.E,hence a · T a ≥ a ·mX . The inequality a · T a ≤ a ·mX we get from X being a (T,V)-category,therefore a · T a = a ·mX .


In conclusion, (f : X −→ Y ) 7−→ (f op : Xop −→ Y op) defines a functor

(−)op : (T,V)-RepCat −→ (T,V)-RepCat

which makes the diagram

(T,V)-RepCat(−)0

(−)op// (T,V)-RepCat

(−)0

V-Cat(−)op

// V-Cat

commutative.

5.6 The ultrafilter monad on Top. We consider the ultrafilter monad with its Barrextension to Rel, so that (, 2)-Cat ∼= Top. By 5.4, extends to a monad on Top, againdenoted by . For a topological space X, βX is the space of all ultrafilters of the setX where, for X ∈ ββX and x ∈ βX, one has X −→ x precisely when mX(X ) ≤ x (seeExamples 5.3.7), that is, precisely when for every open set A ∈ x , one has A = a ∈ βX |A ∈ a ∈ X . Therefore:

5.6.1 Lemma. c©The sets A (with A ⊆ X open) form a base of the topology of βX.

Important note. For a topological space X (usually assumed to be completely regular), thespace βX of ultrafilters on X should not be confused with the Cech–Stone compactificationof X. However, in the discussion following Proposition 5.6.2 we show how the Cech–Stonecompactification of X may be obtained from the space βX defined here.Diagram (5.4.ii) specializes to

Top ' //

OrdCompHaus //

CompHaus

Top // Ord // Set .

Here the category OrdCompHaus of ordered compact Hausdorff spaces and their morphisms(see Examples 5.2.1) appears as the category of Eilenberg–Moore algebras for the ultrafiltermonad c© on both Ord and Top.The inclusion functor CompHaus −→ Top factors as

CompHaus (−)d// Ord K // Top ,

so its left adjoint, usually called Cech–Stone compactification, can be taken as π0 ·M c©whereM a K (see Theorem 5.3.5) and π0 a (−)d. Recall that for an ordered compact Hausdorff


space X = (X,≤, α), the graph R ⊆ X ×X of the order relation ≤ is closed and thereforecompact Hausdorff. The reflection q : X −→ π0(X) is actually the coequalizer

Rp1//

p2// X

q// π0(X) (5.6.i)

in CompHaus which is usually different from the coequalizer in Set (see Exercise 5.A).However, if the graph E ⊆ X ×X of the equivalence relation induced by ≤ is closed, thenE is compact Hausdorff, and the coequalizer (5.6.i) in CompHaus can be constructed as inSet. This is certainly the case when ≤ is confluent: see Exercise 5.A.

5.6.2 Proposition.c© For a topological space X = (X, a), the order relation a = βa ·mXon βX is confluent if and only if for all disjoint closed subsets A,B ⊆ X there exist opensubsets U, V ⊆ X with A ⊆ U , B ⊆ V and U ∩ V = ∅.

Proof. Assume first that a = βa ·mX is confluent. Let A,B ⊆ X be closed subsets of Xwith the property that every open neighborhood of A intersects every open neighborhoodof B. Hence, by Corollary II.1.13.3, there is an ultrafilter x ∈ βX with

U ∩ V | U, V ∈ OX,A ⊆ U,B ⊆ V ⊆ x .

By definition, the filter generated by the filter base A is disjoint from the ideal generatedby U ∈ OX | U /∈ x , so Corollary II.1.13.5 guarantees the existence of an ultrafiltera ∈ βX with A ∈ a and x ≤ a. A similar argument yields an ultrafilter b ∈ βX withB ∈ b and x ≤ b . By hypothesis, there is some y ∈ βX with a ≤ y and b ≤ y . Hence,A ∈ y and B ∈ y , so that A ∩B 6= ∅.Assume now that the condition on disjoint closed subsets is satisfied. Let x , a, b ∈ βXwith x ≤ a and x ≤ b . Let A ∈ a and B ∈ b be closed. Then U ∈ x and V ∈ x , andtherefore U ∩V 6= ∅, for all open subsets U, V ⊆ X with A ⊆ U and B ⊆ V . Consequently,A ∩B 6= ∅. Hence, by Corollary II.1.13.3, there is an an ultrafilter y ∈ βX with

A ∩B | A,B ⊆ X closed, A ∈ a, B ∈ b ⊆ y ,

so that a ≤ y and b ≤ y .

A topological space with the property described in Proposition 5.6.2 is called normal. Bythe discussion preceding this Proposition, we have:

5.6.3 Corollary. The Cech–Stone compactification of a normal space X is isomorphicto the space of connected components of βXc© with respect to the order relation of Exam-ple 5.3.7(1).

A topological space X is -cocomplete if every ultrafilter x has a generic convergence pointx0 ∈ X, so that

x −→ x ⇐⇒ x0 ≤ x


for all x ∈ X, or equivalently, lim x = x0 where lim x denotes the set of limit points ofx . A subset of the form x, for x ∈ X, is a trivial example of an irreducible closed subset,that is, of a non-empty closed subset A ⊆ X with the property that, whenever A ⊆ A1 ∪A2for closed subsets A1, A2 ⊆ X, then A ⊆ A1 or A ⊆ A2. A topological space X is calledsober if every irreducible closed subset A ⊆ X is of the form A = x, for a unique x ∈ X;without this uniqueness requirement X is weakly sober . By Exercise 5.D, every irreducibleclosed subset A ⊆ X is the set A = lim x of limit points of some ultrafilter c©x ∈ βX. Hence,we obtain the following result.

5.6.4 Proposition. c©A topological space X is -cocomplete if and only if

(1) for every x ∈ βX, lim x is irreducible, and

(2) X is weakly sober.

Proof. The result follows immediately from the definitions.

Note that (1) implies in particular that X is compact.To better understand the condition a · βa = a ·mX , we introduce the way-below relation onthe lattice of open subsets of X: for A ⊆ X open, A is way-below B, written as A B, ifevery open cover of B has a finite sub-cover of A; that is, whenever B ⊆ ⋃i∈I Bi with opensubsets Bi, then there exists a finite subset K ⊆ I such that A ⊆ ⋃i∈K Bi. This relationcan be equivalently expressed in the language of ultrafilter convergence: A B preciselywhen every ultrafilter c©x on A has a limit point in B (Exercise 5.E). Furthermore, X iscalled core-compact if, for every point x ∈ X and every open neighborhood B of x, thereexists an open neighborhood A of x with A B.

5.6.5 Remark. We will see in Theorem 5.8.5 below that the core-compact spaces areprecisely the exponentiable objects (see II.4.4) in the category Top.

5.6.6 Proposition. c©A topological space X with convergence relation a : βX −→7 X iscore-compact if and only if a · βa = a ·mX .

Proof. Assume first that X is core-compact. Since a · βa ≤ a ·mX for every topologicalspace, we only have to show a · βa ≥ a ·mX . Let X ∈ ββX and x ∈ X where x is a limitpoint of mX(X ). Hence, B ∈ X for every open neighborhood B of x. Let now A be anopen neighborhood of x, and choose any open neighborhood B of x with B A. Then

lim−1A = x ∈ βX | lim x ∩ A 6= ∅ ⊇ B ∈ X ,

where lim x denotes the set of all limit points of x . Hence, the neighborhood filter f of x isdisjoint from the ideal j = A ⊆ X | lim−1A /∈ X; therefore, by Corollary II.1.13.5, thereexists some x ∈ βX disjoint from j (so that X βa x , see Example 1.10.3(3)) and x containsevery open neighborhood of x, that is, x a x. Assume now that X is not core-compact,


that is, there is some x ∈ X and some open neighborhood B of x so that for every openneighborhood A of x there is some ultrafilter y ∈ βX with A ∈ y and lim y ∩ B = ∅.Consequently, there is some X ∈ ββX containing y ∈ βX | lim y ∩ B = ∅ and A, forevery open neighborhood A of x. Then mX(X ) converges to x, but X βa x implies thatB /∈ x , so x cannot converge to x.

5.6.7 Remark.c© It is often easier to check core-compactness of a topological space X byjust looking at the elements of a subbase for OX (that is, a subset of OX whose set of finiteintersections forms a base for OX, see II.1.9). Given a set X and a subset B ⊆ PX of thepowerset of X (with no further axioms), one defines core-compactness and convergencea : βX −→7 X for (X,B) in the same way as one does for topological spaces, and we notethat the topology 〈B〉 generated by B has the same convergence as B. We wish to concludethat a ·βa = a ·mX implies that (X,B) is core-compact; however, the proof above uses thatthe collection of all neighborhoods of x ∈ X forms a filter, a fact that is not necessarily trueif B is just any subset of PX. For the filter f generated by all B-neighborhoods of x, westill have f ∩ j = ∅ if the convergence a satisfies the following condition: every ultrafilterhas a smallest convergence point with respect to the order relation a · eX . Hence, underthis condition, a topological space X is core-compact if X is core-compact with respect toa subbase since

(X,B) is core-compact =⇒ a · βa = a ·mX for the convergence a of B⇐⇒ a · βa = a ·mX for the convergence a of 〈B〉⇐⇒ (X, 〈B〉) is core-compact.

In fact, this argument works for any other property of a topological space which can beequivalently expressed in terms of opens and in terms of ultrafilter convergence, withoutusing the axioms of a topology. Another important example is compactness: a topologicalspace X is compact if X is compact with respect to a subbase. This result is known asAlexander’s Subbase Lemma.

Combining Propositions 5.4.4, 5.6.4 and 5.6.6 gives

5.6.8 Proposition.c© A topological space X is representable if and only if

(1) X is core-compact,

(2) for every x ∈ βX, lim x is irreducible, and

(3) X is weakly sober.

5.7 Representable topological spaces. In this subsection we will present a moredetailed analysis of representable topological spaces.


First we will see that a representable topological space is not only core-compact but evenlocally compact; a topological space is locally compact if the neighborhood filter of everypoint x ∈ X has a base formed by compact neighborhoods of x.

5.7.1 Lemma. c©Every representable topological space is locally compact.

Proof. By Lemma 5.6.1, the topology on βX is generated by all sets of the form

A = a ∈ βX | A ∈ a ,

where A ⊆ X is open. Furthermore, for any ultrafilter X ∈ ββX with A ∈ X , we havemX(X ) ∈ A, and therefore A is compact. For any x ∈ βX,

A | A ∈ x

is a base of the neighborhood filter of x ; and we conclude that βX is locally compact. If Xis representable, then X is a split subobject of βX (since α : βX −→ X can be chosen suchthat α(eX(x)) = x) and hence also locally compact (Exercise 5.F).

For a core-compact space X, condition (1) of Proposition 5.6.4 is equivalent c©to the followingstability property of the way-below relation : for open subsets U1, . . . , Un and V1, . . . , Vn(n ∈ N) of X with Ui Vi for each 1 ≤ i ≤ n, also ⋂i Ui ⋂

i Vi (Exercise 5.H). Note thatit is enough to consider the cases n = 0 and n = 2, and for n = 0 this condition reads asX X which just means that X is compact. Saying that a topological space X with thisstability property is stable, we obtain the following result.

5.7.2 Theorem. c©A topological space X is representable if and only if X is locally compact,weakly sober and stable.

We remark that representable T0-spaces are also called stably compact in the literaturewhere, however, the stability condition on the way-below relation is usually replaced by therequirement that the compact down-closed subsets of X are closed under finite intersection.To see that these conditions are indeed equivalent, first note that a representable space X iscompact, and the binary intersections of pairs of compact down-closed subsets is compact:if A,B ⊆ X are compact and down-closed and A ∩ B ∈ x ∈ βX, then any smallestconvergence point of x belongs to both A and B and therefore also to A∩B. Consequently,in a representable space the finite intersection of compact down-closed subsets is againcompact. For the converse implication, we use the following description of the way-belowrelation on the lattice of opens of a locally compact space.

5.7.3 Lemma. Let X be locally compact and U, V ⊆ X open. Then U V if and only ifU ⊆ K ⊆ V for some compact and down-closed K ⊆ X.

Proof. Assume first that U V . For every x ∈ V there is a compact neighborhood Kof x with K ⊆ V . Since V = ⋃K ⊆ V | K is a compact neighborhood of some x ∈ V


and U V , there is some compact K with U ⊆ K ⊆ V . Furthermore, if A ⊆ W forsome A ⊆ X and some open subset W ⊆ X, then also ↓A ⊆ W since open subsets aredown-closed. In particular, the down-closure of a compact subset is compact and thereforeK above can be chosen down-closed. Conversely, if U ⊆ K ⊆ V for some compact K ⊆ X,then every open cover of V contains a finite sub-cover of K and hence also of U .

5.7.4 Remark. If the compact open subsets of a locally compact space X form a base ofthe topology of X, then K in the proof of Lemma 5.7.3 can be chosen as a compact opensubset of X.

From Lemma 5.7.3 we deduce at once that, for a locally compact space, stability of theway-below relation under finite intersection follows from stability of compact down-setsunder finite intersection.5.7.5 Theorem.c© A topological space is representable if and only if it is locally compact,weakly sober, and if the finite intersection of compact down-closed subsets is compact.

We turn now our attention to pseudo-homomorphisms. Recall from 5.4 that a pseudo-homomorphism between representable topological spaces is a continuous map f : X −→ Ythat preserves the smallest convergence points of ultrafilters.5.7.6 Proposition.c© Let f : X −→ Y be a continuous map between representable topologicalspaces. Then the following assertions are equivalent:

(i) f is a pseudo-homomorphism;

(ii) for every compact down-closed subset K ⊆ Y , f−1(K) is compact;

(iii) for all open subsets U, V ⊆ Y , U V implies f−1(U) f−1(V ).

Proof. Assume first (i) and let K ⊆ Y be compact, x ∈ βX with f−1(K) ∈ x and x bea smallest convergence point of x . Then f(x) is a smallest convergence point of βf(x )and, since K is compact and K ∈ βf(x ), we have f(x) ∈ K. Therefore, x ∈ f−1(K), andwe have shown that f−1(K) is compact, that is, (i) =⇒ (ii). The implication (ii) =⇒ (iii)follows from Lemma 5.7.3. Assume now (iii) and let x ∈ X be a smallest convergence pointof x ∈ βX. Assume that βf(x ) −→ y ∈ Y . Let V ⊆ Y be any open neighborhood of yand choose some open U ⊆ X with y ∈ U V . Then f−1(U) f−1(V ) and f−1(U) ∈ x ,hence x ∈ f−1(V ) and therefore f(x) ∈ V . We conclude that f(x) ≤ y, so (iii) =⇒ (i).

5.7.7 Corollary.c© Let X be a representable topological space, and 2 as in Examples 5.3.2.A continuous map ϕ : X −→ 2 is a pseudo-homomorphism if and only if the open setϕ−1(0) ⊆ X is compact.

5.7.8 Proposition.c©

(1) Let (X,≤, α) be an ordered compact Hausdorff space and a = (≤) · α its inducedtopology. A subset A ⊆ X is open in (X, a) if and only if A is down-closed and openin the compact Hausdorff space (X,α).


(2) Let X be a representable space, x ∈ βX and x0 ∈ X be a smallest convergence pointof x . For any x ∈ X, x ≤ x0 if and only if x contains all complements of compactdown-sets B with x /∈ B.

Proof. To see (1), let (X,≤, α) be an ordered compact Hausdorff space and A ⊆ X. Letϕ : X −→ 2 be the characteristic map of the complement X \ A of A. Then

A is open ⇐⇒ ϕ : (X, (≤) · α) −→ 2 is continuous⇐⇒ ϕ : (X,≤) −→ 2 is monotone and ϕ : (X,α) −→ 2 is continuous (by 5.3.1)⇐⇒ A is down-closed in (X,≤) and open in (X,α).

To see (2), let first x ≤ x0. Then x cannot contain any compact down-sets B with x /∈ B.Assume now that x contains these subsets. Take a neighborhood B of x0 where B is acompact down-set. Then x ∈ B since otherwise B ∈ x and X \B ∈ x .

5.7.9 Corollary. c©Let X be a representable space. Then the topology of Xop is generated bythe complements of compact down-sets B of X. Furthermore, taking smallest convergencepoints of an ultrafilter on X with respect to its original topology describes the ultrafilterconvergence with respect to the topology generated by all open sets of X and all open sets ofXop.

5.7.10 Corollary. c©Let (X,≤, α) be an ordered compact Hausdorff space with separatedorder. The topology of (X,α) is generated by the open subsets and the complements ofcompact down-closed subsets of the representable space (X, (≤) · α).

5.8 Exponentiable topological spaces. In Section 4 we have seen that the category Topcan be fully embedded into the (locally) cartesian closed category PsTop (see Examples 4.1.3and Corollary 4.5.2), and that Top itself is not cartesian closed (see Exercise 4.G). In thissubsection we will provide a characterization of exponentiable topological spaces, that is, ofthose spaces X where (−)×X : Top −→ Top has a right adjoint.Let X be a topological space with convergence a : βX −→7 X (but we write more intuitivelyx −→ x instead of x a x) and let 2 be the Sierpinski space (see Examples 5.3.2). We formthe exponential 2X in PsTop, and write π1 : 2X ×X −→ 2X and π2 : 2X ×X −→ X for theprojection maps and ε : 2X ×X −→ 2 denotes the evaluation map. In the following we willthink of the elements of 2X as closed subsets of X. For any subset V ⊆ X, we put

V ♦ = A ⊆ X | A closed, A ∩ V 6= ∅ .

The topology of X we denote as OX, and O(x) stands for the collection of open neighbor-hoods of x ∈ X.

5.8.1 Proposition. For every topological space X, the pseudotopological space 2X is dual-izable. The underlying order of 2X is subset inclusion; and p −→ A ⇐⇒ µ(p) ⊆ A forp ∈ β(2X) and A ⊆ X closed, where µ(p) = ⋂

A∈p⋃A.


Proof. Recall from Examples 4.5.3 that the convergence structure of 2X is given by

p −→ A ⇐⇒

for all x ∈ X, w ∈ β(2X ×X) with βπ1(w) = p, x := βπ2(w) :x −→ x =⇒ (βε(w) = 1 =⇒ x ∈ A)

⇐⇒

for all x ∈ X, w ∈ β(2X ×X) with βπ1(w) = p, x := βπ2(w) :(x −→ x& βε(w) = 1) =⇒ x ∈ A

⇐⇒

for all x ∈ X :p (a · (βε)) x =⇒ x ∈ A ;

for p ∈ β(2X) and A ⊆ X closed, where in the last line we interpret ε : 2X ×X −→ 2 asthe membership relation 2X −→7 X. We put

µ(p) = x ∈ X | p (a · (βε)) x ,

for p ∈ 2X , and thenp −→ A ⇐⇒ µ(p) ⊆ A .

Furthermore,

µ(p) = x ∈ X | ∃x ∈ βX : (p (βε) x & x −→ x)= x ∈ X | ∀V ∈ O(x),A ∈ p ∃A ∈ A, y ∈ V : y ∈ A= x ∈ X | ∀V ∈ O(x),A ∈ p V ♦ ∩ A 6= ∅= x ∈ X | ∀V ∈ O(x) V ♦ ∈ p .

We also note that V ♦ ∩ A 6= ∅ is equivalent to V ∩ ⋃A 6= ∅, and therefore

µ(p) = x ∈ X | ∀A ∈ p, V ∈ O(x) V ∩ ⋃A 6= ∅= x ∈ X | ∀A ∈ p x ∈ ⋃A= ⋂

A∈p⋃A .

Consequently, µ(p) is a closed subset of X, for every p ∈ β(2X); and µ(p) = A for p = A

the principal ultrafilter generated by A ⊆ X closed. Therefore A −→ B ⇐⇒ A ⊆ B.

5.8.2 Proposition. For every topological space X, the pseudotopological space (2X)op istopological, where the topology of (2X)op is generated by the sets V ♦ with V ⊆ X open.

Proof. By definition, the convergence of (2X)op is given by

p −→ A ⇐⇒ A ⊆ ⋂A∈p⋃A

⇐⇒ ∀x ∈ A ∀A ∈ p (x ∈ ⋃A)⇐⇒ ∀x ∈ A ∀A ∈ p ∀V ∈ O(x) (V ∩ ⋃A 6= ∅)⇐⇒ ∀x ∈ A ∀V ∈ O(x)∀A ∈ p (V ♦ ∩ A 6= ∅)⇐⇒ ∀V ∈ OX (A ∈ V ♦ =⇒ V ♦ ∈ p) ;


that is, it is generated by the sets

V ♦ = A ⊆ X | A closed, A ∩ V 6= ∅ (V ⊆ X open)

and therefore it is the convergence of the topology generated by these sets.

We find it remarkable that, although 2X is topological if and only if X is exponentiable(see 5.8.4 below), its dual belongs always to Top. The topological space V X := (2X)op isusually referred to as the lower-Vietoris space.

5.8.3 Lemma. Let X be a pseudotopological space. Then (−)X : PsTop −→ PsTop preservesinitial sources (with respect to the canonical forgetful functor to Set).

Proof. Let (fi : Y −→ Yi)i∈I be initial in PsTop. Let p ∈ β(Y X) and h ∈ Y X so that, forall i ∈ I, β(fXi )(p) −→ fXi (h). Let w ∈ β(Y X × X) and x ∈ X with βπ1(w) = p andβπ2(w) −→ x. Then, for all i ∈ I,

βπ1(β(fXi × 1X)(w)) = βπ1(w) −→ x and βπ2(β(fXi × 1X)(w)) = β(fXi )(p) −→ fXi (h) ,

therefore β(fXi × 1X)(w) −→ (fi · h, x) and

βfi(βεY (w)) = βεYi(β(fXi × 1X)(w)) −→ fi(h(x)) .

Hence, by hypothesis, βεY (w) −→ h(x). This proves p −→ h.

5.8.4 Proposition. Let X be a topological space. Then the following assertions are equiv-alent:

(i) X is exponentiable in Top;

(ii) the pseudotopological space 2X is topological;

(iii) for every topological space Y , the pseudotopological space Y X is topological.

Proof. Assume first that X is exponentiable in Top. We write temporarily [X,−] for theright adjoint of (−)×X : Top −→ Top, and ε′ denotes the counit of (−)×X a [X,−]; inparticular, we consider ε′2 : [X, 2]×X −→ 2. By the universal property of ε : 2X ×X −→ 2in PsTop, there is a unique map t : [X, 2] −→ 2X in PsTop making

[X, 2]×X t×1X //

ε′2$$

2X ×X

ε22


commute. For any p −→ A in 2X , we let (2X)p,A be the topological space of all closedsubsets of X where, besides the principal convergence, only p −→ A (see Exercise 5.C).Then there exists a continuous map sp,A : (2X)p,A −→ [X, 2] making

(2X)p,A ×Xs p,A×1X

//

ε2%%

[X, 2]×X

ε2zz2

commute. Hence, for all p −→ A in 2X , t · sp,A is the identity map; and therefores : [X, 2] −→ 2X is an isomorphism. In particular, 2X is topological. Assume now that 2X istopological. Let Y be a topological space. Then the source PsTop(Y, 2) is initial. Hence,also PsTop(Y X , 2X) is initial by Lemma 5.8.3, and therefore Y X is topological. Finally, theimplication (iii) =⇒ (i) is clear.

We can now derive a characterization of exponentiable topological spaces.5.8.5 Theorem.c© Let X be a topological space. Then X is exponentiable if and only if Xis core-compact (see Proposition 5.6.6).

Proof. The space X is exponentiable if and only if (V X)op is topological, which by Propo-sition 5.5.3 is equivalent to V X being core-compact. Finally, V X is core-compact if andonly if X is core-compact (see Exercise 5.I).

We also note that V X is -cocomplete, for any topological space X, but V X is onlyrepresentable if X is core-compact. For X core-compact and K ⊆ X compact, K♦ is acompact down-set in V X and therefore its complement is open in V Xop (see Corollary 5.7.9).Assume now that X is even locally compact. Then one easily verifies that the sets

(K♦) = A ∈ V X | A ∩K = ∅ (K ⊆ X compact)generate the convergence of 2X = (V X)op. Hence, if we interpret the elements of 2X asopen subsets of X, the topology of 2X is generated by the sets

V ⊆ X open | K ⊆ V (K ⊆ X compact) .This topology is known as the compact-open topology.

5.9 Representable approach spaces. Thec© situation for approach spaces is quite similarto the one for topological spaces when one considers the Barr extension of to P+-Rel.This extension then yields a lifting of the ultrafilter monad to App ' (,P+)-Cat. In thiscase, diagram (5.4.ii) becomes

App ' //

MetCompHaus //

CompHaus

App //Met // Set .


Here MetCompHaus denotes the category of metric compact Hausdorff spaces and morphisms(see Example 5.2.1(4)). An approach space X = (X, a) is representable if and only ifa · βa = a ·mX and, moreover, every ultrafilter x has a generic convergence point x0, witha0 = a · eX the underlying metric; this latter condition means

a(x , x) = a0(x0, x)

for all x ∈ X. Similar to the situation in Top, one calls a non-expansive map ϕ : X −→ [0,∞]irreducible if infx∈X ϕ(x) = 0 and, if for any non-expansive maps ϕ1, ϕ2 : X −→ [0,∞]with ϕ(x) ≥ minϕ1(x), ϕ2(x) (for all x ∈ X) one has ϕ ≥ ϕ1 or ϕ ≥ ϕ2. Recallfrom Examples 5.3.2 that we consider [0,∞] as an approach space with convergenceλ(u, u) = u (supA∈u infv∈A v), for u ∈ β[0,∞] and u ∈ [0,∞]. A typical example of anirreducible non-expansive map is ϕ = a0(x,−), for x ∈ X (see Exercise 5.J). An approachspace X is called sober whenever every irreducible non-expansive map ϕ : X −→ [0,∞] is ofthe form ϕ = a0(x,−), for a unique x ∈ X; and X is called weakly sober if such x ∈ X isnot necessarily unique.

5.9.1 Lemma. c©Let X = (X, a) be an approach space and ϕ : X −→ [0,∞] be an irreduciblenon-expansive map. Then there exists some x ∈ βX with ϕ = a(x ,−).

Proof. We freely make use of Exercise 5.J and the notation introduced there. Let ϕ : X −→[0,∞] be irreducible. For every u ∈ [0,∞], u > 0, put Au = x ∈ X | ϕ(x) ≤ u; byhypothesis, Au 6= ∅. Then ϕAu ≤ ϕ since, with A := x ∈ X | ϕAu(x) ≤ ϕ(x), one hasAu ⊆ A and 0 < infϕ(x) | x ∈ X, x /∈ A =: v. Then ϕ(x) ≥ minϕAu(x), v, but v ≤ ϕis not possible since infx∈X ϕ(x) = 0, therefore ϕAu ≤ ϕ. The down-directed set

f = Au | u ∈ [0,∞], u > 0

is disjoint fromj = B ⊆ X | ϕB 6≤ ϕ ,

and j is an ideal since ϕ is irreducible. Hence, by Corollary II.1.13.5 there is some ultrafilterx ∈ βX with f ⊆ x and x ∩ j = ∅. Then

a(x ,−) = supA∈x ϕA ≤ ϕ

and ϕ ≤ a(x ,−) since supA∈x infx∈A ϕ(x) = 0.

Following the example of topological spaces, we call an approach space X = (X, a) core-compact whenever a · βa = a ·mX , and we call X stable whenever a(x ,−) is irreducible, forevery x ∈ βX. With this terminology we have the following result.

5.9.2 Theorem. c©An approach space X = (X, a) is representable if and only if X is weaklysober, stable and core-compact.


Proof. If X is representable, then X is core-compact (see Proposition 5.4.4) and, for everyx ∈ βX, a(x ,−) is irreducible since a(x ,−) = a0(x0,−) for some x0 ∈ X. We also concludethat X is weakly sober since every irreducible non-expansive map ϕ : X −→ [0,∞] is ofthe form ϕ = a(x ,−) for some x ∈ βX (by Lemma 5.9.1). Assume now that X is weaklysober, stable and core-compact. Then every ultrafilter x ∈ β has a generic convergencepoint since a(x ,−) is irreducible and X is weakly sober. Since X is core-compact, theassertion follows from Proposition 5.4.4.

Exercises

5.A Connected components of an ordered set. Show that the left adjoint π0 : Ord −→ Set of(−)d : Set −→ Ord, X 7−→ (X,=) sends an ordered set X = (X,−→) to X/∼, where x ∼ yprecisely when there exists a path x −→ • ←− · · · y. The reflection map q : X −→ X/∼ isthe coequalizer of p1, p2 : R −→ X, where R ⊆ X ×X is the graph of the order relation −→of X. Furthermore, if −→ is confluent, so that for all x −→ y, x −→ y′ there exists z ∈ Xwith y −→ z and y −→ z

x

y

y′

z ,

then x ∼ y if and only if x −→ z ←− y for some z ∈ X.

5.B Connected components of a V-category. Let V be a quantale with k = > the topelement of V. Furthermore, assume that u ⊗ v = ⊥ implies u = ⊥ or v = ⊥, for allu, v ∈ V, so that o : V −→ 2 defined by (o(x) = 1 ⇐⇒ x > ⊥) is a lax homomorphismof quantales (see also Exercise II.1.I). Show that the left adjoint π0 : V-Cat −→ Set of(−)d : Set −→ V-Cat, X 7−→ (X, 1X) is the composite

V-Cat Bo−−→ Ord π0−→ Set ,

where Bo is defined as in 3.5.

5.C A topology with chosen convergence. Let X be a set, x 0 ∈ βX be an ultrafilter on Xand x0 ∈ X. Define a relation a : βX −→7 X by

x a x whenever

x = x 0 and x = x0 , orx = x .

Show that a is the convergence of a topology on X, that is, 1X ≤ a · eX and a · βa ≤ a ·mX .

5.D Irreducible closed sets. Let X be a topological space and A ⊆ X be a non-empty closedsubset of X. Show that A is irreducible if and only if, for all open subsets U, V ⊆ X, if


U ∩ V ∩A = ∅, then U ∩A = ∅ or V ∩A = ∅. Conclude that A is the set of limit pointsof some filter x with A ∈ x . c©Give an example of a compact topological space X with anultrafilter x where lim x is not irreducible.

5.E The way-below relation via ultrafilter convergence. c©Let X be a topological space andA,B ⊆ X be open subsets of X. Show that A B if and only if every ultrafilter x ∈ βXwith A ∈ x has a limit point in B. In particular, X is compact if and only if X X if andonly if every ultrafilter of X converges.

5.F Split subobjects of locally compact spaces. Let f : X −→ Y and g : Y −→ X be continuousmaps between topological spaces with g · f = 1X , and assume that Y is locally compact.Show that X is locally compact.

5.G Local compactness versus core-compactness. c©Let X be a topological space. For B ⊆ Xopen, consider

lim−1(B) = x ∈ βX | lim x ∩B 6= ∅ ,where lim x denotes the set of convergence points of x . Show that lim−1(B) is open inβX if and only if, for every x ∈ B, there is some open neighborhood U of x with U B.Hence, the following statements hold.

(1) For a core-compact space X, the subspace

lim−1(X) = x ∈ βX | x converges to some x ∈ X

of βX is locally compact.

(2) If X is core-compact and every convergent ultrafilter x ∈ βX has a smallest con-vergence point, then the map lim−1(X) −→ X that associates to every convergentultrafilter a (tacitly chosen) smallest convergence point is continuous; therefore, X islocally compact.

(3) If X is Hausdorff, then

X is core-compact ⇐⇒ X is locally compact⇐⇒ every point of X has a compact neighborhood.

5.H Stable spaces. c©Let X be a topological space. Show that

(1) if lim x is irreducible for every x ∈ βX, then X is stable, and

(2) if X is stable and core-compact, then lim x is irreducible for every x ∈ βX.

5.I Local compactness of V X. Consider the space V X of 5.8, for a topological space X.Let x ∈ X and let U,Ui ⊆ X (i ∈ I) be open. Then the following hold:

(1) x ∈ U♦ if and only if x ∈ U ;


(2) (⋃i∈I Ui)♦ = ⋃i∈I U

♦i ;

(3) X is core-compact if and only if V X is core-compact.

Hint. For (3), use Remark 5.6.7.

5.J The approach space [0,∞].c© Consider the Barr extension of the ultrafilter monad toP+-Rel.

(1) Show that ([0,∞], µ, ξ) is a metric compact Hausdorff space where µ(u, v) = v uand ξ : β[0,∞] −→ [0,∞], u 7−→ supA∈u infu∈A u.

(2) Let X = (X, a) be an approach space (= (,P+)-category). Show:

(a) a(x ,−) : X −→ [0,∞] is non-expansive, for every x ∈ βX.(b) ϕA : X −→ [0,∞], x 7−→ infa(x , x) | x ∈ βX,A ∈ x , is non-expansive, for

every A ⊆ X. Furthermore, ϕA∪B(x) = minϕA(x), ϕB(x), for all A,B ⊆ Xand x ∈ X.

(c) a(x,−) : X −→ [0,∞] is irreducible, for every x ∈ X.

(3) Let X = (X, a) be an approach space, x ∈ βX and x ∈ X. Show

a(x , x) = supA∈x ϕA(x) .

Conclude that

a(x ,−) = supϕ | ϕ : X −→ [0,∞] non-expansive, supA∈x infx∈X ϕ(x) = 0 .

5.K Lifting T . Show that the functor T : Set −→ (T,V)-Cat of Proposition 3.3.6 factorsthrough the functor T : (T,V)-Cat −→ (T,V)-Cat of 5.4.

5.L Fundamental adjunctions revisited. With the notations of 3.6 and 5.4, show that thediagrams

(V-Cat)TK

zz

GT

!!

(T,V)-Cat

I $$

Ae // V-Cat

Bp||

Ord

(V-Cat)T

(T,V)-Cat

M::

V-CatAoo

FTaa

OrdJ

dd

Bι

<<

commute (where I denotes the induced-order functor, see 3.3). Describe J := ABι andshow

GTMJ(X,=) = (TX, 1]X) = (TX, T1X) (see the Proof of 5.3.5),KF TBι(X,=) = TX (see 3.3.6).


Notes on Chapter III

The Notes on Chapter IV give some information on the history of the axiomatization of convergence intopology which culminated in the Manes–Barr characterization of a topological space in terms of an abstractultrafilter convergence relation (see [Manes, 1969, 1974], [Barr, 1970]). With the motivation taken fromthese papers and from Lawvere’s landmark contribution [Lawvere, 1973], the theory of (T,V)-categoriesas presented in this Chapter started with [Clementino and Hofmann, 2003] and [Clementino and Tholen,2003]. Whereas the term lax extension of a monad T to V-Rel was first understood in a more restrictivesense than the one used here (dealing only with flat extensions), the set of axioms presented in 1.4 and1.5 was shaped in [Seal, 2005]. The construction of the lax extension of a functor and monad to Rel ofrespectively 1.10 and 1.12 stems from [Barr, 1970]. The deep connection between extensions of functorsto categories of internal relations and the preservation of weak pullbacks was first exposed in [Trnkova,1977]. The Beck–Chevalley condition belongs to the folklore domain of higher-level category theory and iscredited independently to both name givers. The weaker notion of taut monad appears first (under thename Alexandrov monad) in Mobus’ thesis [Mobus, 1981] and was re-introduced in [Manes, 2002]. Thedouble-categorical presentation of lax extensions presented in 1.13 appears in [Cruttwell and Shulman,2010] following a suggestion by Pare.The quantaloid V-Rel described in 1.1 was introduced in a more general form in [Betti, Carboni, Street, andWalters, 1983] and extensively used in a particular case by Rosebrugh and Wood [2002]. In these papers,the quantale V is allowed to be a monoidal category or even a bicategory, and the term V-matrix is usedinstead of V-relation. This term was adopted also in the first studies of lax algebras for a Set-monad laxlyextended to V-Rel, as given in [Clementino and Tholen, 2003] (for V a symmetric monoidal closed category),as well as in the subsequent paper [Clementino, Hofmann, and Tholen, 2004b] (for V a commutative andunital quantale). The latter paper also introduced the Kleisli convolution of Definition 1.7.1 (under thename co-Kleisli composition). The associativity criterion of Proposition 1.9.4 for this operation is original,while the identification of maps (in Lawvere’s sense) in the 2-category V-Rel as given in Proposition 1.2.1goes back to [Freyd and Scedrov, 1990] and [Clementino and Hofmann, 2009].Of course, for a symmetric monoidal closed category V, the notion and theory of (T,V)-categories asinitiated in [Clementino and Tholen, 2003] builds on the theory of V-categories, as introduced in [Eilenbergand Kelly, 1966] and developed further in [Kelly, 1982], after their significance in the context of the subjectof this book had been emphasized in [Lawvere, 1973]. The concept of module (called bimodule by Lawvere)was originally introduced (under the name distributor, but often also called profunctor) by Benabou, see[Benabou, 2000].The presentation of metric spaces as enriched categories (Example 1.3.1(2)) is due to Lawvere [1973].This description motivated numerous works on the reconciliation of order, metric and category theory;see in particular the work of Flagg and his coauthors on continuity spaces [Flagg, 1992, 1997; Flagg andKopperman, 1997; Flagg, Sunderhauf, and Wagner, 1996] and (metric) generalizations of domain theory asin [Bonsangue, van Breugel, and Rutten, 1998; Wagner, 1994]. The probabilistic metric spaces presentedin 2.1 where introduced in [Menger, 1942]; for more information see [Schweizer and Sklar, 1983]. Theywere recognized as enriched categories in [Flagg, 1992] and as such further investigated in [Chai, 2009]and [Hofmann and Reis, 2013]. The monadicity of compact Hausdorff spaces exposed in 2.3.3 is due toManes [1969], and the ensueing presentation of topological spaces as relational algebras (Theorem 2.2.5)was established in [Barr, 1970]. Approach spaces were introduced in [Lowen, 1989], and a comprehensivepresentation of their theory can be found in [Lowen, 1997]. The description of approach spaces as laxalgebras (Theorem 2.4.5) was established in [Clementino and Hofmann, 2003]. The lax-algebraic descriptionof closure spaces of 2.5 together with the introduction of their metric version of Exercise 2.G appeared firstin [Seal, 2005].The easily-established but important property of topologicity of categories of lax algebras over Set along


with the investigation of algebraic functors, change-of-base functors and induced orders was already presentin the initial papers on the subject (see [Clementino and Hofmann, 2003], [Clementino and Tholen, 2003],[Clementino, Hofmann, and Tholen, 2004b], [Seal, 2009]). Of course, these types of functors were previouslystudied in the monad and enriched-category contexts. Universality of coproducts in these categories wasrecognized in [Mahmoudi, Schubert, and Tholen, 2006]; for the study of quotient structures, see [Hofmann,2005].The notions of pseudotopological and pretopological spaces were introduced in [Choquet, 1948] (Exam-ple 4.1.3(2)). Partial products of topological spaces first appeared in Pasynkov’s paper 1965 and werestudied in [Dyckhoff, 1984]. The categorical notion and its linkage with exponentiable morphisms (studiedin the general categorical as well as the topological realm by Niefield [1982]) was established in [Dyckhoffand Tholen, 1987]. The notion of quasitopos (see 4.8) was given by Penon in 1973. Machado proved in 1973that PsTop is cartesian closed and Wyler in 1976 that PsTop is a quasitopos (in fact, the quasi-topos hull ofTop), see Example 4.8.5(1). The corresponding facts about PsApp can be found in [Colebunders and Lowen,1988, 1989]. Under the conditions of Corollary 4.8.2, but in the broader context of a monoidal categoryV, the quasitopos property of (T,V)-Gph was established in [Clementino et al., 2003a]. Final density of(T,V)-Cat in this category as stated in Theorem 4.9.2 originates with [Clementino and Hofmann, 2012].The theme of Section 5 is motivated by the equivalence between Nachbin’s ordered compact Hausdorff spaces(introduced in [Nachbin, 1950]) and stably compact spaces which was first described in Gierz, Hofmann,Keimel, Lawson, Mislove, and Scott [1980]. Another source of inspiration is Hermida’s work 2000; 2001 onrepresentable multicategories which established a similar correspondence in the context of multicategories.The (T,V)-framework presented in the first subsections stems largely from [Tholen, 2009] and is augmentedby crucial ingredients from [Clementino and Hofmann, 2009], such as the functor M of Theorem 5.3.5that in essence facilitates the notion of representability of a (T,V)-category. Representable T0-spacesare known as stably compact spaces; for more information we refer to [Gierz, Hofmann, Keimel, Lawson,Mislove, and Scott, 2003; Jung, 2004; Lawson, 2011] (see also the comment after Theorem 5.7.2). Thenotion of a dual stably compact spaces goes back to [de Groot, 1967; de Groot, Strecker, and Wattel, 1967;Hochster, 1969] (see Corollary 5.7.9). The Cech–Stone compactification of a completely regular topologicalspace, briefly referred to in 5.6, was introduced in [Cech, 1937] and [Stone, 1937] building on [Tychonoff,1930]. The characterization of normal topological spaces in terms of convergence (Proposition 5.6.2) wasfirst obtained in [Mobus, 1981]. The characterization of exponentiable topological spaces as precisely thecore-compact ones (Theorem 5.8.5) is due to [Day and Kelly, 1970]; for more information see [Isbell, 1986].The characterization of core-compactness via convergence (Proposition 5.6.6) stems from [Mobus, 1981,1983] and [Pisani, 1999]. However, the approach taken in this book is quite distinct from the one in thesesources as it makes essential use of the Vietoris construction which has its roots in Vietoris [1922]. Thenotion of sobriety for approach spaces featured in the characterization of representable approach spaces 5.9was introduced in [Banaschewski, Lowen, and Van Olmen, 2006] and further studied in [Van Olmen, 2005]which uses the term approach prime map for what is called (in resemblance to the topological counterpart)irreducible non-expansive map in this book.

Suggestions for further reading: [Bentley, Herrlich, and Lowen, 1991; Birkedal, Støvring, and Thamsborg,2010; Bunge, 1974; Clementino and Hofmann, 2009; Gerlo, Vandersmissen, and Van Olmen, 2006; Herrlich,Colebunders, and Schwarz, 1991; Hofmann and Waszkiewicz, 2011; Kopperman, 1988; Kostanek andWaszkiewicz, 2011; Lowen, 2013; Rutten, 1998; van Breugel, 2001; Van Olmen and Verwulgen, 2010;Waszkiewicz, 2009].

IV Kleisli MonoidsDirk Hofmann, Robert Lowen, Rory Lucyshyn-Wright, Gavin J. Seal

This Chapter revolves around an alternate presentation of (T,V)-Cat as the category T-Monof monoids in the hom-set of a Kleisli category that has the advantage of avoiding explicituse of relations or lax extensions. Our role model is given by the filter monad F for whichF-Mon ∼= Top. After obtaining an isomorphism

T-Mon ∼= (T, 2)-Cat

in Section 1, in Section 2 we use the isomorphisms

(, 2)-Cat ∼= Top ∼= (F, 2)-Cat

as role models to compare (S,V)-categories with (T,V)-categories for a monad morphismα : S −→ T and a general V in lieu of the embedding −→ F and V = 2. In Section 3,we prove that any (T,V)-category obtained from an associative lax extension can also bepresented as a (, 2)-category, in effect passing all needed information provided by V, Tand its lax extension to V-Rel into a new monad = (T,V). In Section 4, we identifythe injective (T, 2)-categories as precisely the T-algebras by exploiting the fact that theforgetful functor SetT −→ (T, 2)-Cat is monadic of Kock-Zoberlein type. Finally, in Section 5we focus on the filter monad and investigate the interplay between (F, 2)-categories andF-algebras in the context of ordered sets.

1 Kleisli monoids and lax algebras

In II.1.9, a topological space is defined as a set equipped with a collection of subsetsclosed under finite intersections and arbitrary unions, and in Exercise II.1.G an equivalentdescription in terms of a set with a finitely additive closure operation is given. In III.2.2,convergence of ultrafilters is used as a defining structure. In this subsection, we present twofilter-based counterparts that avoid the Axiom of Choice: the first focuses on neighborhoodfilters (Proposition 1.1.1) and serves as a model for the Kleisli monoids introduced in 1.3;the second concentrates on filter convergence (Corollary 1.5.4) and is facilitated by a newgeneral construction of a lax extension in 1.4, namely the Kleisli extension of a monad.

281

282 CHAPTER IV. KLEISLI MONOIDS

1.1 Topological spaces via neighborhood filters. A topological space can be entirelydefined in terms of its neighborhood systems by way of the filter monad F = (F,m, e)described in Example II.3.1.1(5). From a categorical viewpoint, it is convenient for the mapτX : PX −→ FX that sends a set A to the principal filter A (see II.1.12) to be monotone;hence, the set FX of filters on X is ordered by the refinement order :

x ≤ y ⇐⇒ x ⊇ y ,

for all x , y ∈ FX. A filter x is finer than y , or y is coarser than x , if x ⊇ y .Given a topology OX and x ∈ X, the collection of all open sets that contain x spans theneighborhood filter ν(x) of x:

A ∈ ν(x) ⇐⇒ ∃U ∈ OX (x ∈ U ⊆ A) ,

for all A ⊆ X. This defines a map ν : X −→ FX that sends a point of a topological spaceto its neighborhood filter and is such that eX(x) = A ⊆ X | x ∈ A contains ν(x) for allx ∈ X, that is,

eX ≤ ν (1.1.i)in the pointwise refinement order. To relate ν with the filter monad multiplication, wedefine for A ⊆ X the set AF of filters that contain A,

AF := a ∈ FX | A ∈ a ,

and recall that an open set is a neighborhood of each of its points (see Exercise II.1.Q);thus, in particular, for all x ∈ X and A ⊆ X, one has

A ∈ ν(x) ⇐⇒ ∃B ∈ ν(x) ∀y ∈ B (A ∈ ν(y))⇐⇒ ∃B ∈ ν(x) ∀y ∈ B (ν(y) ∈ AF)⇐⇒ ∃B ∈ ν(x) (B ⊆ ν−1(AF))⇐⇒ ν−1(AF) ∈ ν(x)⇐⇒ AF ∈ Fν · ν(x)⇐⇒ A ∈ mX · Fν · ν(x) .

This last expression is simply the Kleisli composition of ν : X −→ FX with itself, so theprevious equivalences show that

ν ν ≤ ν . (1.1.ii)By (1.1.i) and (1.1.ii), a topology on X determines a monoid in the ordered hom-setSetF(X,X) of the Kleisli category SetF. Consider now a continuous map f : X −→ Ybetween topological spaces, with ν : X −→ FX and µ : Y −→ FY the correspondingneighborhood filter maps. If B ⊆ Y is a neighborhood of f(x), then there exists an openset U ⊆ B containing f(x); thus, f−1(U) is an open set with x ∈ f−1(U) ⊆ f−1(B), andf−1(B) is an element of ν(x). Thanks to the equivalence

f−1(B) ∈ ν(x) ⇐⇒ B ∈ Ff · ν(x)

1. KLEISLI MONOIDS AND LAX ALGEBRAS 283

(for all B ⊆ Y and x ∈ X), we deduce µ · f(x) ⊆ Ff · ν(x) for all x ∈ X, that is,

Ff · ν ≤ µ · f .

Instead of considering a map f : X −→ Y , one can look at its image f\ = eY · f : X −→ FYunder the left adjoint Set −→ SetF (II.3.6), and this last condition becomes

f\ ν ≤ µ f\ (1.1.iii)

by naturality of e. Not only do the neighborhood filters of topological spaces have propertiesnicely expressible in the language of the Kleisli category of F, but the conditions (1.1.i),(1.1.ii) and (1.1.iii) are sufficient to describe topological spaces and continuous maps.

1.1.1 Proposition. The category Top of topological spaces and continuous maps is iso-morphic to the category F-Mon whose objects are pairs (X, ν), with ν : X −→ FX a monoidin SetF(X,X):

ν ν ≤ ν , eX ≤ ν ,

and whose morphisms f : (X, ν) −→ (Y, µ) are maps f : X −→ Y such that

f\ ν ≤ µ f\ .

Proof. The previous discussion shows that the neighborhood filters of a topological spacedefine a map ν : X −→ FX satisfying the required properties. Conversely, given a monoidν : X −→ FX in SetF(X,X), we define open sets as those U ⊆ X that are neighborhoodsof each of their points:

U ∈ OX ⇐⇒ ∀x ∈ X (x ∈ U =⇒ U ∈ ν(x)) .

Straightforward verifications show that the set OX, ordered by inclusion, is closed underarbitrary suprema as well as under finite infima, and that U ∈ OY implies f−1(U) ∈ OXwhen f : X −→ Y satisfies Ff · ν ≤ µ · f . One therefore has two functors

Top −→ F-Mon and F-Mon −→ Top

whose composites are routinely verified to be the identities on Top and F-Mon.

1.2 Power-enriched monads. The ultrafilter-convergence presentation of topologicalspaces of III.2.2 uses the algebra of relations in an essential way. In turn, relations areprecisely the morphisms of the Kleisli category associated to the powerset monad: SetP = Rel.The passage from neighborhood to convergence structure presented further on in 1.5 exploitsthe interaction of filters and relations via the principal filter monad morphism τ : P −→ Fwhose components τX : PX −→ FX send a set A ∈ PX to the principal filter A ∈ FX.This monad morphism allows us to place the study of neighborhood systems, appearing inProposition 1.1.1 as morphisms of the Kleisli category SetF, in a more general context. The


following Proposition recalls that a monad morphism τ : P −→ T relates SetT with both Reland Sup via functors

Rel −→ SetT −→ Sup(Exercises II.3.H and II.3.I).1.2.1 Proposition. For a monad T = (T,m, e) on Set, one has a one-to-one correspon-dence between:

(i) monad morphisms τ : P −→ T;

(ii) extensions E of the functor FT : Set −→ SetT along the functor (−) : Set −→ Rel ofIII.1.2:

Rel E // SetT

Set

(−)

OO

FT

<<

(iii) liftings L of the functor GT : SetT −→ Set along the forgetful functor Sup −→ Set:

SetT L //

GT ""

Sup

Set

(iv) complete lattice structures on TX such that Tf : TX −→ TY and mX : TTX −→ TXare sup-maps for all maps f : X −→ Y and sets X.

Proof. To simplify the proof, we identify Rel with SetP (Example II.3.6.2), and Sup withSetP via the isomorphism SetP ∼= Sup of Example II.3.2.2(2).(i) ⇐⇒ (ii): This is a direct consequence of Exercise II.3.I. Here, the functor E sends aSetP-morphism r : X −→ PY to the map Er = τY · r : X −→ TY .(i) ⇐⇒ (iii): The equivalence follows from Exercise II.3.H. Note that L sends a mapf : X −→ TY to the P-homomorphism mY · Tf : (TX,mX · τTX) −→ (TY,mY · τTY ).(iii)⇐⇒ (iv): The functor GT of (iii) sends a map g : X −→ TY to mY · Tg : TX −→ TY ,so that (with g = 1TY or g = eY · f) condition (iv) is just an element-wise restatementof (iii).

For a morphism τ : P −→ T of monads on Set, condition (iii) equips the underlying set TXof a free T-algebra with the separated order given by

x ≤ y ⇐⇒ mX · τTX(x , y) = y (1.2.i)

for all x , y ∈ TX. The hom-sets Set(X,TY ) become separated ordered sets via the inducedpointwise order:

f ≤ g ⇐⇒ ∀x ∈ X (f(x) ≤ g(x))


for all f, g : X −→ TY . Composition on the right is always monotone, but compositionon the left (−)T · f : SetT(Y, Z) −→ SetT(X,Z) may fail to be so, see Exercise 1.C (here,(−)T = mZ · T (−) denotes the monad extension operation of II.3.7). To remedy this andtherefore make SetT into a separated ordered category, it suffices that (−)T be monotone:

f ≤ g =⇒ fT ≤ gT ,

for all f, g : X −→ TY . If this condition is satisfied, then the functors E : Rel −→ SetT andL : SetT −→ Sup of Proposition 1.2.1 become 2-functors between ordered categories.

1.2.2 Definition. A power-enriched monad is a pair (T, τ) composed of a monad T on Setand a monad morphism τ : P −→ T such that

f ≤ g =⇒ fT ≤ gT , (1.2.ii)

for all f, g : X −→ TY . A morphism α : (S, σ) −→ (T, τ) of power-enriched monads is amonad morphism α : S −→ T such that τ = α · σ:

Pσ

τ

S α // T ,

so the category of power-enriched monads is a full subcategory of the comma categoryP/MNDSet (see also Exercise 1.A). When working with power-enriched monads (T, τ), wewill often assume a fixed choice of τ , and speak of “the power-enriched monad T”.

1.2.3 Examples.

(1) There are two trivial monads on Set (Exercise II.3.A), but only one is power-enriched,namely the terminal monad 1 whose functor sends all sets to a singleton ?; thecomponents of its structure morphism P −→ 1 are the unique maps !X : PX −→ ?.The other monad does not even have a structure morphism, as there is no map fromP∅ = ? to ∅.

(2) The powerset monad P with the identity structure 1P : P −→ P is power-enriched.Hence, (P, 1P) is an initial object in the category of power-enriched monads and theirmorphisms. The order on the sets PX coming from (1.2.i) is simply subset inclusionbecause the supremum operation is given by arbitrary union.

(3) The filter monad F is power-enriched via the principal filter natural transformationτ : P −→ F which yields a monad morphism τ : P −→ F. The order on FX definedby (1.2.i) is the refinement order introduced in 1.1, and suprema in FX are given byintersections.

(4) The ultrafilter monad is not power-enriched: for the set X = ∅, one observes thatβX = ∅ cannot be a complete lattice.


(5) The up-set monad has at least two different structure morphisms σ, τ : P −→ U, definedcomponentwise for A ∈ PX by

σX(A) = B ⊆ X | A ∩B 6= ∅ and τX(A) = B ⊆ X | A ⊆ B

(τ is just the extension of the principal filter natural transformation). The orderinduced on UX by σ is given by subset inclusion, while the one induced by τ isopposite, that is, τ induces the refinement order on up-sets: for all x , y ∈ UX,

x ≤ y ⇐⇒ x ⊇ y .

These morphisms demonstrate that the morphism P −→ T given with a power-enrichedmonad is indeed a structure and not a property of the monad.

(6) Both monad morphisms of the previous example can be extended to the double-powerset monad to give σ, τ : P −→ P2. However, neither of these satisfy the condition(1.2.ii) (see Exercise 1.C).

1.3 T-monoids. Motivated by Proposition 1.1.1, we introduce the category of monoids inthe hom-sets of a Kleisli category.

1.3.1 Definition. Let T = (T,m, e) be a monad on a category X whose Kleisli categoryXT is a separated ordered category. The category T-Mon of T-monoids (or Kleisli monoids)has as objects pairs (X, ν), where X is an X-object, and its structure ν : X −→ TX is atransitive and reflexive XT-morphism:

ν ν ≤ ν , eX ≤ ν

(where is composition of the Kleisli category XT); a morphism f : (X, ν) −→ (Y, µ) is anX-morphism f : X −→ Y satisfying:

Tf · ν ≤ µ · f or equivalently f\ ν ≤ µ f\ ,

where f\ := eY · f . In the case where T = (T, τ) is a power-enriched monad, the order onthe hom-sets of SetT depends on τ ; however, we will often assume that τ is given implictly,and denote a category of Kleisli monoids by T-Mon rather than by (T, τ)-Mon.

We hasten to remark that in presence of the reflexivity condition, transitivity can beexpressed as an equality ν ν = ν, since

ν = ν eX ≤ ν ν ≤ ν .

Idempotent structures are also preserved by the functor GT = (−)T : XT −→ X:

νT · νT = (ν ν)T = νT . (1.3.i)


1.3.2 Examples.

(1) For T = 1 the terminal monad, Kleisli monoids are simply pairs (X, !X : X −→ ?),and morphisms are maps f : X −→ Y . In other words, the category of Kleisli monoidsis isomorphic to Set:

1-Mon ∼= Set .

(2) In the case of the powerset monad (together with its identity structure 1P), P-Monis the category of ordered sets. Indeed, a map ν : X −→ PX is precisely a relationon X, and the transitivity and reflexivity conditions translate as reflexivity andtransitivity of ν; because the set PX is ordered by set-inclusion, ν is the down-setmap ↓X : X −→ PX (see II.1.7). A map f : X −→ Y is a morphism of P-Mon if andonly if it preserves the relations, that is, if and only if f is a monotone map. Hence,

P-Mon ∼= Ord .

(3) Proposition 1.1.1 and Example 1.2.3(3) show that when F is equipped with theprincipal filter morphism τ : P −→ F, F-Mon is the category of topological spaces andcontinuous maps:

F-Mon ∼= Top .

(4) With the principal filter morphism τ : P −→ U of Example 1.2.3(5), the category ofU-monoids is isomorphic to the category of interior spaces:

U-Mon ∼= Int

(one can proceed as in the proof of Proposition 1.1.1 or more syntactically as inExercise 1.D). In fact, the monad morphism σ : P −→ U of Example 1.2.3(5) yields

U-Mon ∼= Cls ,

so that the structures τ and σ return isomorphic categories of U-monoids.

1.3.3 Proposition. A morphism of power-enriched monads α : (S, σ) −→ (T, τ) induces afunctor

S-Mon −→ T-Mon

that sends (X, ν) to (X,αX · ν) and commutes with the underlying-set functors.

Proof. Thanks to the functor Setα : SetS −→ SetT that sends ν to αX · ν (Exercise II.3.I),the claim easily follows from the fact that αX is monotone (Exercise 1.A).


1.4 The Kleisli extension. Categories of lax algebras depend upon the lax extensionof a monad T on Set to V-Rel. The Barr extension (III.1.10) provides a construction of alax extension by viewing a relation r : X −→7 Y as a composite r = q · p (where p, q areprojection maps); the Kleisli extension introduced below exploits relations as morphismsof the Kleisli category SetP = Rel (Example II.3.6.2). Hence, we will often be workingwith maps r : X −→ PY representing relations r : X −→7 Y , and will indifferently use thenotations P = (P,⋃, −) or (P, (−)P, −) for the powerset monad, and T = (T,m, e) or(T, (−)T, e) for an arbitrary monad on Set, together with the corresponding expressions formonad morphisms τ : P −→ T (see II.3.7).Let us denote by (−)[ : Relop −→ SetP the functor that is identical on sets and sends a relationr : X −→7 Y to the map r[ : Y −→ PX representing the opposite relation r : Y −→7 X:

x r y ⇐⇒ x ∈ r[(y) .

By composition with the functors E : SetP −→ SetT and L : SetT −→ Sup of Proposition 1.2.1,one obtains a functor

(−)τ : Relop (−)[// SetP E // SetT L // Sup

that sends a set X to TX, and a relation r : X −→7 Y to the map rτ : TY −→ TX defined by

rτ := mX · T (τX · r[) = (τX · r[)T .

1.4.1 Definition. Given a power-enriched monad (T, τ), the Kleisli extension T of T toRel (with respect to τ) is described by the functions T = TX,Y : Rel(X, Y ) −→ Rel(TX, TY )(indexed by sets X and Y ), with

x (T r) y ⇐⇒ x ≤ rτ (y ) (1.4.i)

for all relations r : X −→7 Y , and x ∈ TX, y ∈ TY , or, equivalently,

(T r)[ = ↓TX ·rτ : TY −→ PTX .

1.4.2 Examples.(1) In the case of the terminal power-enriched monad (1, !), the Kleisli extension of a

relation r : X −→7 Y is ? −→7 ? with constant value >.

(2) To obtain an explicit description of the Kleisli extension of the powerset monad (P, 1P),observe that

A ⊆ r1P(B) ⇐⇒ A ⊆ ⋃X ·Pr[(B) ⇐⇒ ∀x ∈ A ∃y ∈ B (x ∈ r[(y)) ⇐⇒ A ⊆ r(B) ,

for a relation r : X −→7 Y , and A ∈ PX, B ∈ PY , where r(B) = x ∈ X | ∃y ∈B (x ∈ r[(y)), as in Example III.1.10.3(2). Hence,

A (P r) B ⇐⇒ A ⊆ r(B) .

Here, we obtain the lax extension P introduced in Example III.1.4.2(2).


(3) Let T = F be the filter monad and τ : P −→ T the principal filter natural transformation.For a relation r : X −→7 Y , A ⊆ X and y ∈ FY , we have

A ∈ mX · F (τX · r[)(y ) ⇐⇒ (τX · r[)−1(AF) ∈ y⇐⇒ y ∈ Y | A ∈ τX · r[(y) ∈ y⇐⇒ y ∈ Y | r[(y) ⊆ A ∈ y⇐⇒ ∃B ∈ y (r(B) ⊆ A) .

This shows precisely that

rτ (y ) = ↑PXr(B) | B ∈ y ,

so x (F r) y ⇐⇒ x ⊇ rτ (y ) or, if we use the notation r[y ] = r(B) | B ∈ y,

x (F r) y ⇐⇒ x ⊇ r[y ] . (1.4.ii)

The Kleisli extension of the filter monad returns the lax extension F of Exam-ple III.1.10.3(4).

(4) The Kleisli extension of the up-set monad U, equipped with the principal filter naturaltransformation τ : P −→ U, is obtained as for the filter monad in the previous example,so that

x (Ur) y ⇐⇒ x ⊇ r[y ]

for all maps r : X −→ PY and up-sets x ∈ UX, y ∈ UY .

To prove that T : Rel −→ Rel is indeed a lax extension of the Set-functor T , it is convenientto express the former as a composite of lax functors. In view of this, we remark that T r(for a relation r : X −→7 Y ) can be written as

T r = (rτ )∗ : TX −→7 TY ,

where (−)∗ : Ord −→ Modop is the functor that sends a monotone map f : X −→ Y to themodule f ∗ = f · (≤Y ) : Y −→7 X (see II.1.4 and II.2.2). The Kleisli extension is therefore afunctor

T : Relop (−)τ// Sup // Ord (−)∗

//Modop

(here, Sup −→ Ord is the forgetful functor). There is moreover a lax functor Mod −→ Relthat assigns to a module its underlying relation: composition of modules is compositionof relations, identity modules are order relations and 1X ≤ (≤X) for any ordered set X.Hence, with E : SetP −→ SetT and L : SetT −→ Sup denoting the functors from (ii) and (iii)of Proposition 1.2.1, the Kleisli extension T op can be decomposed as the top line of the


commutative diagram

Relop (−)[// SetP E // SetT L //

GT""

Sup //

Ord (−)∗//

||

Modop // Relop

Set

(−)cc

FP

OO

FT

;;

T // Set

(1.4.iii)

in which all arrows except Modop −→ Relop are functors, and Modop −→ Relop is a lax functorthat fails only to preserve identities (the unnamed arrows are all forgetful).1.4.3 Proposition. Given a power-enriched monad (T, τ), the Kleisli extension T of T toRel yields a lax extension T = (T ,m, e) of T = (T,m, e) to Rel.

Proof. The fact that T : Rel −→ Rel is a lax functor follows from its decomposition as laxfunctors preserving composition in the first line of (1.4.iii). The lax extension condition(Tf) ≤ T (f ) can be deduced from the diagram

Relop (−)τ// Ord

(−)∗//Modop // Relop

1Relop

Set

(−)OO

T // Set (−)//

≤

Relop

in which the first line is T op. The second lax extension condition T (h · r) = (Th) · T r forall relations r : X −→7 Y and maps h : Z −→ Y (see Proposition III.1.4.3(3)) comes from theequivalences

x (T (h · r)) z ⇐⇒ x ≤ rτ · (h)τ (z) ⇐⇒ x ≤ rτ · Th(z) ⇐⇒ x ((Th) · T r) z

for all x ∈ TX, z ∈ TZ. To verify oplaxness of e : 1Rel −→ T , we use that τX =mX · τTX · PeX = ∨

TX ·PeX . Given a relation r : X −→7 Y , and x ∈ X, y ∈ Y with x r y,one then has

eX(x) ≤ ∨x′∈r[(y) eX(x′) = τX · r[(y) = (τX · r[)T · eY (y) = rτ · eY (y) ,

as required. For proving oplaxness of m : T T −→ T , recall that mX ·τTX ·↓TX = ∨TX · ↓TX =

1TX , and note that(rτ )T = (rτ · 1TY )T = ((τX · r[)T · 1TY )T = (τX · r[)T · 1TTY = rτ ·mY .

Thus, if X ∈ TTX and Y ∈ TTY are such that X (T T r) Y , or equivalently X ≤ (T r)τ (Y ),then

mX(X ) ≤ mX · (T r)τ (Y )= 1TTX · (τTX · ↓TX ·rτ )T(Y )= (1TTX · τTX · ↓TX ·rτ )T(Y )= (rτ )T(Y )= rτ ·mY (Y ) ,

which concludes the proof.


1.4.4 Remark. Since the Kleisli extension provides the monad T with a lax extension,there is a natural order on TX associated with T (see III.3.3); on TX there is also theorder (1.2.i) induced by the monad morphism τ : P −→ T. Since the first order T1X isdefined via the second:

x (T1X) y ⇐⇒ x ≤ y

(Definition (1.4.i)), the orders are equivalent. Let us emphasize that T fails to preserveidentity relations unless T = 1 is the terminal power-enriched monad.

1.5 Topological spaces via filter convergence. In this subsection, we show that(F, 2)-Cat is isomorphic to F-Mon ∼= Top (Proposition 1.1.1), that is, we present topologicalspaces as sets equipped with a transitive and reflexive convergence relation a : FX −→7 X.The correspondence between convergence and neighborhoods can be formalized as in III.2.2via maps

conv : Set(X,FX) −→ Rel(FX,X) and nbhd : Rel(FX,X) −→ Set(X,FX) .

In fact, one can without further thought replace the filter monad F with a power-enrichedmonad (T, τ). By identifying Rel(TX,X) with Set(X,PTX), isomorphic as ordered sets,we define

conv(ν) = ↓TX ·ν and nbhd(r) = ∨TX ·r[

for all maps ν : X −→ TX and relations r : TX −→7 X. In pointwise notation, these mapsmay be written as

x conv(ν) y ⇐⇒ x ≤ ν(y) and nbhd(r)(y) =∨x ∈ TX | x ∈ r[(y) ,

for all y ∈ X and x ∈ TX, as a direct generalization of the fact that, in a topological spaceX, a filter x converges to a point y precisely when x is finer than the neighborhood filterof y.

1.5.1 Proposition. With Set(X,TX) and Rel(TX,X) ordered pointwise, the monotonemaps defined above form an adjunction nbhd a conv : Set(X,TX) −→ Rel(TX,X) for allsets X.Moreover, the fixpoints of (conv · nbhd) are precisely the unitary relations, and conv is fullyfaithful, so that the fixpoints of (nbhd · conv) are the maps ν : X −→ TX.

Proof. The equivalencenbhd(r) ≤ ν ⇐⇒ r[ ≤ conv(ν)

(for all maps ν : X −→ TX and relations r : TX −→7 X) follows directly from the adjunction∨TX a ↓TX . Similarly, from ∨

TX · ↓TX = 1TX follows that nbhd · conv = 1, that is, conv isfully faithful (see Corollary II.1.5.2).


Remark 1.4.4 shows that (T1X)[ = ↓TX . Hence, if r : TX −→7 X is unitary, then

r[ = (↓TX)P · r[ (r right unitary in III.1.7.3)= (↓TX)P · (eX · T r ·mX)[ (r left unitary)= (↓TX)P · ((mX)[)P · (T r)[ · eX ((eX)[ = −TX · eX)= (↓TX ·mX)P · (T r)[ · eX (gP · fP = (gP · f)P)= (↓TX ·mX)P · ↓TTX ·(τTX · r[)T · eX (definition of T )= (↓TX ·mX)P · ↓TTX ·τTX · r[ (fT · eX = f)≥ (↓TX ·mX)P · −TTX · τTX · r[ (↓TTX ≥ −TTX)= ↓TX ·mX · τTX · r[ (fP · −X = f)= conv · nbhd(r) (mX · τTX = ∨

TX)≥ r[ (nbhd a conv)

and we may conclude that conv · nbhd(r) = r (via the understood identification ofRel(TX,X) with Set(X,PTX)). Conversely, if r is a fixpoint of conv · nbhd, then r isof the form conv(ν) for some ν : X −→ TX, and one sees that r · T1X ≤ r, so r isright unitary. To prove that r is left unitary, we must verify that eX · T r ≤ r · mX .Thus, suppose that X (T r) eX(y) holds. By definition of T , we have X ≤ rτ · eX(y),or equivalently X ≤ τTX · r[(y). Applying mX to each side of this inequality, we obtainmX(X ) ≤ mX · τTX · r[(y). This means precisely that mX(X ) ≤ ∨TX ·r[(y) = nbhd(r)(y),or mX(X ) (conv · nbhd(r)) y which is mX(X ) r y by the fixpoint condition.

1.5.2 Proposition. The adjoint maps nbhd and conv defined above are monoid homomor-phisms between SetT(X,X) and (T,V)-URelop(X,X), that is, they satisfy

nbhd(s r) = nbhd(r) nbhd(s) , conv(µ) conv(ν) = conv(ν µ)nbhd(1]X) = eX , conv(eX) = 1]X

for all unitary relations r, s : TX −→7 X, and maps µ, ν : X −→ TX.

Proof. The equality nbhd(1]X) = eX follows immediately from the definition of 1]X , as

x 1]X y ⇐⇒ x (eX · T1X) y ⇐⇒ x ≤ eX(y)

for all x ∈ TX and y ∈ X. The multiplication mX = 1TTX of the monad T is a sup-mapand 1TTX · τTX = ∨

TX (Proposition 1.2.1), so

1TTX · τTX · PmX · ↓TTX = ∨TX ·PmX · ↓TTX = mX ·

∨TTX · ↓TTX = 1TTX .

By definition of T , we then obtain

1TTX · (τTX · r[)T = 1TTX · τTX · PmX · ↓TTX ·(τTX · r[)T = 1TTX · τTX · PmX · (T r)[ . (1.5.i)


Therefore,

nbhd(r) nbhd(s) = (1TTX · τTX · r[)T · 1TTX · τTX · s[ (1TTX · τTX = ∨TX)

= ((1TTX · τTX · r[)T)T · τTX · s[ (gT · fT = (gT · f)T)= (1TTX · (τTX · r[)T)T · τTX · s[ ((gT · f)T = gT · fT)= (1TTX · τTX · PmX · (T r)[)T · τTX · s[ (by (1.5.i) above)= 1TTX · (τTX · PmX · (T r)[)T · τTX · s[ ((gT · f)T = gT · fT)= 1TTX · τTX · (PmX · (T r)[)P · s[ (naturality of τ)= ∨

TX ·((mX)[)P · (T r)[)P · s[ (((mX)[)P = PmX)= ∨

TX ·(s · T r ·mX)[ (Rel = SetP)= nbhd(s r) .

The equalities for conv then follow directly from the fact that conv and nbhd are inverse ofeach other on fixpoints (Proposition 1.5.1).

1.5.3 Theorem. Given a power-enriched monad (T, τ) equipped with its Kleisli extensionT , there is an isomorphism

(T, 2)-Cat ∼= T-Monthat commutes with the underlying-set functors.

Proof. For a (T, 2)-algebra (X, r), Proposition 1.5.2 implies that (X, nbhd(r)) is a T-monoid,and conversely, if (X, ν) is a T-monoid, then (X, conv(ν)) is a (T, 2)-algebra (one also usesthe fact that nbhd and conv are monotone). Moreover, Proposition 1.5.1 entails that theseobjects are in bijective correspondence.We are therefore left to show that this correspondence is functorial. Consider first a(T, 2)-functor f : (X, r) −→ (X, s), so that r · (Tf) ≤ f · s. We have

Tf · nbhd(r) = Tf · ∨TX ·r[= ∨

TX ·PTf · r[ (Tf sup-map)= ∨

TX ·(r · (Tf))[ (PTf = (((Tf))[)P)≤ ∨TX ·(f · s)[ (f a (T, 2)-functor)= ∨

TX ·(s[)P · dX · f ((f )[ = −X · f)= ∨

TX ·s[ · f (gP · −X = g)= nbhd(s) · f ;

hence, f is a morphism of T-monoids. Consider now f : (X, ν) −→ (Y, µ) satisfyingTf · ν ≤ µ · f . Then x conv(ν) y means x ≤ ν(y), so we have Tf(x ) ≤ Tf · ν(y) ≤ µ · f(y),and can therefore conclude that Tf(x ) conv(µ) y; that is, f is a (T, 2)-functor between thecorresponding (T, 2)-categories.


1.5.4 Corollary. The category Top of topological spaces is isomorphic to the category(F, 2)-Cat whose objects are pairs (X, a), with a : FX −→7 X a relation representingconvergence and, when a and F a are denoted by −→, satisfying

X −→ y & y −→ z =⇒ ∑X −→ z and x −→ x ,

for all x, z ∈ X, y ∈ FX, and X ∈ FFX; here X −→ y ⇐⇒ X ⊇ a[y ]. The morphismsare the convergence-preserving maps f : X −→ Y :

x −→ y =⇒ f [x ] −→ f(y)

for all y ∈ X, x ∈ FX.

Proof. Proposition 1.1.1 together with the previous Theorem yield an isomorphism betweenTop and (F, 2)-Cat, and the statement is just an explicit description of the latter categoryusing the Kleisli extension of the filter monad (1.4.ii).

Naturally, the same statement holds for the category Cls of closure spaces, for which wegive the following compact form.

1.5.5 Corollary. For the up-set monad U equipped with the Kleisli extension associatedwith the principal filter natural transformation, there is an isomorphism

Cls ∼= (U, 2)-Cat


Proof. This is another application of Theorem 1.5.3 with Exercise 1.D in the case of theup-set monad.

Exercises

1.A The slice under P. Let S = (S, n, d), T = (T,m, e) be monads on Set, and α : S −→ Ta monad morphism. The following statements are equivalent for objects σ : P −→ S andτ : P −→ T of the comma category P/MNDSet:

(i) α is a morphism σ −→ τ ;

(ii) αX : SX −→ TX is a sup-map for all sets X.

1.B Constructing the power-enrichment. Let T = (T,m, e) be a monad on Set. Supposethat the sets TX are equipped with an order that make them complete lattices, and is suchthat Tf : TX −→ TY and mX : TTX −→ TX are sup-maps for all maps f : X −→ Y and


sets X. The monad morphism τ : P −→ T of Proposition 1.2.1 is then given componentwiseby

τX(A) = ∨x∈A eX(x) .

1.C The double-powerset monad is not power-enriched. Consider the maps f, g : a, b −→P 2? given by

f(a) = ∅ , f(b) = ?g(a) = ∅ , g(b) = ∅, ? .

One has f(x) ⊆ g(x) for all x ∈ a, b, but if x = a, b ∈ P 2a, b, then fP2(x ) 6⊆

gP2(x ), since

fP2(x ) = ∅, ? , gP

2(x ) = ? ,where (−)P2 = mX · P 2(−) comes from the double-powerset monad P2. Hence, neitherthe principal filter natural transformation τ : P −→ P2, nor the natural transformationσ : P −→ P2 of Example 1.2.3(6) make P2 into a power-enriched monad.

1.D Kleisli monoids from the double-powerset monad. The adjunction P • a (P •)op :Setop −→ Set of Example II.2.5.1(6) yields a one-to-one correspondence

Xf−→ P 2Y

PXP •f←− PY .

(1.5.ii)

Both hom-sets Set(X,P 2Y ) and Set(PY, PX) inherit a pointwise order, the first fromthe refinement order on P 2Y and the second from the inclusion order on PX. If f, g ∈Set(X,P 2Y ) and P •f, P •g ∈ Set(PY, PX) are maps obtained via (1.5.ii), then

f ≤ g ⇐⇒ P •g ≤ P •f .

If we denote by PSet the category whose objects are sets and morphisms from X to Y aremaps f : PX −→ PY , then the correspondence (1.5.ii) describes an isomorphism of orderedcategories:

SetP2 ∼= PSetco op .

Moreover, a map ν : X −→ P 2X factors through UX → P 2X (where UX is the set of up-sets on X) precisely when P •ν is monotone, and ν : X −→ UX factors through FX → UX(where FX is the set of filters on X) if and only if P •ν preserves finite intersections. Thereare therefore isomorphisms (see II.2.1)

U-Mon ∼= Int and F-Mon ∼= Top .

1.E Finitary up-sets. An up-set a ∈ UX is finitary if for all A ⊆ X,

A ∈ a =⇒ ∃F ∈ a (F finite & F ⊆ A) ;


the set of all finitary up-sets on a set X is denoted by UfinX. Show that the components ofthe up-set monad U restrict to such elements to yield the finitary-up-set monad Ufin on Set.With every set UfinX ordered by subset inclusion, Ufin is power-enriched.The category of Ufin-monoids is isomorphic to the full subcategory Clsfin of closure spacesCls (see III.2.5):

Ufin-Mon ∼= Clsfin .

1.F The clique monad and closure spaces. A clique a on a set X is a subset of PX suchthat

(1) A,B ∈ a =⇒ A ∩B 6= ∅,

(2) A ∈ a, A ⊆ B =⇒ B ∈ a,

for all A,B ∈ PX. The set of all cliques on X is denoted by CX, and the double-powersetmonad P2 restricts to such sets to yield the clique monad C. The Kleilsi category of theclique monad is a separated ordered category, and there is an isomorphism

U-Mon ∼= C-Mon .

Since U-Mon is isomorphic to the category Cls of closure spaces, cliques provide an alternativestructure for the study of closure spaces via the isomorphism Cls ∼= C-Mon.

1.G The Kleisli extension to Rel is associative. For a power-enriched monad (T, τ), themonoid homomorphisms nbhd and conv of 1.5 can be extended to yield monotone maps

nbhd = nbhdX,Y : Rel(TX, Y ) −→ Set(Y, TX)conv = convY,X : Set(Y, TX) −→ Rel(TX, Y )

that form an adjunction nbhd a conv for all sets X, Y . When T is equipped with its Kleisliextension, one has

nbhd(s r) = nbhd(r) nbhd(s) , conv(µ) conv(ν) = conv(ν µ)nbhd(1]X) = eX , conv(eX) = 1]X

for all unitary relations r, s : TX −→7 Y , and maps µ, ν : Y −→ TX. As a consequence, theKleisli extension T is associative, and one can form the category (T, 2)-URel.The maps nbhd and conv define functors nbhd : SetT −→ (T, 2)-URelop and conv : (T, 2)-URelop −→SetT that determine a 2-isomorphism

SetT ∼= (T, 2)-URelop .

1.H Unitary relations as sup-maps. For a power-enriched monad (T, τ) and a relationr : TX −→7 Y , the following statements are equivalent:

(i) r is unitary;


(ii) r[ = ↓TX∨r[;

(iii) r(−, y) : TX −→ 2op is a sup-map for all y ∈ Y .


2 Lax extensions of monads

In Corollary 1.5.4 we effectively established an isomorphism (F, 2)-Cat −→ (, 2)-Cat, bothcategories being isomorphic to Top. It turns out that this isomorphism may be thoughtof as induced by the monad morphism −→ F. More generally, in this subsection we seeksufficient conditions for a monad morphism α : S −→ T into a power-enriched monad T toinduce an isomorphism

Aα : (T,V)-Cat −→ (S,V)-Catwhen S and T are equipped with adequate lax extensions; here, Aα is the algebraic functorof α, see III.3.4. For this, we proceed in two steps: we first obtain isomorphisms

(S, 2)-Cat ∼= T-Mon ∼= (T, 2)-Cat ,

and then consider the case where 2 is replaced by an arbitrary quantale V (Theorems 2.3.3and 2.5.3). Each of these steps is facilitated by the construction of lax extensions inducedby a lax extension T of T to Rel. Specifically, in 2.1 we “transfer” the lax extension from Tto S along α, and in 2.4 we describe a process of generating a lax extension of T to V-Relfrom a lax extension to Rel.

2.1 Initial extensions. In III.1.10 and 1.4, two constructions of a lax extension of amonad to Rel are given: the Barr extension and the Kleisli extension. In practice, the Barrextension can be extracted from the Kleisli extension of a larger monad. For example, theBarr extension of the ultrafilter functor β : Set −→ Set:

x (βr) y ⇐⇒ x ⊇ r[y ]

(for all relations r : X −→7 Y , x ∈ βX, y ∈ βY ) is the restriction to ultrafilters of the Kleisliextension of the filter functor

x (F r) y ⇐⇒ x ⊇ r[y ]

(for all x ∈ FX, y ∈ FY ).More generally, if α : S −→ T is a natural transformation of Set-functors, and T is a laxextension of T to V-Rel, the initial extension of S induced by α is the lax extension S givenby

Sr := αY · T r · αX ,

for any V-relation r : X −→7 Y . In pointwise notation, the definition becomes

Sr(x , y ) = T r(αX(x ), αY (y )) ,

for all x ∈ SX, y ∈ SY . Before showing that S is indeed a lax extension of S if T is oneof T (Proposition 2.1.1), we briefly discuss the “initial” terminology.

2. LAX EXTENSIONS OF MONADS 299

Recall from III.3.4 that a morphism of lax extensions α : (S, S) −→ (T, T ) is a naturaltransformation α : S −→ T that extends to an oplax transformation S −→ T :

αY · Sr ≤ T r · αX ,

for all V-relations r : X −→7 Y . If U : V-LXT −→ SetSet denotes the forgetful functor from thecategory of lax extensions to V-Rel (see III.3.A) that sends T to T , then the initial extensionis an U -initial morphism in the sense of II.5.6. Indeed, consider a natural transformationλ : R −→ S with a lax extension R of R; then λ is a morphism of lax extensions if and onlyif α · λ : R −→ T is one:

αY · λY · Rr ≤ T r · αX · λX ⇐⇒ λY · Rr ≤ αY · T r · αX · λX ⇐⇒ λY · Rr ≤ Sr · λX

for all relations r : X −→7 Y .

2.1.1 Proposition. For a lax extension T to V-Rel of a Set-functor T , and a naturaltransformation α : S −→ T , the initial extension S of S induced by α is a lax extension of S.Furthermore, if T belongs to a lax extension to V-Rel of a monad T = (T,m, e) andα : S −→ T is a monad morphism, then S also belongs to a lax extension of S = (S, n, d).

Proof. Since T preserves the order on the hom-sets V-Rel(X, Y ), it is immediate that Sdoes too. If r : X −→7 Y and s : Y −→7 Z are V-relations, then

Ss · Sr = αZ · T s · αY · αY · T r · αX≤ αZ · T s · T r · αX≤ αZ · T (s · r) · αX = S(s · r) ,

because T is a lax functor. As α is a natural transformation, we have αY · Sf = Tf · αXwhich can be written Sf ≤ αY · Tf · αX , or equivalently (Sf) ≤ αX · (Tf) · αY , in V-Rel;the extension conditions for S then follow because they are satisfied for T .Finally, suppose that T yields a lax extension of the monad T. Since e : 1V-Rel −→ T is oplax,then so is d : 1V-Rel −→ S: indeed, α is a monad morphism, so we have α · d = e, and

r ≤ eY · T r · eX = dY · Sr · dX ,

as expected. To verify oplaxness of n, we use that m · Tα · αS = α · n:

S Sr = αSY · T (αY · T r · αX) · αSX= αSY · (TαY ) · T T r · TαX · αSX≤ αSY · (TαY ) ·mY · T r ·mX · TαX · αSX = nY · Sr · nX ,

as required.


From this last result one infers that in presence of the initial extension S of S, the mapsαX : SX −→ TX become order-embeddings with respect to the orders III.3.3 induced bythe lax extensions:

x ≤ y ⇐⇒ αX(x ) ≤ αX(y )for all x , y ∈ SX. In fact, this condition witnesses the smooth interaction of the initial andKleisli extensions, as we will see next.2.1.2 Proposition. A morphism α : (S, σ) −→ (T, τ) of power-enriched monads becomes amorphism α : S −→ T of the Kleisli extensions to Rel.When the sets SX and TX are equipped with the orders (1.2.i) induced by σ and τ respectively,the components αX are order-embeddings if and only if the initial extension of S induced byα is the Kleisli extension of S.

Proof. Observe that αX · rσ = rτ · αY for any relation r : X −→7 Y . Therefore,

x ≤ rσ(y ) =⇒ αX(x ) ≤ αX · rσ(y ) = rτ · αY (y )

for all x ∈ SX, y ∈ SY (αX is monotone by Exercise 1.A). This implies that α is amorphism between the respective Kleisli extensions. If αX is an order-embedding, theimplication above is an equivalence, so that α is initial. Conversely, if the initial extensionof S is the Kleisli extension, the equivalence also holds, and we can conclude that αX is anorder-embedding by choosing r = 1X .

2.1.3 Examples.

(1) For every monad S on Set, there is a unique monad morphism ! : S −→ 1 intothe terminal monad. When the latter is equipped with its largest lax extension?> : V-Rel −→ V-Rel (such that ?>r(?, ?) = > for all V-relation r : X −→7 Y as inExample III.1.4.2(3)), the initial extension S of ?> induced by ! is given by

Sr(x , y ) = > ,

for all x ∈ SX, y ∈ SY ; hence, S = S>.

(2) It is obvious from Example 1.4.2(4) that the Kleisli extension F of F can be obtainedas the restriction to filters of the Kleisli extension U of the up-set functor U ; inthis case, the components αX : FX −→ UX of α are the embeddings, and F is theinitial extension of U induced by α. Similarly, the Barr extension of the ultrafilterfunctor can be obtained by restriction of the Kleisli extension F of the filter functor,and the lax extension of the identity monad can also be seen as a restriction of theultrafilter functor (via the principal ultrafilter natural transformation). Thus, thechain of natural transformations, whose respective components are all embeddings, asdescribed in Example II.3.1.1(5):

1Set −→ β −→ F −→ U


yields the following chain of initial extensions:

1Rel −→ β −→ F −→ U .

2.2 Sup-dense and interpolating monad morphisms. The isomorphism (S, 2)-Cat ∼=(T, 2)-Cat that we are aiming for requires that the monads S and T be sufficiently compatible.The conditions we present continue to be guided by the case where S = , T = F andα : → F is the inclusion of the set of ultrafilters into the set of filters.Hence, consider a monad morphism α : S −→ T, where T = (T,m, e) is a power-enrichedmonad with structure τ : P −→ T and equipped with its Kleisli extension T , and S = (S, n, d)a monad equipped with its initial extension S induced by α. To be able to exploitthe adjunction ∨

TX a ↓TX as in Proposition 1.5.2, we introduce the transformationα : PS −→ T via

αX := ∨PαX = mX · τTX · PαX = αT

X · τSX ,

or equivalently, αX(A) = ∨αX(A) for all A ⊆ SX. Each αX preserves suprema, and

therefore has a right adjoint, denoted by α↓X : TX −→ PSX, so that

α↓X( f ) = x ∈ SX | αX(x ) ≤ f = α−1X · ↓TX( f )

for all f ∈ TX. The maps α↓X allow for a convenient description of the initial extension Sof S. Indeed, for a relation r : X −→7 Y , we have

(Sr)[ = α↓X · rτ · αY ,

so the order relation S1X on SX is given by (S1X)[ = α↓X · αX .The monad morphism α : S −→ T is sup-dense if one has

α · α↓ = 1TX ; (2.2.i)

in pointwise notation this says that every element of TX can be expressed as a supremumof αX-images of elements of SX:

∀f ∈ TX ∃A ⊆ SX ( f = ∨αX(A)) .

When S is a submonad of T and the embedding is sup-dense, we simply say that S issup-dense in T.The morphism α : S −→ T is interpolating for a relation r : SX −→7 X if

α↓X · αX · r[ ≤ (↓SX ·nX)P · (Sr)[ · dX (2.2.ii)

holds. This condition expands to α↓X · αX · r[ ≤ (α↓X · αX · nX)P · α↓SX · τSX · r[ and can bewritten pointwise as

αX(x ) ≤∨αX(y ) | y r y =⇒ ∃X ∈ SSX (x ≤ nX(X ) & αSX(X ) ≤ τSX · r[(y))


for all x ∈ SX, y ∈ X. If S is a submonad of T, the previous condition naturally has asimpler expression, and may be represented graphically by

x ≤ mX · τSX · r[(y) =⇒ ∃X :X_

≤ τSX · r[(y)

x ≤mX(X )A monad morphism α : S −→ T is interpolating if it is interpolating for all relationsr : SX −→7 X. If S is a submonad of T and the embedding is interpolating, we may simplysay that S is interpolating in T.Note that α is interpolating whenever it is a morphism of power-enriched monads α :(S, σ) −→ (T, τ). Indeed, since −SSX ≤ α↓SX · αSX , we have

α↓X · αX · r[ = α↓X · αTX · αSX · σSX · r[

= (α↓X · αTX · αSX)P · −SSX · σSX · r[

≤ (α↓X · αTX · αSX)P · α↓SX · αSX · σSX · r[

= (α↓X · αX · nX)P · α↓SX · τSX · r[ .

2.2.1 Examples.(1) Any power-enriched monad T = (T,m, e) comes with the interpolating monad mor-

phism α = e : I −→ T. Indeed, using that αTX = 1TX and −X ≤ α↓X · αX , we

haveα↓X · αX · r[ = α↓X · τX · r[ = (−X)P · α↓X · τX · r[ ≤ (α↓X · αX)P · α↓X · τX · r[

for all relations r : X −→7 X.

(2) If S = P is the powerset monad embedded in T = F via the principal filter morphismτ : P −→ F, then the interpolation condition is immediate since τ is a morphism ofpower-enriched monads.

(3)c© Consider the filter monad F with the principal monad morphism τ : P −→ F. Ev-ery filter is the supremum (that is, the intersection) of all ultrafilters finer than it(Corollary II.1.13.4), so the ultrafilter monad is sup-dense in F.Let us verify that is interpolating in F. For ultrafilters x , y on X and a relationr : βX −→7 X, suppose x ≤ ∑ rτ (y ) (with ∑ denoting the monad multiplication of ),that is,

∀B ∈ y (r[(B) ⊆ A =⇒ A ∈ x ) ,for all A ⊆ X (where A = z ∈ βX | A ∈ z, see III.2.2). If there existed A ∈ x andB ∈ y with A ∩ r[(B) = ∅, we would have r[(B) ⊆ (A) = (A), so that A ∈ x , acontradiction. Therefore, A∩ r[(B) 6= ∅ for all A ∈ x and B ∈ y , and there exists anultrafilter X on βX that refines both A | A ∈ x and rτ (y ). In particular, ∑X = x .By setting y = y, we observe that rτ (y ) = τβX · r[(y), so the interpolation conditionis verified.


2.3 (S, 2)-categories as Kleisli monoids. Given a lax extension S to V-Rel of a monadS on Set, and a non-trivial quantale V (see III.1.2), recall that (see III.4.1)

(S,V)-UGph

is the category whose object are pairs (X, r) with r : SX −→7 X a unitary V-relation, so

eX · Sr ·mX ≤ r and r · S1X ≤ r ,

and whose morphisms f : (X, r) −→ (Y, s) are maps f : X −→ Y satisfying

f · r ≤ s · Sf .

In addition, given a functor T : Set −→ Set that lifts tacitly along the forgetful functorOrd −→ Set, we can consider the lax comma category

(1Set ↓T )≤

whose objects are pairs (X, ν) with a map ν : X −→ TX, and whose morphisms are mapsf : X −→ Y with Tf · ν ≤ µ · f :

Xf//

ν

≤

Y

µ

TXTf// TY .

We now present conditions which will give us an isomorphism between the two categories(S, 2)-UGph and (1Set ↓T )≤ that restricts to an isomorphism

(S, 2)-Cat ∼= T-Mon .

Hypotheses. We consider a monad morphism α : S −→ T, where T = (T,m, e) is a monadpower-enriched by τ : P −→ T and equipped with its Kleisli extension T , and S = (S, n, d)is a monad equipped with the initial extension S induced by α. The sets SX and TX areequipped with the orders III.3.3 induced by the respective lax extensions, and hom-sets ofSet are ordered pointwise.Following 1.5, we define for all sets X an adjunction

Set(X,TX)conv //> Rel(SX,X) ,

nbhdoo

by exploiting the Ord-isomorphism Rel(SX,X) ∼= Set(X,PSX). More concretely, we set

conv : Set(X,TX) −→ Set(X,PSX) , ν 7−→ α↓X · ν ,nbhd : Set(X,PSX) −→ Set(X,TX) , r[ 7−→ αX · r[ .


The adjunction nbhd a conv follows from αX a α↓X ; to verify this, one can use the pointwise

notation of conv and nbhd:

x conv(ν) y ⇐⇒ αX(x ) ≤ ν(y) and nbhd(r)(y) = ∨TX αX(r[(y)) ,

for all relations r : SX −→7 X, and maps ν : X −→ TX. Naturally, conv and nbhd restrictto mutually inverse isomorphisms between the sets of fixpoints of (conv · nbhd) and of(nbhd · conv).The fixpoints of (nbhd · conv) are exactly the maps ν : X −→ TX such that:

∀y ∈ X ∃A ⊆ SX (ν(y) = ∨αX(A)) .

Hence, (nbhd · conv) = 1Set(X,TX) precisely when α is sup-dense. In turn, a relationr : SX −→7 X is a fixpoint of (conv · nbhd) precisely when

∀y ∈ X (αX(x ) ≤ ∨αX(r[(y)) =⇒ x r y) .

One obtains the following generalizations of Proposition 1.5.1, Proposition 1.5.2, andTheorem 1.5.3.

2.3.1 Lemma. For the adjunction nbhd a conv : Set(X,TX) −→ Rel(SX,X) definedabove, the following hold:

(1) Fix(nbhd · conv) = Set(X,TX) if and only if α is sup-dense;

(2) a relation r : SX −→7 X is a fixpoint of (conv · nbhd) if and only if it is unitary and αis interpolating for r.

Proof. The first point is immediate from the previous discussion.For a unitary relation r : SX −→7 X, we obtain as in the proof of Proposition 1.5.1

r[ = (↓SX ·nX)P · (Sr)[ · dX . (2.3.i)

If moreover α is interpolating for r, then

α↓X · αX · r[ ≤ (↓SX ·nX)P · (Sr)[ · dX = r[ ,

and r is indeed a fixpoint of (conv · nbhd).Conversely, if r : SX −→7 X is a fixpoint of (conv · nbhd), it is of the form conv(ν) for amap ν : X −→ TX, so r · S1X ≤ r, and r is right unitary. Moreover, if X Sr dX(y) holds,we can apply mX · TαX to each side of the inequality αSX(X ) ≤ rτ · eX(y) to conclude thatnX(X ) r y by the fixpoint condition, that is, r is left unitary. As r is unitary, (2.3.i) holdsand implies that α is interpolating for r.


2.3.2 Proposition. The adjoint maps nbhd and conv defined above satisfy

nbhd(s r) ≤ nbhd(r) nbhd(s) , conv(µ) conv(ν) ≤ conv(ν µ)nbhd(1]X) = eX , conv(eX) = 1]X

for all relations r, s : SX −→7 X, and maps µ, ν : X −→ TX (where 1]X = dX · S1X).Moreover, if α is sup-dense, then nbhd(s r) = nbhd(r) nbhd(s) for all relations r, s :TX −→7 X. If in addition α is interpolating, then conv(µ) conv(ν) = conv(ν µ) alsoholds.

Proof. The displayed equalities follow from the fact that

x (1]X) y ⇐⇒ x (dX · αX · T1X · αX) y ⇐⇒ αX(x ) ≤ eX(y)

for all x ∈ SX and y ∈ X.To show that nbhd(s r) ≤ nbhd(r) nbhd(s), we first note that

αX · PnX = (αX · 1SSX)T · τSSX = (αTX · αSX)T · τSSX = αT

X · αSX ;

by composing these equalities with α↓SX · rτ · αX on the right, we obtain

αX · PnX · α↓SX · rτ · αX ≤ αT

X · rτ · αY . (2.3.ii)

We can now proceed as in Proposition 1.5.2:

nbhd(r) nbhd(s) = (αX · r[)T · αX · s[

= αTX · (τSX · r[)T · αT

X · τSX · s[ ((gT · f)T = gT · fT)= (αT

X · rτ · αX)T · τSX · s[ (gT · fT = (gT · f)T)≥ (αX · PnX · (Sr)[)T · τSX · s[ (by (2.3.ii))= αT

X · (τSX · PnX · (Sr)[)T · τSX · s[ ((gT · f)T = gT · fT)= αT

X · τSX · (PnX · (Sr)[)P · s[ (τ natural transformation)= αX · (((nX)[)P · (Sr)[)P · s[ (PnX = ((nX)[)P)= αX · (s · Sr · nX)[ (Rel = SetP)= nbhd(s r) .

The inequality for conv follows from the adjunction nbhd a conv.If α is sup-dense, then the inequality in (2.3.ii) becomes an equality, so that nbhd(r) nbhd(s) = nbhd(s r). For maps µ, ν : X −→ TX, the (S, 2)-relations conv(µ) and conv(ν)are unitary, and therefore so is conv(µ) conv(ν) (Exercise III.1.N). The claim for convthen follows from nbhd(r) nbhd(s) = nbhd(s r) and the fact that nbhd and conv form abijection between Set(X,TX) and the set of all unitary (S, 2)-relations r : X −7 X (seeLemma 2.3.1).


2.3.3 Theorem. Let (T, τ) be a power-enriched monad together with a monad morphismα : S −→ T, and suppose that T is equipped with its Kleisli extension T , and S with the initialextension of T induced by α. If α is sup-dense, then there is a full reflective embedding(1Set ↓T )≤ → (S, 2)-UGph that commutes with the underlying-set functors and restricts to afull reflective embedding

T-Mon → (S, 2)-Cat .If α is also interpolating, then this functor is an isomorphism.

Proof. For every map ν : X −→ TX, the relation conv(ν) is a fixpoint of (conv · nbhd), andis therefore unitary by Lemma 2.3.1. Similarly, a unitary relation r : SX −→7 X yields amap nbhd(r) : X −→ TX. Thus, we can consider the functors

C : (1Set ↓T )≤ −→ (S, 2)-UGphN : (S, 2)-UGph −→ (1Set ↓T )≤

defined on objects by C(X, ν) = (X, conv(ν)) and N(X, r) = (X, nbhd(r)), and leavingmaps untouched (the fact that C and N send morphisms to morphisms follows easily fromthe definitions; see also Exercise 1.G). The adjunction nbhd a conv yields an adjunctionN a C. Lemma 2.3.1 shows that if α is sup-dense, then C is a full reflective embedding,and Proposition 2.3.2 yields that C restricts to a functor T-Mon → (S, 2)-Cat. Finally, if αis also interpolating, then C is an isomorphism by Lemma 2.3.1 again.

Theorem 1.5.3 now appears as a direct consequence of this more general result, since α = 1Tis both sup-dense and interpolating. Moreover, the category of Kleisli monoids provides alink between presentations of lax algebras.2.3.4 Proposition. If α : S −→ T is sup-dense as in Theorem 2.3.3, then the algebraicfunctor

Aα : (T, 2)-Cat −→ (S, 2)-Catof III.3.4 is a full reflective embedding. If α is also interpolating, then Aα is an isomorphism.

Proof. The isomorphism (T, 2)-Cat ∼= T-Mon of Theorem 1.5.3 composed with the fullreflective embedding T-Mon → (S, 2)-Cat of Theorem 2.3.3 sends (X, a : TX −→7 X) to(X, a ·αX : SX −→7 X). Hence, this composition is precisely the algebraic functor Aα. Whenα is interpolating, Aα is an isomorphism by Theorem 2.3.3.

2.3.5 Examples.

(1) Depending on whether a relation r on a set X is seen as a map

r : X ×X −→ 2 , r : X −→ PX , or r : PX ×X −→ 2 ,

the category Ord of ordered sets is described respectively as any of the three categories

2-Cat ∼= P-Mon ∼= (P, 2)-Cat .


(2) c©Whether ultrafilter convergence, neighborhood systems, or filter convergence is chosenas defining structure, the category Top of topological spaces appears as

(, 2)-Cat ∼= F-Mon ∼= (F, 2)-Cat .

2.4 Strata extensions. To pass from (T, 2)-categories to (T,V)-categories, one views aV-relation r : X −→7 Y as a family of 2-relations (rv : X −→7 Y )v∈V indexed by V , via

x rv y ⇐⇒ v ≤ r(x, y) (2.4.i)

for all x ∈ X, y ∈ Y , and v ∈ V. The relation rv is referred to as the v-stratum ofr. Conversely, given a family (rv : X −→7 Y )v∈V of relations, one can define a V-relationr : X −→7 Y by setting

r(x, y) :=∨v ∈ V | x rv y .

Starting with a V-relation r : X −→7 Y , the V-relation obtained from the family (rv : X −→7Y )v∈V is r again. The family (rv)v∈V obtained from r is not arbitrary, as one has for exampleif u, v ∈ V

u ≤ v =⇒ rv ≤ ru .

In fact, a V-indexed family comes from a V-relation precisely when the family can beidentified with an inf-map r(−) : Vop −→ Rel(X, Y ).2.4.1 Lemma. For any sets X and Y , if the set Inf(Vop,Rel(X, Y )) is ordered pointwise,then the preceding correspondence between V-relations and V-indexed families of relationsdescribes an Ord-isomorphism

V-Rel(X, Y ) ∼= Inf(Vop,Rel(X, Y )) .

Proof. If (rv)v∈V is obtained from a V-relation r, then one has

x (∧v∈A rv) y ⇐⇒ ∀v ∈ A (x rv y) ⇐⇒ ∀v ∈ A (v ≤ r(x, y)) ⇐⇒ ∨A ≤ r(x, y)

for all A ⊆ V, x ∈ X, y ∈ Y , so r(−) is an indeed an inf-map r(−) : Vop −→ Rel(X, Y ). Astraightforward verification shows that if r was originally the image of a family (r′v)v∈V then(rv)v∈V = (r′v)v∈V , and the discussion preceding the statement allows us to conclude.

Suppose that a lax extension T : Rel −→ Rel of a functor T : Set −→ Set is given. For everyV-relation r : X −→7 Y , we set

TVr(x , y ) :=∨v ∈ V | x (T rv) y . (2.4.ii)

The assignment

TV : V-Rel −→ V-Rel , (r : X −→7 Y ) 7−→ (TVr : TX −→ TY ) ,

is called the strata extension of T along Rel −→ V-Rel.2.4.2 Proposition. Let T be a Set-functor. If T is a lax extension of T to Rel, then thestrata extension TV of T is a lax extension of T to V-Rel.


Proof. We first notice that in V-Rel

Tf ≤ T f ≤ TVf , (Tf) ≤ T (f ) ≤ TV(f )

because both f and f take values in ⊥, k. For V-relations r, r′ : X −→7 Y with r ≤ r′

one has rv ≤ r′v, and then T rv ≤ T r′v for all v ∈ V, which implies TVr ≤ TVr′. To verify

TVs · TVr ≤ TV(s · r) for all V-relations r : X −→7 Y , and s : Y −→7 Z, consider x ∈ TX,y ∈ TY , and z ∈ TZ. If there are u, v ∈ V with x T ru y and y T sv z, then x T (sv · ru) zholds by the hypothesis on T ; but sv · ru ≤ (r · s)u⊗v, so x (T (r · s)u⊗v) z holds. Hence,

TVr(x , y )⊗ TVs(y , z) = ∨u ∈ V | x (T ru) y ⊗ ∨v ∈ V | y (T sv) z≤ ∨u⊗ v ∈ V | x (T (r · s)u⊗v) z ≤ TV(r · s)(x , z)

concludes the proof.

2.4.3 Proposition. For V completely distributive, the strata extension TV of a lax extensionT of T to Rel yields a lax extension of T to V-Rel.

Proof. For any V-relation r : X −→7 Y , one has rv ≤ eY · T rv · eX for all v ∈ V, so thatr ≤ eY · TVr · eX . To see that the monad multiplication is oplax, let us first show that foreach v ∈ V and u v, we have (TVr)v ≤ T ru. Indeed, consider x ∈ TX, y ∈ TY , v ∈ Vsuch that x (TVr)v y , that is, v ≤ TVr(x , y ); by definition of in a completely distributivelattice, there is for each u v a u′ ∈ V with u ≤ u′ and x (T ru′) y , so that x (T ru) y . AsT is monotone, and V is completely distributive, we can therefore write for X ∈ TTX andY ∈ TTY :∨v ∈ V | X T (TVr)v Y ≤

∨u ∈ V | X (T T ru) Y ≤

∨u ∈ V | mX(X ) (T ru) mY (Y )

These inequalities then yield TV TVr(X ,Y ) ≤ TVr(mX(X ),mY (Y )), as required.

2.4.4 Remarks.

(1) For a lax extension T of T to Rel, one has for V = 2:

T2r(x , y ) = > ⇐⇒ x r y

for all relations r : X −→7 Y , and x ∈ TX, y ∈ TY , that is, the strata extension of Tto 2-Rel = Rel returns T :

T2 = T .

(2) The strata extension to V-Rel of the identity functor 1Rel, which is a lax extension of1Set to Rel (see Example III.1.5.2), returns the identity functor 1V-Rel.


The Barr extension of the ultrafilter monad to P+-Rel given in III.2.4 is simply thestrata extension of the Barr extension to Rel along Rel −→ P+-Rel. More generally, fora completely distributive V, the Barr extension of β to V-Rel is given by the strataextension to V-Rel of its Barr extension to Rel:

βr(x , y ) := βVr(x , y ) =∨v ∈ V | x (βrv) y ,

and one obtains the corresponding equivalent expression

βr(x , y ) =∧

A∈x ,B∈y

∨x∈A,y∈B

r(x, y)

for all V-relations r : X −→7 Y , and x ∈ βX, y ∈ βY . Proposition 2.4.3 then offers ageneralization of Proposition III.2.4.3, as follows.2.4.5 Corollary. For V completely distributive, the Barr extension = (β,m, e) is a flatlax extension to V-Rel of the ultrafilter monad = (β,m, e).

Proof. By Proposition 2.4.3, we only need to verify that the Barr extension is flat. Forv ∈ V with ⊥ < v, one has

(1X)v(x, y) =

1X(x, y) if v ≤ k

⊥ otherwise,

for all x, y ∈ X. Moreover, if r = ⊥X : X −→7 X is the constant relation to ⊥, that is, theempty relation, then βr = ⊥X . By definition, one therefore has βV1X(x , y ) = β1X(x , y )for all x , y ∈ βX, so βV is flat.

For a power-enriched monad T and completely distributive V , the Kleisli extension T of Tto V-Rel is given by the strata extension to V-Rel of its Kleisli extension to Rel:

T r(x , y ) := TVr(x , y ) =∨v ∈ V | x ≤ (rv)τ (y ) ,

for all V-relations r : X −→7 Y and x ∈ TX, y ∈ TY . As in the case of the Barr extensionabove, there are alternate descriptions of the lax extensions given in Examples 1.4.2 whenV is completely distributive; see Exercise 2.H.The strata extension has a very good behavior with respect to the order induced on thesets TX and to initial extensions.2.4.6 Proposition. Let T be a lax extension to Rel of a Set-functor T . The order inducedon TX by the strata extension TV of T is the order induced by T .

Proof. Suppose first that x , y ∈ TX are such that x T1X y . Since 1X = (1X)k for1X : X −→ X seen as a V-relation, we have k ≤ TV1X(x , y ) by definition of the strataextension to V-Rel. If on the other hand, one has k ≤ TV1X(x , y ), because V is non-trivial (by convention, see 2.3) there exists v ∈ V such that ⊥ < v and x T (1X)v y ; as(1X)v = (1X)k = 1X because 1X takes values in ⊥, k, we can conclude that x T1X y .


2.4.7 Proposition. If α : (S, S) −→ (T, T ) is a morphism of lax extensions, then α :(S, SV) −→ (T, TV) is one too. Moreover, if S is the initial extension of S to Rel induced byα : S −→ T , then SV is the initial extension of S to V-Rel induced by α (where T is equippedwith its respective lax extensions T and TV).

Proof. By definition of a morphism of lax extensions and of the strata extension,SVr(x , y ) =

∨v ∈ V | x (Srv) y ≤

∨v ∈ V | αX(x ) (T rv) αY (y ) = TVr(αX(x ), αY (y ))

for all x ∈ SX, y ∈ SY and V-relation r : X −→7 Y . Hence, α is a morphism between therespective strata extensions and if S is initial, then SV is initial.

2.5 (S,V)-categories as Kleisli towers. The strata extension of a lax extension 2.4constructs a lax extension TV to V-Rel from a lax extension T to Rel of a monad T onSet. Such a lax extension TV then determines a category of (T,V)-categories. In contrast,the tower construction that we describe here determines (T,V)-categories directly from acategory of (T, 2)-categories by exploiting the isomorphism

V-Rel(X, Y ) ∼= Inf(Vop,Rel(X, Y ))of Lemma 2.4.1. Under this isomorphism, reflexive and transitive (T,V)-relations a : X −7 Xcorrespond to certain inf-maps Vop −→ Rel(TX,X) whose characterization depends onlyon the original lax extension T to Rel (Proposition 2.5.2 below). In fact, these inf-mapscorestrict to fixpoints of nbhd · conv so that Proposition 1.5.1 allows us to consider insteadinf-maps Vop −→ Set(X,TX), in effect relating (T,V)-category structures with T-monoidstructures. In this subsection, we pursue this idea in the presence of a sup-dense andinterpolating monad morphism α : S −→ T, considering Rel(SX,X) in lieu of Rel(TX,X)and using Lemma 2.3.1 instead of Proposition 1.5.1. We then obtain Theorem 2.5.2 as ageneralization of Theorem 2.3.3.The following Lemma will be used throughout without explicit mention in our proofs.2.5.1 Lemma. For all maps f : X −→ Y , g : Y −→ Z, V-relations r : X −→7 Y , s : Y −→7 Zand u, v ∈ V,

sv · f = (s · f)v , g · rv = (g · r)v and su · rv ≤ (s · r)v⊗u .

Proof. The expressions follow directly from the definition of the strata of a relation(see 2.4).

2.5.2 Proposition. Let T be a lax extension of T to Rel and V a quantale. Via theisomorphism V-Rel(TX,X) ∼= Inf(Vop,Rel(TX,X)) of Lemma 2.4.1, there is a one-to-onecorrespondence between transitive and reflexive (T,V)-relations r : X −7 X and inf-mapsr(−) : Vop −→ (T,V)-URel(X,X) that satisfy

rv ru ≤ ru⊗v and eX ≤ rk

for all u, v ∈ V.


Proof. For a transitive V-relation r : TX −→ X, we show rv · (TVr)u ≤ ru⊗v ·mX for allu, v ∈ V: since T ru ≤ (TVr)u by definition, ru rv ≤ rv⊗u will follow. If z ∈ X andX ∈ TTX are such that X (rv · (TVr)u) z, then there exists y ∈ TX with v ≤ r(y , z) andu ≤ TVr(X , y ). Therefore,

u⊗ v ≤∨

y∈TXTVr(X , y )⊗ r(y , z) = r · TVr(X , z) ≤ r ·mX(X , z) ,

and (r ·mX)u⊗v = ru⊗v ·mX yields the required inequality. If r : TX −→7 X is a reflexiveV-relation, then eX ≤ rk because eX takes values in ⊥, k.Conversely, assume that rv ru ≤ ru⊗v holds for all u, v ∈ V. One has to show thatTVr(X , y ) ⊗ r(y , z) ≤ r(mX(X ), z) for all z ∈ X, y ∈ TX, X ∈ TTX. Choose elementsz, y ,X and set v := r(y , z), A := u ∈ V | X (T ru) y, so that TVr(X , y ) ⊗ r(y , z) =∨u∈A u⊗ v. Since by hypothesis rv · T ru ≤ ru⊗v ·mX , we have u⊗ v ≤ r(mX(X ), z) for all

u ∈ A. Hence, r is transitive. Reflexivity of r follows from eX ≤ rk ≤ r.

Let now (T, τ) be a power-enriched monad, so that all hom-sets SetT(X,X) are com-plete lattices when equipped with the pointwise order (see Proposition 1.2.1). To extendTheorem 2.3.3 to V-relations, we consider the category

(T,V)-Mon

whose objects are Kleisli (T,V)-towers, that is, pairs (X, ν(−)) with X a set and ν(−) :Vop −→ SetT(X,X) an inf-map satisfying

νu νv ≤ νu⊗v and eX ≤ νk

for all u, v ∈ V. A morphism of Kleisli (T,V)-towers f : (X, ν(−)) −→ (Y, µ(−)) is a mapf : X −→ Y such that

Tf · νv ≤ µv · ffor all v ∈ V .

2.5.3 Theorem. Let (T, τ) be a power-enriched monad, α : S −→ T a sup-dense andinterpolating monad morphism, and suppose that V is completely distributive. With respectto the initial extension SV of S induced by α (where T is equipped with its Kleisli extensionTV), there is an isomorphism

(S,V)-Cat ∼= (T,V)-Mon


Proof. By Lemma 2.3.1, a Kleisli (T,V)-tower (X, ν(−)) corresponds to a pair (X, r(−)),with r(−) : Vop −→ (S, 2)-URel(X,X) an inf-map (set rv := conv(νv) so that νv = nbhd(rv)).By Proposition 2.3.2, the conditions for ν(−) to be a Kleisli (T,V) tower translate as

rv ru ≤ ru⊗v and eX ≤ rk


for all u, v ∈ V. Proposition 2.5.2 then yields a bijective correspondence between Kleisli(T,V)-towers (X, ν(−)) and (T,V)-categories (X, r).Consider now a morphism f : (X, ν(−)) −→ (Y, µ(−)) of Kleisli (T,V)-towers. By Theo-rem 2.3.3, f · conv(νv) ≤ conv(µv) · Tf for all v ∈ V; hence, by definining r and s as the(S,V)-relations corresponding respectively to the families (conv(νv) : SX −→7 X)v∈V and(conv(µv) : SY −→7 Y )v∈V via Lemma 2.4.1, one obtains that f : (X, r) −→ (Y, s) is an(S,V)-functor. Conversely, if f : (X, r) −→ (Y, s) is an (S,V)-functor, then

f · rv = (f · r)v ≤ (s · Tf)v = sv · Tf

for all v ∈ V . This proves that the mentioned correspondence on objects is functorial.

2.5.4 Corollary. Let (T, τ) be a power-enriched monad and suppose that V is completelydistributive. With respect to the Kleisli extension TV , there is an isomorphism

(T,V)-Cat ∼= (T,V)-Mon .

Proof. Apply the Theorem to α = 1T which is both sup-dense and interpolating.

2.5.5 Examples.

(1) The category Met of metric spaces (see Example III.1.3.1(2)) is isomorphic to any ofthe categories

P+-Cat ∼= (P,P+)-Mon ∼= (P,P+)-Cat ,(with P equipped with its Kleisli extension to P+-Rel).

(2)c© The category App of approach spaces can equivalently be described as

(,P+)-Cat ∼= (F,P+)-Mon ∼= (F,P+)-Cat ,

(with equipped with its Barr extension and F with its Kleisli extension to P+-Rel).We use here the isomorphism (,P+)-Cat ∼= App of Theorem III.2.4.5, and therefore theAxiom of Choice, to prove these results. Nevertheless, the isomorphism (F,P+)-Mon ∼=App can be established without the use of the Axiom of Choice, see Exercise 2.K.

(3)c© The category of “many-valued neighborhood spaces” is obtained as

(, [0, 1])-Cat ∼= (F, [0, 1])-Mon ∼= (F, [0, 1])-Cat ,

where the frame [0, 1] is considered a quantale (see II.1.10). Analogous results holdfor F replaced by U, giving rise to extensions of the category of interior (or closure)spaces.

(4)c© (, 22)-Cat is the category BiTop of bitopological spaces and bicontinuous maps (Ex-ercise III.2.D). This category can also be described as (F, 22)-Mon, or as (F, 22)-Cat,avoiding the use of the Axiom of Choice.


(5) Tower extensions allow us to effectively describe (T,V)-categories for different latticesV . For instance, (F, 0, 1, 2)-Cat, where the frame 0, 1, 2 is considered a quantale, isisomorphic to the category whose objects are triples (X,O0X,O1X), with topologiesO0X ⊆ O1X, and whose morphisms are maps which are continuous with respect toboth topologies.

(6) For the quantale 3 = (⊥, k,>) (see Example II.1.10.1(2)), there are isomorphisms

3-Cat ∼= (P, 3)-Mon ∼= (P, 3)-Cat .

Objects of 3-Cat are sets X with a map r(−) : 3 −→ Rel(X,X) such that r⊥ satisfiesx r⊥ x for all x ∈ X, rk is a reflexive and transitive relation, and r> is a relationsuch that r> ≤ rk, r> · rk ≤ r>, and rk · r> ≤ r>; morphisms of 3-Cat are maps thatpreserve the relations rk and r>. Therefore, 3-Cat is isomorphic to the category whoseobjects are ordered sets (X,≤X) equipped with an auxiliary relation, that is, a modulea : X −→7 X such that a ≤ (≤X) (see II.1.4), and whose morphisms are monotonemaps preserving the auxiliary relation.

Exercises

2.A The category of lax extensions to V-Rel. The forgetful functor U : V-LXT −→ SetSet

(that sends a lax extension T : V-Rel −→ V-Rel to its underlying functor T : Set −→ Set) istopological: if (Ti, Ti)i∈I is a family of lax extensions to V-Rel, then every family (αi : S −→Ti)i∈I of natural transformations admits a U -initial lifting (αi : (S, S) −→ (Ti, Ti))i∈I with

Sr :=∧i∈I

(αi)Y · Tir · (αi)X ,

or equivalently,Sr(x , y) =

∧i∈I

T r((αi)X(x ), (αi)Y (y)) ,

for all V-relations : X −→7 Y , x ∈ TX, and y ∈ TY . Moreover, if all αi are morphismsof monads αi : S −→ Ti, and all Ti are lax extensions of Ti = (Ti,mi, ei), then S is a laxextension of S = (S, n, d).

2.B The Zariski topology on a set of ultrafilters. Consider the set βX of all ultrafilters on aset X, and the closure operation c : PβX −→ PβX given by

x ∈ c(A) ⇐⇒ x ≤ ∨A ⇐⇒ x ⊇ ⋂Afor all x ∈ βX, A ⊆ βX. This closure operation defines the Zariski topology on βX whoseclosed sets can be identified with the set of filters FβX.A relation r : βX −→7 Y is therefore a fixpoint of (conv · nbhd) : Rel(βX,X) −→ Rel(βX,X)exactly when r[(y) is closed in this topology on βX for all y ∈ X.


2.C The neighborhood map as a monoid homomorphism. For a power-enriched monad Tequipped with its Kleisli extension T to Rel, a monad morphism α : S −→ T inducing theinitial extension on S, and relations r, s : SX −→7 X, the map

nbhd : Set(X,PSX) −→ Set(X,TX)

satisfies nbhd(s r) = nbhd(r) nbhd(s) if and only if one of the following equivalentconditions hold:

(i) αTX · rτ · αX = αX · PnX · (Sr)[;

(ii) αTX · rτ · αX ≤ αX · PnX · (Sr)[;

(iii) αTX · rτ · αX(y ) ≤ ∨αX · nX(X ) | X ∈ SSX : αSX(X ) ≤ rτ · αX(y ) for all y ∈ SX.

Hint. For (ii) =⇒ (i), use that αTX · rτ · αX = αX · PnX · (Sr)[.

2.D Full coreflective subcategories of T-Mon. Consider a monad morphism α : S −→ T intoa power-enriched monad (T, τ) equipped with its Kleisli extension T , where S is equippedwith its initial extension S induced by α. If α is interpolating and satisfies

αTX · rτ · αX ≤ αX · PnX · (Sr)[

(see Exercise 2.C) for all for all unitary relations r : SX −→7 X, then there is a full coreflectiveembedding (S, 2)-UGph → (1Set ↓T )≤ that commutes with the underlying-set functors andrestricts to

(S, 2)-Cat → T-Mon .By composing this embedding with the isomorphism T-Mon ∼= (T, 2)-Cat, one obtains acoreflective embedding (S, 2)-Cat −→ (T, 2)-Cat that sends (X, r : SX −→7 X) to (X, r :TX −→7 X), where

r( f , y) =∧r(x , y) | x ∈ SX : αX(x ) ≤ f ,

for all f ∈ TX, y ∈ X. In particular, the principal filter monad morphisms P −→ F −→ Uyield full coreflective embeddings

Ord // Top // Cls .

2.E Ultracliques and closure spaces. With the set of cliques CX of Exercise 1.F orderedby reverse inclusion, an ultraclique is either the empty clique or a minimal element of CX.Alternatively, a non-empty ultraclique x is an up-set in PX such that

A ∈ x ⇐⇒ A /∈ x

for all A ∈ PX. Although the existence of “sufficiently many” ultracliques requires theAxiom of Choice (to obtain for example a clique version of Proposition II.1.13.2), these


structures appear to be less elusive than ultrafilters. For example, if the set X has at leastthree distinct elements x, y, z ∈ X, then a non-principal ultraclique is given by

↑PXx, y, y, z, z, x .

The clique monad restricts to ultracliques to form the ultraclique monad = (κ,m, e).With the clique and ultraclique monads both equipped with their initial extensions inducedby their inclusions in U, there are isomorphisms, there are isomorphisms

(C, 2)-Cat ∼= U-Mon ∼= (, 2) .

2.F A flat extension of the ultraclique monad. The initial extension of the non-emptyultraclique functor (see Exercise 2.E) is flat, although the functor does not satisfy theBeck–Chevalley condition.Hint. For the first statement, investigate the induced order on a set of non-empty ultracliques.For the second, consider the maps

x f// x, y x, y, z ,g

oo

where g(y) = g(z) = y and g(x) = x.

2.G Unitary V-relations are sup-maps. Let V be a completely distributive quantale and(T, τ) a power-enriched monad. For a unitary (T,V)-relation r : X −→7 Y (with respect tothe Kleisli extension of T to V-Rel), the map r(−, y) : TX −→ Vop preserves suprema for ally ∈ Y (see also Exercise 1.H).

2.H Alternative descriptions of lax extensions. If the lattice V is completely distributive,the Kleisli extensions to V-Rel of the powerset, filter, and up-set functors of Examples 1.4.2can equivalently be expressed as

P r(A,B) = ∧x∈A

∨y∈B r(x, y) ,

F r( f , g) = ∧B∈g

∨A∈f

∧x∈A

∨y∈B r(x, y) ,

Ur(x , y ) = ∧B∈y

∨A∈x

∧x∈A

∨y∈B r(x, y) ,

respectively, where r : X −→7 Y is a V-relation, A ∈ PX, B ∈ PY , f ∈ FX, g ∈ FY ,x ∈ UX and y ∈ UY .

2.I Monad retractions induce equivalences of categories. Let T be a lax extension of T andα : S −→ T a retraction of monads (so there exists a monad morphism ρ : T −→ S such thatα · ρ = 1T). If S is the initial extension induced by α, then (S,V)-Cat and (T,V)-Cat areequivalent categories.

2.J Topologicity of the functor (T, 2)-UGph −→ Set. For a lax extension T of a monad T onSet, the forgetful functor U : (T,V)-UGph −→ Set is topological.


2.K Approach spaces as (F,P+)-categories. Consider the filter monad F with the principalfilter natural transformation τ : P −→ F and the extended real half-line P+. The category ofreflexive and transitive P+-towers of (F, 2)-UGph is isomorphic to the category of reflexiveand transitive P+-towers of PSet (see Exercise 1.D). Moreover, P+-towers of PSet (cv :PX −→ PX)v∈V and maps δ : X × PX −→ P+ determine each other via

δ(x,A) := infv ∈ [0,∞] | x ∈ cv(A) and cv(A) = x ∈ X | δ(x,A) ≤ v ,

and this correspondence induces an isomorphism (F,P+)-Cat ∼= App.

2.L Quantales as (T,V)-categories. Let (T, τ) be a power-enriched monad, with T =(T,m, e), and V a completely distributive quantale (with totally below relation, see II.1.11).The map α : V −→ TV defined by

α(v) :=∨x ∈ TV | ∀u ∈ V (u v =⇒ x ≤ τV(↑u))

is an inf-map α : Vop −→ TV . Hence, α has a left adjoint ξ : TV −→ Vop in Ord. Verify thatthe pair (V , α) is a T-monoid, and therefore (V , ξ∗) is a (T, 2)-category (see II.1.4).The map α extends to α : Vop −→ Set(V , TV), via

α(u)(v) := α(u⊗ v) ,

so that (V , α) is a V-tower of T-Mon, that is, an object of (T,V)-Mon. The structureV-relation of the corresponding (T,V)-category (X, rα) is given by

rα(x , v) = ξ(x ) v

(see II.1.10).

3. LAX ALGEBRAS AS KLEISLI MONOIDS 317

3 Lax algebras as Kleisli monoids

In this subsection, we show that an associative lax extension to V-Rel of a monad T alwayshas an associated power-enriched monad that, when equipped with its Kleisli extensionto Rel, returns the same category of lax algebras:

(T,V)-Cat ∼= -Mon

(Theorem 3.2.2). In 3.3, an alternate description of the monad is given in the case ofapproach spaces, that is, when T = and V = P+.

3.1 The ordered category (T,V)-URel. Given an associative lax extension to V-Relof a monad T on Set, recall from III.1.9 that sets with unitary (T,V)-relations form thecategory (T,V)-URel, with Kleisli convolution as composition and the identity on X givenby 1]X = eX · T1X . Moreover, (T,V)-URel forms an ordered category when the hom-sets(T,V)-URel(X, Y ) ⊆ V-Rel(TX, Y ) are equipped with the pointwise order inherited fromV-Rel, since the Kleisli convolution preserves this order on the left and right.

3.1.1 Lemma. Let T be a lax extension of T to V-Rel. For a family of unitary (T,V)-relations (ϕi : X −7 Y )i∈I , the (T,V)-relation ∧

i∈I ϕi is unitary.

Proof. The statement follows from the inequalities

(∧i∈I ϕi) · T1X ≤∧i∈I(ϕi · T1X) = ∧

i∈I ϕi ,

eY · T (∧i∈I ϕi) ·mX ≤ ∧i∈I(eY · Tϕi ·mX) = ∧i∈I ϕi

since the inequalities in the other direction always hold.

This Lemma implies that the ordered category (T,V)-URel has complete hom-sets. However,in general (T,V)-URel is not a quantaloid since ϕ (−) typically does not preserve suprema.The situation is better for composition on the right, and when the maps (−) ϕ preservesuprema we say that (T,V)-URel is a right-sided quantaloid.

3.1.2 Proposition. Let T be an associative lax extension of T to V-Rel. Then, for every(T,V)-relation ϕ : X −7 Y and every set Z, the monotone map

(−) ϕ : (T,V)-Rel(Y, Z) −→ (T,V)-Rel(X,Z)

has a right adjoint (−) ϕ : (T,V)-Rel(X,Z) −→ (T,V)-Rel(Y, Z) that sends ψ : X −7 Z toψ (Tϕ ·mX) (see Exercise III.1.E). Moreover, if ϕ : X −7 Y and ψ : X −7 Z are unitary,then ψ ϕ is also unitary, and consequently (T,V)-URel is a right-sided quantaloid.

Proof. Consider a (T,V)-relation ϕ : X −7 Y . For all γ : Y −7 Z and ψ : X −7 Z,

γ ϕ ≤ ψ ⇐⇒ γ · Tϕ ·mX ≤ ψ ⇐⇒ γ ≤ ψ (Tϕ ·mX) ,


hence, the map

(−) ϕ : (T,V)-Rel(X,Z) −→ (T,V)-Rel(Y, Z) , ψ 7−→ (ψ ϕ) := ψ (Tϕ ·mX)

is right adjoint to (−) ϕ. Suppose now that ϕ : X −7 Y and ψ : X −7 Z are unitary. Byassociativity of the Kleisli convolution and Proposition III.1.9.4,

(1]Z (ψ ϕ)) ϕ ≤ 1]Z ψ = ψ and((ψ ϕ) 1]Y ) ϕ ≤ (ψ ϕ) ϕ ≤ ψ ;

therefore, 1]Z (ψ ϕ) ≤ ψ ϕ and (ψ ϕ) 1]Y ≤ ψ ϕ.

Recall from Exercise III.1.M that a map f : X −→ Y gives rise to a unitary (T,V)-relationf ] : Y −7 X via

f ] = f · eY · T1Y = eX · (Tf) · T1Y = eX · T (f ) ,

where 1]X = (1X)] is the identity morphism on X in (T,V)-URel.

3.1.3 Lemma. Let T be a lax extension of T to V-Rel. For a unitary (T,V)-relationϕ : X −7 Y , one has

f ] ϕ = f · ϕfor all maps f : Z −→ Y .

Proof. One computes f ] ϕ = f · eY · T1Y · Tϕ ·mX = f · (1]X ϕ) = f · ϕ.

If T is an associative lax extension of T to V-Rel, then f ] g] = (g · f)] for all mapsf : X −→ Y and g : Y −→ Z in Set:

f ] g] = f · g] = f · g · eZ · T1Z = (g · f)] .

Hence, there is a functor(−)] : Set −→ (T,V)-URelop

that maps objects identically. We now proceed to show that this functor is left adjoint tothe contravariant hom-functor

(T,V)-URel(−, 1) : (T,V)-URelop −→ Set ,

where 1 = ? denotes a singleton. We identify elements x ∈ X with maps x : 1 −→ X, andwith a unitary (T,V)-relation ψ : X −7 Y we associate the map ψ[ : Y −→ (T,V)-URel(X, 1)defined by

ψ[(y) := y] ψ = y · ψ = ψ(−, y)for all y ∈ Y (the second equality follows from Lemma 3.1.3, and the third by definition ofcomposition in V-Rel); here, ψ(−, y)(x , ?) := ψ(x , y).


The next Lemma shows that unitariness of a (T,V)-relation ϕ : X −7 Y can be tested justby using elements of Y .

3.1.4 Lemma. Let T be an associative lax extension of T to V-Rel, and ϕ : X −7 Y a(T,V)-relation. Then ϕ is unitary if and only if y · ϕ is unitary for all y ∈ Y .

Proof. If ϕ is unitary, then y ·ϕ = y] ϕ is unitary as well (see III.1.9). To verify the otherimplication, suppose that y · ϕ is unitary for all y ∈ Y . Then one has

y · (ϕ eX) = y · ϕ · T1X = y · ϕ

and

y · (eY ϕ) = y · eY · Tϕ ·mX = e1 · (Ty) · Tϕ ·mX = e1 · T (y · ϕ) ·mX = y · ϕ

for all y ∈ Y , so ϕ eX = ϕ and eY ϕ = ϕ.

3.1.5 Proposition. Let T be an associative lax extension of T to V-Rel.

(1) For a set X, the product 1X = ∏x∈X 1x in (T,V)-URel (with 1x = 1 for all x ∈ X)

exists, and can be chosen as 1X = X with projections πx = x] : X −7 1 (x ∈ X).

(2) The contravariant hom-functor

(T,V)-URel(−, 1) : (T,V)-URelop −→ Set

has (−)] : Set −→ (T,V)-URelop as left adjoint. The unit and counit of the associatedadjunction are given by the Yoneda maps

yX = (1]X)[ : X −→ (T,V)-URel(X, 1) , x 7−→ x]

and the evaluation relations

εX : X −7 (T,V)-URel(X, 1) , εX(x , ψ) = ψ(x , ?) ,

respectively.

Proof.

(1) For a family (φx : Y −7 1)x∈X of unitary (T,V)-relations, one can define a (T,V)-relation φ : Y −7 X by setting φ(y , x) = φx(y , ?) for all y ∈ TY . Since x · φ = φxis unitary for all x ∈ X, then so is φ by Lemma 3.1.4. Unicity of the connectingmorphism φ : Y −7 X follows from its definition.

(2) By Exercise II.2.L, the functor H = (T,V)-URel(−, 1) : (T,V)-URelop −→ Set has a leftadjoint F = 1(−) that sends a set X to its product 1X = X, and a map f : X −→ Y tothe unitary (T,V)-relation f ] : Y −7 X. The same exercise yields that y is indeed theunit of the adjunction and ε its counit.


3.2 The discrete presheaf monad. For an associative lax extension T of T to V-Rel,the adjunction described in Proposition 3.1.5 induces a monad

= (T,V) = (T,V , T) = (Π,m,y)

on Set, where

ΠX = (T,V)-URel(X, 1) , Πf(ψ) = ψ f ] ,mX(Ψ) = Ψ εX , yX(x) = x] ;

for all x ∈ X, f : X −→ Y , and unitary (T,V)-relations ψ : X −7 1, Ψ : ΠX −7 1. We call the discrete presheaf monad associated to T.The fully faithful comparison functor L : Set −→ (T,V)-URelop (Proposition II.3.6.1) isbijective on objects since the left adjoint (−)] : Set −→ (T,V)-URelop is so. That is, theKleisli category Set of is isomorphic to (T,V)-URelop. Explicitly, LX = X for each set X,and L sends a morphism r : X −→ ΠY in Set to the unitary (T,V)-relation r]εY : Y −7 X.By Lemma 3.1.3, r] εY (y , x) = r(x)(y , ?) for all x ∈ X and y ∈ TY , so the inverse of Lsends a unitary (T,V)-relation ϕ : Y −7 X to ϕ[ : X −→ ΠY .Since (T,V)-URel is a right-sided quantaloid (Proposition 3.1.2), ΠX = (T,V)-URel(X, 1) isa complete lattice when equipped with its pointwise order:

ψ1 ≤ ψ2 ⇐⇒ ∀x ∈ TX (ψ1(x ) ≤ ψ2(x )) ,

for all unitary (T,V)-relations ψ1, ψ2 : X −7 1. By Proposition 3.1.2, the map (−) ϕ :ΠY −→ ΠX preserves suprema for every unitary (T,V)-relation ϕ : X −7 Y ; in particular,Πf = (−) f ] and mX = (−) εX preserve suprema for every set X and every mapf : X −→ Y . The order on the sets ΠX therefore corresponds to a monad morphismτ : P −→ from the powerset monad P (Proposition 1.2.1). As a consequence, the hom-setsof Set are complete ordered sets when ordered pointwise:

f1 ≤ f2 ⇐⇒ ∀x ∈ X (f1(x) ≤ f2(x))

for all maps f1, f2 : X −→ ΠY . One then has for unitary (T,V)-relations ϕ1, ϕ2 : Y −7 X:

ϕ1 ≤ ϕ2 ⇐⇒ ∀x ∈ X ∀y ∈ TY (ϕ1(y , x) ≤ ϕ2(y , x)) ⇐⇒ ϕ[1 ≤ ϕ[2 .

We therefore obtain an isomorphism (see also the discussion after Lemma 3.1.3)

(T,V)-URel(Y,X) −→ Set(X, (T,V)-URel(Y, 1)) , ϕ 7−→ (ϕ[ : x 7−→ ϕ(−, x))

in Ord. Since (T,V)-URel is an ordered category, this shows that Set is also an orderedcategory.3.2.1 Proposition. Let T be an associative lax extension to V-Rel of a monad T on Set.The discrete presheaf monad = (T,V) is power-enriched, and the comparison functor

L : Set −→ (T,V)-URelop

is a 2-isomorphism. Moreover, for every f : X −→ Y in Set one has (f ])[ = yY · f .


Proof. The functor L and its inverse have been described above. Since the Kleisli comparisonfunctor L makes the diagram

Set L // (T,V)-URelop

SetF

``

(−)]

99

commute, (f ])[ = L−1(f ]) = Ff = yY · f .

3.2.2 Theorem. Given an associative lax extension T of T to V-Rel, there is an isomor-phism

(T,V)-Cat ∼= -Monthat commutes with the underlying-set functors.

Proof. Recall that every reflexive and transitive (T,V)-relation a : X −7 X is also unitary.By Proposition 3.2.1, sending a to a[ defines a bijection between reflexive and transitive(T,V)-relations a : X −7 X and maps ν : X −→ ΠX satisfying yX ≤ ν and ν ν ≤ ν.Furthermore, a map f : X −→ Y is a (T,V)-functor f : (X, a) −→ (Y, b) if and only ifa f ] ≤ f ] b; this condition is equivalent to (yY · f) a[ ≤ b[ (yY · f) in Set, that is,f : (X, a[) −→ (Y, b[) is a morphism of Kleisli monoids.

Since is power-enriched, we can consider the corresponding Kleisli extension of toRel. Explicitly, the extension Π of the Set-functor Π to Rel sends a relation r : X −→7 Y tothe relation Πr : ΠX −→7 ΠY defined by

ψ1 Πr ψ2 ⇐⇒ ψ1 ≤ ψ2 · T r

for all unitary (T,V)-relations ψ1 : X −7 1, ψ2 : Y −7 1 (Exercise 3.B). With respect tothis extension of we consider the category (, 2)-Cat.

3.2.3 Corollary. Given an associative lax extension T of T to V-Rel, there is an isomor-phism

(T,V)-Cat ∼= (, 2)-Catthat commutes with the underlying-set functors, where = (T,V) is equipped with itsKleisli extension .

Proof. Theorem 3.2.2 yields the isomorphism (T,V)-Cat ∼= -Mon, and Theorem 1.5.3 theisomorphism -Mon ∼= (, 2)-Cat.

3.2.4 Remark. The Kleisli extension of a power-enriched monad = (T,V , T) is inparticular associative (see Exercise 1.G). Hence, the transition from T to preservesassociativity while maintaining the same categories of lax algebras (up to isomorphism).


3.2.5 Proposition. Let T be an associative lax extension of T to V-Rel. Then the naturaltransformation Y defined componentwise by

YX : TX −→ ΠX , x 7−→ T1X(−, x )

is a monad morphism Y : T −→ (T,V).

Proof. The left adjoint functor (−)] : Set −→ (T,V)-URelop extends to a functor

(−)] : SetT −→ (T,V)-URelop

sending r : X Y to r] := r · T1Y : Y −7 X. Indeed, the identity 1X : X X in SetT(which is the map eX : X −→ TX) goes to the identity 1]X = eX · T1X in (T,V)-URel; andfor r : X Y and s : Y Z one has

r] s] = r · T1Y · T (s · T1Z) ·mZ = r · T (s · T1Z) ·mZ= r · (Ts) · T T1Z ·mZ = (mZ · Ts · r) · T1Z ,

hence (−)] preserves composition. By definition, the diagram

SetT(−)]

// (T,V)-URelop

SetFT

bb

(−)]

88

commutes, and therefore (−)] : SetT −→ (T,V)-URelop induces a monad morphism Y : T −→(T,V). Its component YX is the composite (see II.3.I)

TXyTX // (T,V)-URel(TX, 1) (−)1TX]

// (T,V)-URel(X, 1) ,

hence YX(x ) = x · eTX · T1TX · T T1X ·mX = x · T1X = T1X(−, x ).

3.2.6 Examples.

(1) For the identity monad I on Set extended to the identity monad on Rel, one has

(I, 2)-URel ∼= Rel

and the monad = (I, 2) is isomorphic to the powerset monad P. Hence, Ord ∼=(P, 2)-Cat by 3.2.3. The monad morphism Y : I −→ P is necessarily given by the unitof P.


(2) More generally, for the identity monad I on Set extended to the identity monad onV-Rel, one has

(I,V)-URel ∼= V-Rel

and the monad = (I,V) is isomorphic to the V-powerset monad PV (see Exer-cise III.1.D). Hence, V-Cat ∼= (PV , 2)-Cat by 3.2.3. As above, the monad morphismY : I −→ PV is necessarily given by the unit of PV .The monoid (V ,⊗, k) induces a monad V on Set with functor V × (−) (see Exer-cise II.3.B), and for each set X, there is a map αX : V ×X −→ VX defined by

αX(u, x)(y) =

u if x = y,⊥ else.

One easily verifies that these maps yield a monad morphism α : V −→ PV (Exercise 3.D).It is also clear that α is sup-dense; therefore, when considering the V-powerset monadPV with its Kleisli extension and V with the initial extension induced by α, we obtain(see Proposition 2.3.4) a full reflective embedding

Aα : V-Cat → (V, 2)-Cat .

However, Aα is not an equivalence in general. Indeed, for V = P+ and a metric spaceX = (X, a), one has

(u, x) (a · αX) y ⇐⇒ a(x, y) ≤ u ,

for all x, y ∈ X and u ∈ [0,∞]. Consider X = a, b with the relation −→: [0,∞]×X −→7 X defined by

(u, a) −→ b ⇐⇒ 0 < u

and (u, a) −→ a, (u, b) −→ b for all u ∈ [0,∞]. Then X is indeed a (V, 2)-categorybut −→ is not induced by a metric on X.

(3) For the powerset monad P on Set with its Kleisli extension P to Rel, one has anisomorphism

(−)] : SetP −→ (P, 2)-URelop

commuting with the left adjoints from Set, and therefore Y : P −→ (P, 2) is anisomorphism.

(4) Consider the ultrafilter monad on Set with its Barr extension to Rel (Exam-ple III.1.10.3(3)). In this case, the monad = (, 2) is isomorphic c©to the filtermonad F on Set. Indeed, a unitary (, 2)-relation ψ : X −7 1 may be identified witha set A ⊆ βX of ultrafilters on X with the property that x ⊇ ⋂A implies x ∈ A.Therefore, the map

δX : (, 2)-URel(X, 1) −→ FX , A 7−→ ⋂A


is a bijection. Let us show that δ = (δX) is indeed a monad morphism δ : −→ F.Recall from Example II.3.1.1(4) that the filter monad on Set is induced by theadjunction

SLatopSLat(−,2)

//> Set .

(P •)opoo

Since SLat is equivalent to the category SLatco of join-semilattices and their homo-morphisms, F is also induced by the adjunction

SLatcoopSLat(−,2)

//> Set .

(P •)opoo

In the former case, a filter f ∈ FX corresponds to the characteristic map χf : PX −→ 2with

χf (A) = 1 ⇐⇒ A ∈ f ,

whereas in the latter case f corresponds to the map χ′f : PX −→ 2

χ′f (A) = 1 ⇐⇒ A /∈ f .

The covariant hom-functor (, 2)-URel(1,−) : (, 2)-URel −→ Set lifts to a functorL : (, 2)-URel −→ SLatco. since, for every ψ : X −7 Y in (, 2)-URel, the map

ψ (−) : PX −→ PY , A 7−→ y ∈ Y | ∃x ∈ βX (A ∈ x & x ψ y)

preserves finite suprema (we use here the identification (, 2)-URel(1, X) ∼= PX). Itfollows from Lemma 3.1.3 that the diagram

(, 2)-URelop Lop// SLatcoop

Set(−)]

ff

(P •)op

::

commutes up to natural isomorphism, and therefore Lop induces an isomorphism ofmonads −→ F whose component at X is precisely δX . The composite δ ·Y : −→ −→ F is the canonical monad morphism −→ F.

(5) By Corollary 3.2.3 and Theorem III.2.4.5,

App ∼= (,P+)-Cat ∼= (, 2)-Cat .c©

In the next subsection we provide an alternative description of = (,P+).


3.3 Approach spaces. For an alternative description of (,P+) (where the ultrafiltermonad is provided with the extension β of β to P+-Rel of III.2.4), we consider the category

Met(∨,⊥,+)

of separated metric spaces X = (X, d), the underlying order of which, given by

x ≤ y ⇐⇒ 0 = d(x, y) ,

has finite suprema, and which admit an action + : X × [0,∞] −→ X (denoted here as aright action) satisfying

d(x+ u, y) = d(x, y) u , (3.3.i)for all x, y ∈ X and u ∈ [0,∞] (recall from II.1.10.1(3) that “” denotes truncatedsubtraction); a morphism of Met(∨,⊥,+) is a non-expansive map that preserves finite supremaand the action of [0,∞]. The equation (3.3.i) implies immediately that the monotone mapd(x,−) : X −→ P+ has x + (−) : P+ −→ X as a left adjoint, therefore a separated metricspace admits at most one such action.The metric space [0,∞] = ([0,∞], µ) with µ(u, v) = v u (u, v ∈ [0,∞]) belongs toMet(∨,⊥,+) since its underlying order is the natural ≥, which has finite suprema, and theaction of [0,∞] is given by the usual addition +. Similarly, for every set X, the set [0,∞]Xwith the metric

[ϕ, ϕ′] = supµ(ϕ(x), ϕ′(x)) | x ∈ X(for all ϕ, ϕ′ ∈ [0,∞]X) and the action given by pointwise addition belongs to Met(∨,⊥,+).In fact, in Met(∨,⊥,+) one has

[0,∞]X ∼=∏x∈X [0,∞]x ,

where [0,∞]x = ([0,∞], µ) for all x ∈ X. Therefore, as in Proposition 3.1.5, the contravari-ant hom-functor

Met(∨,⊥,+)(−, [0,∞]) : Metop(∨,⊥,+) −→ Set

has a left adjoint[0,∞](−) : Set −→ Metop

(∨,⊥,+) .

The functor J of the monad J induced on Set by this adjunction is given by

JX = Met(∨,⊥,+)([0,∞]X , [0,∞]) ,

with the unit X −→ JX defined by evaluation x 7−→ (ϕ 7−→ ϕ(x)), and the multiplicationJJX −→ JX defined by Ψ 7−→ (ϕ 7−→ Ψ(evϕ)), with evϕ(Φ) = Φ(ϕ) for all Φ ∈ JX,ϕ ∈ [0,∞]X .We show now that J is isomorphic to the monad . To this end, we first observe that a mapϕ : X −→ [0,∞] can be interpreted as a unitary (,P+)-relation ϕ : 1 −7 X; in particular,every element u ∈ [0,∞] can be seen as a unitary (,P+)-relation u : 1 −7 1. From this


perspective, the distance [ϕ, ϕ′] ∈ [0,∞] (where ϕ, ϕ′ ∈ [0,∞]X) is precisely the liftingϕ( ϕ′ of ϕ′ along ϕ in (,P+)-URel (see Example II.4.8.3), and the action ϕ + u is thecomposite ϕ u. Every unitary (,P+)-relation ψ : X −7 Y defines a mapping

ψ (−) : (,P+)-URel(1, X) −→ (,P+)-URel(1, Y )

that clearly preserves the action of [0,∞], and from

ψ ϕ (ϕ( ϕ′) ≥ ψ ϕ′

follows that ϕ ( ϕ′ ≥ (ψ ϕ) ( (ψ ϕ′), for all ϕ, ϕ′ : 1 −7 X. To see that ψ (−)preserves finite suprema, note that

ψ ϕ = infx ∈βX ψ(x ,−) + ξ(βϕ(x )) = ψ · ϕ ,

where ϕ : 1 −→7 βX is defined as ϕ(x ) = ξ(βϕ(x )) with

ξ(a) := supA∈a infu∈A u = infA∈a supu∈A u ,

for all a ∈ β[0,∞]. Being left adjoint, ψ ·(−) preserves all suprema, and ϕ 7−→ ϕ preserves thebottom element; ϕ 7−→ ϕ also preserves binary suprema since min : [0,∞]× [0,∞] −→ [0,∞]is continuous. All said, the covariant hom-functor K := (,P+)-URel(1,−) takes values inMet(∨,⊥,+). Moreover, it follows from Lemma 3.1.3 that the diagram

(,P+)-URelop Kop//Metop

(∨,⊥,+)

Set(−)]

ff

[0,∞](−)

::

commutes up to natural isomorphism, and therefore Kop induces a monad morphismδ : −→ J. A small computation shows that, for every set X, the map

δX : ΠX −→ Met(∨,⊥,+)([0,∞]X , [0,∞])

sends ψ : X −7 1 to ψ (−) : [0,∞]X −→ [0,∞].

3.3.1 Theorem.c© δ : −→ J is an isomorphism.

Proof. Every subset A ⊆ X can be seen as an element of [0,∞]X , namely as the functionA = θA : X −→ [0,∞] sending x ∈ A to 0 and everything else to∞. With this interpretation,for every Φ : [0,∞]X −→ [0,∞] in Met(∨,⊥,+) we set

Γ(Φ)(x ) = supA∈x Φ(A) .

Then ψ = Γ(Φ) : X −7 1 is indeed unitary since one obtains

ξ(βψ(X )) ≥ ψ(mX(X ))


from the inequality Φ(A) ≤ Γ(Φ)(x ), for all A ∈ x . If Φ is of the form Φ = ψ (−) forsome unitary ψ : X −7 1, then

Γ(Φ)(x ) = supA∈x ψ A = supA∈x ψ · A ≤ ψ · supA∈x A .

Since A(y ) = 0 if A ∈ y and A(y ) = ∞ if A /∈ y , one obtains supA∈x A(x ) = 0 andsupA∈x A(y ) =∞ for y 6= x , and consequently Γ(Φ) ≤ ψ. To see that

ψ(x ) ≤ Γ(Φ)(x ) = supA∈x inf y∈βA ψ(y ) ,

we use Lemma III.2.4.2 c©, which guarantees the existence of some X ∈ ββX with

βA | A ∈ x ⊆ X and supA∈x inf y∈βA ψ(y ) ≥ ξ(βψ(X )) .

Since ψ : X −7 1 is unitary,

ξ(βψ(X )) ≥ ψ(mX(X )) = ψ(x ) .

Let now Φ : [0,∞]X −→ [0,∞] in Met(∨,⊥,+). We show first that Γ(Φ) (−) coincides withΦ on subsets B ⊆ X of X. Indeed,

Γ(Φ) B = infx ∈βX supA∈x Φ(A) + B(x ) = infx ∈βB supA∈x Φ(A) ≥ Φ(B) .

To see that infx ∈βB supA∈x Φ(A) ≤ Φ(B), we apply Corollary II.1.13.5 c©(if Φ(B) <∞) tothe filter base B and the ideal j = A ⊆ X | Φ(A) > Φ(B). Hence, there is someultrafilter x ∈ βX with B ∈ x and x ∩ j = ∅, and therefore

supA∈x Φ(A) ≤ Φ(B) .

To finish the proof, we show that any Φ : [0,∞]X −→ [0,∞] is completely determined by itsrestriction to subsets B ⊆ X of X. Since

Φ(ϕ) = supu<∞minΦ(ϕ), (Φ(0)+u) = supu<∞minΦ(ϕ),Φ(u) = supu<∞Φ(minϕ, u) ,

Φ is determined by its restriction to bounded maps ϕ : X −→ [0,∞]. Let now ϕ : X −→ [0,∞]be a bounded map, ε > 0, and N be any natural number with ϕ(x) < N · ε, for all x ∈ X.For every natural number n with 0 ≤ n < N we set An = x ∈ X | n ·ε ≤ ϕ(x) < (n+1) ·εand un = n · ε, and define

ϕε = minAn + un | 0 ≤ n < N

(note that finite suprema in [0,∞]X are given by pointwise minima). Then Φ(ϕε) ≤ Φ(ϕ) ≤Φ(ϕε) + ε and Φ(ϕε) = minΦ(An) + un | 0 ≤ n < N, which proves that Φ is determinedby its effect on subsets of X.


3.4 Revisiting change-of-base. In 3.2 we devised a construction that incorporates aquantale into a monad. In this subsection we extend this construction to quantale morphismsand show that certain change-of-base functors correspond to algebraic functors.Recall from III.3.5 that every lax homomorphism ϕ : V −→ W of quantales induces a laxfunctor

ϕ : V-Rel −→W-Relthat sends the V-relation r : X×Y −→ V to ϕr : X×Y −→W . Moreover, ϕ : V-Rel −→W-Relis actually a functor if ϕ : V −→ W is a homomorphism of quantales.Let now T = (T,m, e) be a monad on Set together with associative lax extensions

T : V-Rel −→ V-Rel and T :W-Rel −→W-Rel.

A lax homomorphism of quantales ϕ : V −→ W is compatible with these lax extensionsif T (ϕr) ≤ ϕ(T r) for all r : X −→7 Y in V-Rel, and we say that ϕ : V −→ W is strictlycompatible with T and T if T (ϕr) = ϕ(T r) for all r : X −→7 Y in V-Rel.3.4.1 Proposition. If ϕ : V −→ W is a lax homomorphism of quantales compatible withassociative lax extensions T and T of a monad T = (T,m, e) to V-Rel andW-Rel respectively,then r 7−→ ϕr defines a lax functor

ϕ : (T,V)-URel −→ (T,W)-URel

such that (−)] ≤ ϕop · (−)]. Moreover, ϕ : (T,V)-URel −→ (T,W)-URel is even a functormaking

(T,V)-URelop ϕop// (T,W)-URelop

Set(−)]

dd

(−)]

99

commute provided that ϕ : V −→ W is a homomorphism of quantales strictly compatiblewith the lax extensions T and T of T.

Proof. Let r : X −7 Y and s : Y −7 Z be morphisms in (T,V)-URel. Then

(ϕs) (ϕr) = (ϕs) · T (ϕr) ·mX≤ (ϕs) · (ϕT r) · (ϕmX)≤ ϕ(s r)

with equality if ϕ : V −→ W is a homomorphism of quantales strictly compatible with thelax extensions T and T of T. If f : X −→ Y is map, then

f ] = eY · T (f ) ≤ ϕ(eY · T (f )) = ϕf ] ,

and, as above, we have equality if ϕ : V −→ W is a homomorphism of quantales strictlycompatible with the lax extensions T and T of T. Note that this applies in particular tothe identity 1]X in (T,V)-URel.


3.4.2 Lemma. Let ϕ : V −→ W be a homomorphism of quantales strictly compatible withassociative lax extensions T and T of T = (T,m, e) to V-Rel and W-Rel respectively. Thenthe right adjoint ψ : W −→ V of ϕ is a lax homomorphism of quantales compatible withthese lax extensions of T.

Proof. For u, v ∈ W ,

ψ(u)⊗ ψ(v) ≤ ψϕ(ψ(u)⊗ ψ(v)) = ψ(ϕψ(u)⊗ ϕψ(v)) ≤ ψ(u⊗ v) andk ≤ ψϕ(k) = ψ(l) .

Similarly, for aW-relation r : X −→7 Y we obtain T (ψr) ≤ ψϕT (ψr) = ψT (ϕψr) ≤ ψ(T r).

3.4.3 Proposition. Every homomorphism ϕ : V −→ W of quantales strictly compatiblewith lax extensions T and T of T = (T,m, e) to V-Rel and W-Rel induces a morphism

(T, ϕ) : (T,V) −→ (T,W)

of power-enriched monads. The component of (T, ϕ) at a set X is given by

(T,V)-URel(X, 1) −→ (T,W)-URel(X, 1) , r 7−→ ϕr .

Proof. Firstly, ϕ : V −→ W induces a functor ϕ : (T,V)-URel −→ (T,W)-URel so that

(T,V)-URelop ϕop// (T,W)-URelop

Set(−)]

dd

(−)]

99

commutes. A quick computation shows that the X-component of the monad morphism(T, ϕ) induced by ϕop has the form described above. Secondly, the right adjoint ψ of ϕinduces a lax functor ψ : (T,W)-URel −→ (T,V)-URel, therefore the map s 7−→ ψs defines aright adjoint to

(T,V)-URel(X, 1) −→ (T,W)-URel(X, 1) , r 7−→ ϕr .

Hence, each component of (T, ϕ) preserves suprema.

3.4.4 Theorem. Let ϕ : V −→ W be a homomorphism of quantales strictly compatible withthe lax extensions T and T of T = (T,m, e) to V-Rel and W-Rel and let ψ :W −→ V be theright adjoint of ϕ. Then the change-of-base functor Bψ corresponds to the algebraic functorA(T,ϕ), that is, the diagram

(T,W)-Cat //

Bψ

((T,W), 2)-CatA(T,ϕ)

(T,V)-Cat // ((T,V), 2)-Cat

commutes. Here the horizontal arrows represent the isomorphism of Corollary 3.2.3.


Proof. Let (X, a) be a (T,W)-category, ρ ∈ Π(T,V)(X) and x ∈ X. Then ρ −→ x in the((T,V), 2)-category obtained via the upper-right path of the diagram if and only if

ϕρ ≤ a(−, x) ,

which is equivalent toρ ≤ ψa(−, x) ;

and this means precisely ρ −→ x in the ((T,V), 2)-category obtained via the lower-leftpath of the diagram.

Exercises

3.A The functor (−)] : Set −→ (T,V)-URel. Let T be a lax extension to V-Rel of a monadT = (T,m, e) on Set. There is a functor (−)] : Set −→ (T,V)-URel that sends a mapf : X −→ Y to the unitary (T,V)-relation f] : X −7 Y given by

f] := eY · T1X · Tf = eY · T f .

One has in particularϕ f] = ϕ · Tf

for all unitary (T,V)-relations ϕ : Y −7 Z. If T is associative, then f] and f ] form anadjunction f] a f ] in the ordered category (T,V)-URel.

3.B The Kleisli extension of . For an associative lax extension T of T to V-Rel and itsassociated power-enriched monad = (Π,m,y), the power-enrichment τ : P −→ is givencomponentwise by

τX(A) = ∨x∈A x

] .

For a relation r : X −→7 Y , one has

τX · r[ = (r · 1]X)[ ,

where the (−)[ operation on the left comes from the construction of the Kleisli extension 1.4,and the (−)[ on the right is the one defined in 3.1. Hence, rτ (ψ2) = ψ2 (r · 1]X) = ψ2 · T r,so that

ψ1 (Πr) ψ2 ⇐⇒ ψ1 ≤ ψ2 · T r

for all unitary (T,V)-relations ψ1 : X −7 1, ψ2 : Y −7 1.

3.C The discrete presheaf monad for the list monad. Since the list monad L = (L,m, e)on Set is cartesian (see Exercise 1.Q), the Barr extension L of L to Rel is associative and,moreover, every (L, 2)-relation ϕ : X −7 Y is unitary. With = (L, 2) denoting thediscrete presheaf monad associated to L, the functor Π can be identified with the composite


PL of the powerset functor P with L. Via this identification, yX sends x ∈ X to (x)and Πf sends A ⊆ LX to Lf(x1, . . . , xk) | (x1, . . . , xk) ∈ A, and the multiplication sendsA ∈ PLPLX to

(x1,1, . . . , x1,i1 , . . . , xk,1, . . . , xk,ik) | (A1, . . . , Ak) ∈ A, (xj,1, . . . , xj,ij) ∈ Aj ∈ PLX .

3.D A monad morphism α : V −→ PV . The maps αX of 3.2.6(2) form the components of amonad morphism α : V −→ PV that is sup-dense but not interpolating.

3.E Continuous P+-actions on compact Hausdorff spaces. The convergence map

ξ : β[0,∞] −→ [0,∞] , x 7−→ supA∈x infx∈A x

yields the standard topology on [0,∞], with respect to which addition + : [0,∞]× [0,∞] −→[0,∞] is continuous. There is a distributive law δ (see II.3.8) of the ultrafilter monad over the monad V induced by the monoid ([0,∞],+, 0) (see Example 3.2.6(2)) defined by

δX : β([0,∞]×X) −→ [0,∞]× βX , w 7−→ (ξ(βπ1(w)), βπ2(w)) ,

for each set X. The induced monad = ([0,∞]× β, m, e) has its multiplication given by

mX : [0,∞]× β([0,∞]× βX) −→ [0,∞]× βX , (v,W ) 7−→ (v + ξ(a),mX(X ))

(with a = βπ1(W ), X = βπ2(W ) and m the multiplication of the ultrafilter monad) and itsunit by

eX : X −→ [0,∞]× βX , x 7−→ (0, eX(x))(with e the unit of the ultrafilter monad). Describe the category of -algebras.

3.F Approach spaces as (, 2)-categories. There is a monad morphism α : −→ from themonad of Exercise 3.E to the monad of Example 3.2.6(5) whose component at a set Xis given by

αX(u, x )(y ) =

u if x = y ,⊥ otherwise.

This monad morphism α : −→ is sup-dense and interpolating, so the algebraic functor

Aα : (, 2)-Cat −→ (, 2)-Cat c©

is an isomorphism, and App ∼= (, 2)-Cat.Hint. To see that α is interpolating, observe that for each S ⊆ [0,∞] and u ∈ [0,∞] withS 6= ∅ and u = inf S, there is some a ∈ β[0,∞] with S ∈ a and ξ(a) = u.


4 Injective lax algebras as Eilenberg–Moore algebras

Our primary goal in this subsection is to show that certain injective objects in T-Mon are theEilenberg–Moore algebras of the monad T. The motivating example is given by continuouslattices, which are the injective objects of the category Top0 of T0-spaces; however, we donot assume any prior knowledge of these particular ordered structures and will derive thenecessary concepts in 4.4. A consequence of the formal approach is a unified treatment forthe description of Sup as a category of injectives for Ord, Cnt for Top, Dst for Cls, Frm forClsfin, and SupP+ for Met.

4.1 Eilenberg–Moore algebras as Kleisli monoids. The embedding SetT → (T,V)-Catof Proposition III.1.6.5 describes T-algebras as (T,V)-algebras when the lax extension of Tis flat. Here, we show that a similar situation can occur when the extension is not flat.By Exercise II.3.H, a morphism τ : P −→ T from the powerset monad P to a monadT = (T,m, e) on Set yields a functor

Setτ : SetT −→ SetP , (X, a) 7−→ (X, a · τX)

that commutes with the respective underlying-set functors. Via the isomorphism SetP ∼= Sup(Example II.3.2.2(2)), every P-algebra structure s : PX −→ X represents a supremumwith right adjoint ↓X := sa : X −→ PX the down-set map of X, that is, the separatedorder relation on X (II.1.7). Hence, every T-algebra morphism becomes a sup-map f :(X, a · τX) −→ (Y, b · τY ) that has a right adjoint

fa : Y −→ X

in Ord. In particular, a T-algebra structure a : TX −→ X has a right adjoint aa : X −→ TXwhich satisfies

aa aa ≤ aa and eX ≤ aa . (4.1.i)

Indeed, the second inequality follows by adjunction from a · eX = 1X ; the first is aconsequence of a · Ta = a ·mX , since this identity implies mX ≤ aa · a · Ta and therefore

mX · T (aa) · aa ≤ aa · a · Ta · T (aa) · aa = aa

(a · aa = 1X because 1X = a · eX ≤ a · aa ≤ 1X), that is, (X, aa) is a T-monoid. Iff : (X, a) −→ (Y, b) is a T-algebra homomorphism, then b · Tf = f · a yields

Tf · aa ≤ ba · f

by adjunction, so f : (X, aa) −→ (Y, ba) is a morphism of T-monoids. Hence, when (T, τ) ispower-enriched, there is a functor

SetT −→ T-Mon , (X, a) 7−→ (X, aa)

4. INJECTIVE LAX ALGEBRAS AS EILENBERG–MOORE ALGEBRAS 333

that commutes with the respective underlying-set functors. In particular, (TX,maX) isa T-monoid, and every T-monoid structure ν on X becomes a T-monoid morphism ν :(X, ν) −→ (TX,maX), since ν ν ≤ ν equivalently means

Tν · ν ≤ maX · ν .

We note that, unlike the functor Set → (T,V)-Cat of Proposition III.1.6.5, the functorSetT −→ T-Mon fails to be full in general, as the following Example shows.

4.1.1 Example. For the powerset monad P, the functor SetP −→ P-Mon simply describesthe forgetful functor

SetP ∼= Sup −→ Ord ∼= P-Mon

(Example 1.3.2.(2)).

4.2 Monads on categories of Kleisli monoids. In order to characterize injectiveobjects of T-Mon, it is convenient to introduce a monad T′ derived from the original monadT on Set, as follows. Consider a morphism α : S −→ T of power-enriched monads S = (S, n, d)and T = (T,m, e). By composing the functor Setα : SetT −→ SetS (Exercise II.3.H) withG : SetS −→ S-Mon (4.1), we obtain a functor

GT′ : SetT −→ S-Mon

that sends a T-algebra (X, a) to the S-monoid (X, (a · αX)a) and leaves maps unchanged.This functor GT′ is right adjoint. Rather than describing its left adjoint explicitly, weconstruct below the induced monad T′ on S-Mon via its Kleisli triple components T ′, e′X ,and (−)T′ (Proposition 4.2.2). Theorem 4.3.2 then shows that the new monad has the sameEilenberg–Moore category as T, a fact that justifies a posteriori the notation for the functorGT′ .In 4.4, we exploit the case where α = τ : P −→ F is the principal filter monad morphismto identify the category SetF as the category Cnt of continuous lattices via the monadicfunctor SetF −→ Ord ∼= P-Mon. In 4.6, we show that when α = 1T : T −→ T is the identity,the monadic functor SetT −→ T-Mon is Kock–Zoberlein; this property then facilitates ourstudy of injective objects in 4.6.

Construction.

(1) Given an S-monoid structure µ : X −→ SX, the map ν := αX · µ yields a T-monoidstructure on X (Proposition 1.3.3). We define the set T ′X of νT-invariants as theequalizer in Set of the pair (νT, 1TX):

T ′XqX // TX

νT //

1TX// TX .


The universal property of qX implies the existence of a map pX : TX −→ T ′X suchthat

qX · pX = νT and pX · qX = 1T ′X .

Indeed, as νT · νT = νT (by (1.3.i)), there is a map pX : TX −→ T ′X with qX · pX = νT;therefore, we have qX · pX · qX = νT · qX = qX , so that pX · qX = 1T ′X by unicity ofthe induced map T ′X −→ T ′X.The set T ′X can be equipped with the S-monoid structure ωX : T ′X −→ ST ′X givenby

ωX := SpX · (mX · αTX)a · qX .

Lemma 4.2.1 below shows that qX : (T ′X,ωX) −→ (TX, (mX · αTX)a) is also anequalizer in S-Mon. This ensures that the maps e′X and fT

′ defined in the followingpoints are morphisms of S-monoids.

(2) Since νT · ν = ν, there exists a map e′X : X −→ T ′X such that qX · e′X = ν:

Xe′X

ν

##

T ′X qX// TX

νT //

1TX// TX .

This yields a morphism of S-monoids e′X : (X,µ) −→ (T ′X,ωX). Since qX · e′X = ν andpX · qX = 1T ′X , one can equivalently obtain e′X as either

e′X = pX · ν or e′X = pX · eX

because pX · ν = pX · νT · eX = pX · qX · pX · eX = pX · eX .

(3) If (Y, µY ) is another S-monoid, and f : (Y, µY ) −→ (T ′X,ωX) is an S-monoid morphism,then

νT · (qX · f)T = (νT · qX · f)T = (qX · f)T .

Hence, there exists a unique map fT′ : T ′Y −→ T ′X making the following diagram

commute:T ′Y

fT′

(qX ·f)T·qY

##

T ′X qX// TX

νT //

1TX// TX .

This yields a morphism of S-monoids fT′ : (T ′Y, ωY ) −→ (T ′X,ωX) that can also beobtained directly as

fT′ = pX · (qX · f)T · qY .


4.2.1 Lemma. For a morphism α : S −→ T of power-enriched monads S = (S, n, d) andT = (T,m, e), the map

qX : (T ′X,ωX) −→ (TX, (mX · αTX)a)

(defined in the previous construction) is an equalizer in S-Mon. As a consequence,

e′X : (X,µ) −→ (T ′X,ωX) and fT′ : (T ′Y, ωY ) −→ (T ′X,ωX)

are morphisms of S-monoids.

Proof. To verify that ωX : T ′X −→ ST ′X is an S-monoid structure, observe that

dT ′X = dT ′X · pX · qX = SpX · dTX · qX ≤ SpX · (mX · αTX)a · qX = ωX

because (mX · αTX) · dTX = 1TX , and

ωSX · ωX = SpX · nTX · S(mX · αTX)a · SνT · (mX · αTX)a · qX (qX · pX = νT)

≤ SpX · nTX · (S(mX · αTX))a · SνT · (mX · αTX)a · qX (mX · αTX a retraction)≤ SpX · nTX · (SmX · SαTX)a · (mX · αTX)a · νT · qX (νT morphism in S-Mon)= SpX · nTX · (mX · αTX · nTX)a · νT · qX (α monad morphism)= SpX · (mX · αTX)a · qX = ωX (nTX · naTX = 1STX).

One can reason similarly to obtain

SqX · ωX = SνT · (mX · αTX)a · qX ≤ (mX · αTX)a · νT · qX = (mX · αTX)a · qX ,

so that qX : (T ′X,ωX) −→ (TX, (mX · αTX)a) is a morphism in S-Mon. Suppose now thatg : (Y, µY ) −→ (TX, (mX · αTX)a) is a morphism in S-Mon satisfying νT · g = g. SinceqX : T ′X −→ TX is an equalizer of (νT, 1TX) in Set, there exists a unique map h : Y −→ T ′Xwith g = qX · h; moreover,

Sh · µY = SpX · Sg · µY ≤ SpX · (mX · αTX)a · g = ωX · h ,

which shows that h : (Y, µY ) −→ (T ′X,ωX) is a morphism in S-Mon. As a consequence, qXis an equalizer in S-Mon, and e′X , fT′ are the underlying maps of the corresponding uniqueS-monoid morphisms into (T ′X,ωX).

4.2.2 Proposition. If α : S −→ T is a morphism of power-enriched monads, then theconstruction described in (1)–(3) above defines a Kleisli triple (T ′, (−)T′ , e′) on S-Mon.Hence, T′ = (T ′,m′, e′) is a monad on S-Mon with multiplication m′X = 1T′T ′X = pX ·qTX ·qT ′Y .

Proof. Lemma 4.2.1 ensures that the components T ′, (−)T′ , and e′ are of the appropriatetype to yield a Kleisli triple on S-Mon. For an S-monoid morphism h : (X,µ) −→ (TY, (αTY ·mY )a), we observe that if ν = αX · µ, then

hT · ν = h


since

mY · Th · αX · µ ≤ mY · αTY · (mY · αTY )a · h = h = mY · Th · eX ≤ mY · Th · αX · µ .

The conditions II(3.7.i) are now easily verified by using this observation. The monad T′ isthen obtained via the Kleisli triple (see II.3.7).

4.2.3 Examples.

(1) If S = T = P, then a P-monoid is an ordered set (X, ↓X) (Example 1.3.2(2)). In thiscase, an element A ∈ PX is a ↓TX-invariant precisely when ↓PX(A) = A, that is, whenA is down-closed: ⋃

↓X x | x ∈ A = A .

The construction in 4.2 therefore describes the down-set monad Dn = (Dn ,∨Dn , ↓) onOrd (Example II.4.9.3).

(2) If S = T = F is the filter monad, then an F-monoid is a topological space (X, ν), whereν : X −→ FX is the neighborhood filter map (1.1). A filter a ∈ FX is νF-invariant ifand only if a is spanned by open sets of X:

A ∈ νF(a) ⇐⇒ ν−1(AF) ∈ a ⇐⇒ x ∈ X | A ∈ ν(x) ∈ a ,

so νF(a) = a means that for every A ∈ a the interior must also be in a. Theopen-filter monad on Top ∼= F-Mon, that is, the monad F′ = (F ′,m′, e′) obtained viaProposition 4.2.2, can therefore be described as follows. The functor F ′ : Top −→ Topsends a topological space X to the set F ′X of filters in OX equipped with itsScott topology; F ′ also sends a continuous map f : X −→ Y to the continuous mapF ′f : F ′X −→ F ′Y given by O ∈ F ′f(a) ⇐⇒ f−1(O) ∈ a for all a ∈ F ′Y . Theunit ν = e′X : X −→ F ′X sends a point x to its set of open neighborhoods, and themultiplication m′X : F ′F ′X −→ F ′X is the restriction of the Kowalsky sum.

4.3 Eilenberg–Moore algebras over S-Mon. For power-enriched monads S and T, themonad T′ on S-Mon is derived from T via the morphism α : S −→ T. Using the notationsof 4.2, we now show that the category of T′-algebras is isomorphic to SetT.

4.3.1 Lemma. Let U : S-Mon −→ Set denote the forgetful functor. The maps pX of 4.2form the components of a natural transformation p : TU −→ UT ′ that defines a lifting of Uthrough (GT′ , GT) (Exercise II.3.H).

Proof. An S-monoid morphism f : (Y, µY ) −→ (X,µ) yields a T-monoid morphism f :(Y, νY ) −→ (TX, ν) (where νY = αY · µY and ν = αX · µ), so that

ν · f = (ν · f)T · eY ≤ (ν · f)T · νY = νT · Tf · νY ≤ νT · ν · f = ν · f .


Therefore, (ν · f)T · νY = ν · f , and using T ′f = (e′X · f)T′ we obtain

T ′f · pY = pX · (ν · f)T · νTY = pX · ((ν · f)T · νY )T = pX · (ν · f)T = pX · νT · Tf = pX · Tf ,

so that p : TU −→ UT ′ is a natural transformation. Moreover, for m′X = (1T ′X)T′ =pX · (qX)T · qT ′X ,

m′X · pT ′X · TpX = pX · (qX)T · (αT ′X · ωX)T · TpX= pX · (mX · TνT · αTX · (mX · αTX)a · qX)T · TpX= pX · (νT ·mX · αTX · (mX · αTX)a · qX)T · TpX= pX · (νT · qX)T · TpX= pX · (qX)T · TpX= pX · νT ·mX

= pX ·mX .

Since e′X = pX · eX we conclude that r does indeed define a lifting of U through (GT′ , GT).

4.3.2 Theorem. If α : S −→ T is a morphism of power-enriched monads, then there is anisomorphism

SetT ∼= S-MonT′

of Eilenberg–Moore categories that commutes with the underlying-set functors.

Proof. Suppose first that (X, a) is a T-algebra. One obtains an S-monoid (X,µ), withµ = (a · αX)a, that can be equipped with the structure a′ : (T ′X,ωX) −→ (X,µ) defined by

a′ := a · qX .

Since a : (TX,mX) −→ (X, a) is a T-algebra morphism, its GT′-image is an S-monoidmorphism, and therefore so is a′. To see that a′ satisfies the algebra conditions for themonad T′, we use the definition of e′X :

a′ · e′X = a · qX · e′X = a · αX · (a · αX)a = 1X .

Suppose now that f, g : (Y, µY ) −→ (T ′X,ωX) are morphisms in S-Mon satisfying a′ · f =a′ · g, or equivalently, a · qX · f = a · qX · g; since a is a T-algebra structure, one hasa · (qX · f)T = a · (qX · g)T (Exercise II.3.G), so that

a′ · fT′ = a · qX · fT′ = a · (qX · f)T · qY = a · (qX · g)T · qY = a · qX · gT

′ = a′ · gT′ .

Therefore, ((X,µ), a′) is a T′-algebra. A T-algebra morphism f : (X, a) −→ (Y, aY ) yields amorphism f : (X, (a ·αX)a) −→ (Y, (aY ·αY )a) in S-Mon. To verify that a′Y · (e′Y ·f)T′ = f ·a′,we first observe

aY · (αY · (aY · αY )a · f)T = aY ·mX · T (αY · (aY · αY )a) · Tf= aY · TaY · T (αY · (aY · αY )a) · Tf= aY · Tf .


Hence,

a′Y · (e′Y · f)T′ = aY · (qY · e′Y · f)T · qX= aY · (αY · (aY · αY )a · f)T · qX= f · a · qX= f · a′ ,

which proves that f : ((X,µ), a′) −→ ((Y, µY ), a′Y ) is a morphism of T′-algebras. Thus, theassignment of ((X, (a · αX)a), a · qX) to a T-algebra (X, a) yields a functor K : SetT −→S-MonT′ that commutes with the underlying-set functors.The lifting p : TU −→ UT ′ of U (see Lemma 4.3.1) yields a functor U : S-MonT′ −→ SetTthat sends a T′-algebra ((X,µ), a′) to (X, a), where a : TX −→ X is defined by

a := a′ · pX ,

and is invariant on maps.Given a T-algebra (X, a), the structure of UK(X, a) is described by

a · qX · pX = a ·mX · T (αX · (a · αX)a) = a · T (a · αX) · T (a · αX)a = a .

To study the image of a T′-algebra ((X,µ), a′) via KU , note first that a′ : (T ′X,ωX) −→(X,µ) is a morphism in S-Mon. Thus, after setting ν = αX · µ and observing that(mX · αTX) · S(αX · µ) = νT · αX , one obtains

1SX = S(a′ · pX) · S(αX · µ)≤ S(a′ · pX) · (mX · αTX)a · νT · αX= Sa′ · ωX · pX · αX≤ µ · (a′ · pX · αX) .

This inequality, combined with (a′ · pX · αX) · µ = 1X yields

µ = (a′ · pX · αX)a .

Hence, the image under KU of the T′-algebra ((X,µ), a′) is the T′-algebra with underlyingS-monoid (X, (a′ · pX · αX)a) = (X,µ) and structure

a′ · pX · qX = a′ .

One concludes that K and U are inverse of each other, and SetT ∼= S-MonT′ .

In general, a T-monoid (X, ν) on a category X is separated whenever ν is a monomorphism.For a power-enriched monad T, this condition amounts to the requirement that the initialorder induced on X by ν : X −→ TX,

x ≤ y ⇐⇒ ν(x) ≤ ν(y) (4.3.i)


for all x, y ∈ X, is separated. The full subcategory of T-Mon whose objects are the separatedKleisli monoids is denoted by T-Monsep.

4.3.3 Corollary. Given a morphism α : S −→ T of power-enriched monads, the monad T′

of Proposition 4.2.2 restricts to a monad on S-Monsep, and there is an isomorphism

SetT ∼= (S-Monsep)T′


Proof. The results of 4.2 leading up to Theorem 4.3.2 can be reproduced by replacing S-Monby S-Monsep. Indeed, the functor GT′ : SetT −→ S-Mon factors through S-Monsep → S-Mon,and (T ′X,ωX) is separated for all S-monoids (X,µ): setting ν = αX · µ, one has

pX · (mX · αTX) · SqX · ωX = pX · (mX · αTX) · SνT · (mX · αTX)a · qX= pX · νT · (mX · αTX) · (mX · αTX)a · qX = 1T ′X ,

that is, ωX is a section.

4.3.4 Examples.

(1) The initial order on a P-monoid (X, ↓) is given by

x ≤ y ⇐⇒ ↓x ≤ ↓ y

for all x, y ∈ X. Hence, if ↓ : X −→ PX is a monomorphism, then ↓x = ↓ y impliesx = y, so X is a separated ordered set, and we have P-Monsep ∼= Ordsep. Hence, withT = S = P, Theorem 4.3.2 and Corollary 4.3.3 state that the forgetful functors

Sup −→ Ord and Sup −→ Ordsep

are strictly monadic (see Example 4.1.1).

(2) The initial order induced on a topological space, seen as an F-monoid, by its neighbor-hood map ν : X −→ FX is its underlying order (see II.1.9). Hence, the isomorphismF-Mon ∼= Top identifies separated F-monoids with T0-spaces (Exercise 4.B). WithTop0 the corresponding full category of Top, we therefore have F-Monsep ∼= Top0. WithS = F or S = P, we obtain strictly monadic functors

SetF −→ Top , SetF −→ Top0 , SetF −→ Ord , and SetF −→ Ordsep ,

thanks to Theorem 4.3.2 and Corollary 4.3.3. The identification of SetF as a categoryof lattices requires more work and is considered in the next section.


4.4 Continuous lattices. For a monad morphism α : P −→ T, the study of T-algebrascan be facilitated by the study of the monad T′ on Ord ∼= P-Mon, as a consequence ofTheorem 4.3.2. This approach leads us here to the identification of continuous lattices asthe Ini-injective objects of Top, for Ini the class of U -initial morphisms (with U : Top −→ Setforgetful).A complete lattice X is continuous if the restriction of the supremum map ∨X : DnX −→ Xto the set IdlX of ideals in X has a left adjoint

X :

X a∨X : IdlX −→ X

(compare with complete distributivity II.1.11). The category of continuous lattices andinf-maps which preserve up-directed suprema is denoted by

Cnt .

Since our focus in this subsection is on convergence of filters rather than that of ideals,from now on we will concentrate on the dual notion of cocontinuous lattices.A complete lattice X is cocontinuous if the restriction of the infimum map ∧X : Up X −→ Xto the set FilX of filters in X has a right adjoint X :∧

X a X : X −→ FilX .

The order on FilX is induced by the order on Up X and is therefore given by reverseinclusion, see II.1.7; hence, one has∧

S ≤ a ⇐⇒ S ⊇ a

for all S ∈ FilX, a ∈ X, and

a =⋂S ∈ FilX | ∧S ≤ a

by Proposition II.1.8.2. Note also that ∧ a = a for all a ∈ X, that is, X : X −→ FX is afully faithful embedding. Cocontinuous lattices with sup-maps that preserve down-directedinfima form a 2-category which is 2-isomorphic to Cntco (the isomorphism sends f : X −→ Yto f op : Xop −→ Y op). Hence, in the sequel, we will write Cntco to designate the category ofcocontinuous lattices.The function Fil defined on ordered sets is the object part of a 2-functor Fil : Ord −→ Ordthat is the restriction of the up-set 2-functor Up to filters: for a map f : X −→ Y , one has

Filf(A) = ⋃x∈A ↑ f(x)

for all filters A ⊆ X. The up-set map of an ordered set X corestricts to a monotone map↑X : X −→ FilX, and union yields the infimum map ∧FilX : FilFilX −→ FilX to form theordered-filter monad

Fil = (Fil ,∧Fil , ↑)


on Ord. There is a distributive law λ : DnFil −→ FilDn of the down-set monad

Dn = (Dn ,∨Dn , ↓)

(Example II.4.9.3) over Fil whose components are given by

λX(x ) = A ∈ DnX | ∀B ∈ x (A ∩B 6= ∅)

for all ordered sets X and x ∈ DnFilX (Exercise 4.A). The resulting monad is thedown-set-filter monad

FilDn = (FilDn ,∧FilDn ·Fil ∨FilDn , ↑ ↓) .

4.4.1 Proposition. The category Cntco is strictly monadic over Ord. More precisely, thereis a 2-isomorphism

Cntco ∼= OrdFilDn

which commutes with the forgetful functors to Ord.

Proof. For a cocontinuous lattice X, the diagram

X

X

//⊥ FilX

∧Xoo

Fil ↓X//⊥ FilDnX

Fil∨Xoo (4.4.i)

motivates us to define a map a := ∧X ·Fil ∨X . Let us verify that a yields the structure

morphism of an FilDn-algebra. The unit condition a · Fil ↓X · ↑X = 1X is immediate, so weonly need to verify the multiplication condition a · FilDna = a · ∧FilDnX ·Fil ∨FilDnX :

a · FilDna = ∧X ·Fil(∨X ·Dna)

= ∧X ·Fil(a · ∨FilDnX) (a left adjoint by (4.4.i))

= ∧X ·∧

FilX ·FilFil ∨X ·Fil ∨FilDnX (∧X preserves infima)= ∧

X ·Fil ∨X ·∧FilDnX ·Fil ∨FilDnX (∧Fil natural transformation).

Consider now an FilDn-algebra (X, a : FilDnX −→ X). Since there is a monad morphism↑Dn : Dn −→ FilDn, the ordered set X is a Dn-algebra, that is, a complete lattice withsupremum given by ∨

X = a · ↑DnX . The FilDn-algebra morphism a is a sup-map anda retraction, so it has a fully faithful right adjoint aa : X −→ FilDnX and we have theadjunctions

Xaa

//⊥ FilDnXFil (a·↑DnX)

//>aoo FilX .

Fil ↓Xoo

Since the components of the ordered-filter monad multiplication are ∧FilX , we have inparticular ∧FilDnX ·Fil ↑DnX = 1FilDnX . By naturality of ↓ and the algebra multiplicationcondition,

a · Fil ↓X ·Fil(a · ↑DnX) = a · ∧FilDnX ·Fil ∨FilDnX ·Fil ↓FilDnX ·Fil ↑DnX = a .


One deduces that a · Fil ↓X admits Fil(a · ↑DnX) · aa as a right adjoint. As the infimumoperation on X (obtained via the monad morphism Fil ↓ : Fil −→ FilDn) is precisely∧X = a · Fil ↓X , the ordered set X is a cocontinuous lattice (and a = ∧

X ·Fil ∨X by theprevious displayed identity).Finally, a cocontinuous lattice morphism f : X −→ Y is also an FilDn-algebra morphism,and a morphism f : (X, a) −→ (Y, b) of FilDn-algebras naturally preserves both supremaand down-directed infima because it is both a Dn-algebra and an Fil-algebra morphism.

The monad FilDn on Ord is the monad F′ on P-Mon obtained in 4.2 from the filter monadF on Set. Indeed, consider the principal filter natural transformation τ : P −→ F and aP-monoid (X, ↓X) (Example 1.3.2(2)). The construction of 4.2 associates to this orderedset the topological space (X, ν) whose neighborhood map is given at each x ∈ X by theprincipal filter of ↓X x ∈ PX:

ν(x) = ↑PX(↓X x) .Hence, (X, ν) is the Alexandroff space associated with the ordered set X (II.5.10.5), andopen sets are the down-closed sets. Thanks to Example 4.2.3(2), the set of νF-invariant filterscan be identified with the set of filters on DnX, so that F′ = FilDn is the down-set-filtermonad on Ord. Proposition 4.4.1 and Theorem 4.3.2 now yield the isomorphism

Cntco ∼= SetF .

The functor GF′ : SetF −→ F-Mon of 4.2 sending a T-algebra (X, a) to the T-monoid (X, aa)therefore describes the functor Cntco −→ Top that equips a cocontinuous lattice with itsScott topology and sends a filtered-inf-preserving sup-map to a continuous map (see alsoExercise 4.D).Moreover, separated F-monoids are precisely T0-spaces and separated P-monoids areseparated ordered sets (Examples 4.3.4) so Corollary 4.3.3 tells us that the forgetful functor

Cntco −→ Ordsep

is strictly monadic.

4.5 Kock–Zoberlein monads on T-Mon. The powerset monad P induces the Kock–Zoberlein monad Dn on Ord (Example 4.2.3(1)), and we will see that F yields such amonad on Top (Example 4.2.3(2)). Before proving in Theorem 4.5.3 that an arbitrarypower-enriched monad T induces a Kock–Zoberlein monad T′ on T-Mon, as in the casesT = P or T = F, we must first show that T-Mon is an ordered category.4.5.1 Lemma. If T = (T,m, e) is a power-enriched monad, the initial order (4.3.i) inducedby the Kleisli monoid structures yields functors

T-Mon −→ Ord and T-Monsep −→ Ordsep

that commute with the underlying-set functors. As a consequence, T-Mon is an ordered andT-Monsep a separated ordered category.


Proof. To see that morphisms in T-Mon are monotone, consider f : (X, ν) −→ (Y, µ) andsuppose that x, y ∈ X are such that x ≤ y, or equivalently ν(x) ≤ ν(y). Then

eY · f(x) = Tf · eX(x) ≤ Tf · ν(x) ≤ Tf · ν(y) ≤ µ · f(y) ;

by composing with µT on the left, one obtains µ · f(x) ≤ µ · f(y), so that f(x) ≤ f(y) bydefinition of the order on Y . Hence, T-monoid morphisms are monotone, and we have theclaimed functors into Ord and Ordsep. Since hom-sets of T-Mon are ordered pointwise, wehave monotone maps

f · (−) : T-Mon(Z,X) −→ T-Mon(Z, Y ) and (−) · g : T-Mon(X, Y ) −→ T-Mon(Z, Y )

for all f ∈ T-Mon(X, Y ) and g ∈ T-Mon(Z,X), and T-Mon is a separated ordered category.

The initial order allows us to refine our analysis of the maps qX and pX defined in 4.2.

4.5.2 Lemma. Let T = (T,m, e) be a power-enriched monad, T′ its induced monad onT-Mon (with S = T and α = 1T in 4.2), and (X, ν) a T-monoid. Then, with respect tothe initial order on T ′X induced by ωX , the section qX : (T ′X,ωX) −→ (TX,maX) is anorder-embedding and the retraction pX : (TX,maX) −→ (T ′X,ωX) is a T-monoid morphism.In fact, pX a qX .

Proof. By Lemmata 4.2.1 and 4.5.1, the maps qX are monotone. Suppose that for aT-monoid (X, ν) there are x , y ∈ T ′X with qX(x ) ≤ qX(y ), or equivalently, such thatmaX · qX(x ) ≤ maX · qX(y ). One can compose each side of this last inequality with themonotone map TpX on the left to obtain ωX(x ) ≤ ωX(y ), that is, x ≤ y by definition ofthe order on T ′X.Since T is power-enriched, eX ≤ ν so 1TX = eTX ≤ νT = qX · pX and

TpX ·maX ≤ TpX ·maX · qX · pX = ωX · pX .

Hence, pX is a T-monoid morphism.The last statement then follows from 1TX ≤ νT = qX · pX and pX · qX = 1T ′X .

4.5.3 Theorem. For a power-enriched monad T, the derived monad T′ on T-Mon (orT-Monsep) is of Kock–Zoberlein type.

Proof. Since T is power-enriched, one has for T-monoid morphisms f, g : (X, ν) −→ (Y, µ)

f ≤ g =⇒ (e′Y · f)T′ = pY · (qY · e′Y · f)T · qX ≤ pY · (qY · e′Y · g)T · qX = (e′Y · g)T′ ,

so T ′ is a 2-functor (since (e′Y · f)T′ = T ′f).


To verify that m′X a e′T ′X for all T-monoids (X, ν), it suffices to verify 1T ′X ≤ e′T ′X ·m′X(because m′X · e′T ′X = 1T ′X always holds). In view of this, we write

qT ′X ≤ TpX ·maX ·mX · TqX · qT ′X= TpX ·maX · (qX)T · qT ′X= TpX ·maX · (νT · qX)T · qT ′X= TpX ·maX · qX · pX · (qX)T · qT ′X= ωX ·m′X = qT ′X · e′T ′X ·m′X .

Hence, 1T ′X ≤ e′T ′X ·m′X because qT ′X is an order-embedding (Lemma 4.5.2). The sameproof obviously holds if T-Mon is replaced by T-Monsep.

4.5.4 Examples.

(1) If T = P, Example 4.2.3(1) shows that P′ = Dn is the down-set monad on Ord. Theo-rem 4.5.3 states that this monad is Kock–Zoberlein and we recover Example II.4.9.3.

(2) The open-filter monad of Example 4.2.3(2) is the monad F′ on F-Mon ∼= Top derivedfrom the filter monad F, and is therefore Kock–Zoberlein.

4.6 Eilenberg–Moore algebras and injective Kleisli monoids. We are now readyto look into the identification of injective objects that motivated this Section 4.Let U : T-Mon −→ Set be the forgetful functor associated to a power-enriched monad T. Wedenote by Ini the class IniU of all U -initial T-monoid morphisms, and by RegMono theclass RegMono(T-Mon) of all U -initial T-monoid morphisms whose underlying maps aremonomorphisms (Exercise II.5.D).In Theorem 4.6.3 below, we identify M-injective Kleisli monoids when M = Ini orM = RegMono as T-algebras (see Exercise II.5.M). If A is an ordered category, we denoteby M-Inj(A) the category of M-injective A-objects with left adjoint A-morphisms.When T is power-enriched, every map Tf is a morphism of T-monoids Tf : (TX,maX) −→(TY,maY ). In fact, its right adjoint (Tf)a is a right adjoint (Tf)a : (TY,maY ) −→ (TX,maX)in T-Mon. Indeed, (−)T applied to Tf · (Tf)a ≤ 1TX yields the inequality in

Tf ·mX · T (Tf)a = mY · TTf · T (Tf)a ≤ mY ,

so that T (Tf)a ·maY ≤ maX · (Tf)a by adjunction.For the associated monad T′ = (T ′,m′, e′) on T-Mon (induced as in 4.2 by the identitymonad morphism α = 1T : T −→ T), if f : (X, ν) −→ (Y, µ) is a T-monoid morphism, onecan define the right adjoint T-monoid morphism (T ′f)a : (T ′Y, ωY ) −→ (T ′X,ωX) by

(T ′f)a := pX · (Tf)a · qY .


Indeed, (T ′f)a is a composite of T-monoid morphisms (Lemma 4.5.2) and one verifies that

1T ′X ≤ (T ′f)a · T ′f and T ′f · (T ′f)a ≤ 1T ′Y

by using the identity T ′f = pY · Tf · qX .

4.6.1 Lemma. If T is a power-enriched monad, the U-initial T-monoid structure on Xinduced by f : X −→ U(Y, µ) is

ν := (Tf)a · µ · f .

As a consequence, e′X : (X, ν) −→ (T ′X,ωX) is U-initial.

Proof. The fact that (Tf)a · µ · f defines an initial T-monoid morphism f : (X, ν) −→ (Y, µ)follows from straightforward verifications. The initial structure on X induced by e′X : X −→U(T ′X,ωX) is therefore

(Te′X)a · ωX · e′X = (TpX · Tν)a · TpX ·maX · qX · pX · eX = (Tν)a ·maX · ν .

Furthermore, (νT)a = eTX · (νT)a ≤ νT · (νT)a ≤ 1TX so

ν ≤ (νT)a · νT · ν = (νT)a · ν ≤ ν ,

that is, (Tν)a ·maX · ν = ν so the last claim is verified.

4.6.2 Lemma. Let T be a power-enriched monad. In the notations of 4.2 (with S = T andα = 1T), a morphism of T-monoids f : (X, ν) −→ (Y, µ) satisfies

ν = (Tf)a · µ · f ⇐⇒ e′X = (T ′f)a · e′Y · f .

Proof. Suppose that ν = (Tf)a · µ · f holds. Then

(T ′f)a · e′Y · f = pX · (Tf)a · qY · e′Y · f = pX · (Tf)a · µ · f = pX · ν = e′X .

Conversely, if e′X = (T ′f)a · e′Y · f , then, recalling that T ′f · pX = pY · Tf by Lemma 4.3.1and that qX = paX by Lemma 4.5.2, we can write

ν = qX · e′X = paX · (T ′f)a · e′Y · f = (pY · Tf)a · e′Y · f = (Tf)a · qY · e′Y · f = (Tf)a · µ · f .

4.6.3 Theorem. If T is power-enriched, then there is an isomorphism of categories

Ini-Inj(T-Mon) ∼= (T-Mon)T′

which commutes with the forgetful functors to T-Mon. In particular, the Ini-injectiveT-monoids are precisely the T′-algebras.


Proof. Consider a T-monoid (X, ν) that is an Ini-injective object. By Lemma 4.6.1, theT-monoid morphism e′X : (X, ν) −→ (T ′X,ωX) is U -initial, so there is a T-monoid morphisma′ : (T ′X,ωX) −→ (X, ν) that extends 1X : (X, ν) −→ (X, ν) along e′X :

(X, ν)e′X //

1X &&

(T ′X,ωX)

a′

(X, ν) ,

(4.6.i)

that is, such that a′ · e′X = 1X . Moreover, by using that T′ is Kock–Zoberlein and e′ anatural transformation, we get

1T ′X = T ′a′ · T ′e′X ≤ T ′a′ · e′T ′X = e′X · a′ ,

so a′ a e′X . Also, since T′ is a Kock–Zoberlein monad, a′ · T ′a′ ' a′ · m′X (that is,a′ · T ′a′ ≤ a′ ·m′X and a′ ·m′X ≤ a′ · T ′a′, see II.4.9); this equivalence is induced by theseparated order on T ′X, so it is actually an equality a′ · T ′a′ = a′ ·m′X . Hence, (X, a′) isa T′-algebra. Let us finally return to the definition of a′: by the adjunction a′ a e′X , themorphism a′ is determined up to equivalence in X; since a′ · pX · ν = 1X implies that theorder on X is separated, a′ is really uniquely determined.Suppose that f : (X, ν) −→ (Y, µ) is a morphism between Ini-injective T-monoids that has aright adjoint fa : (Y, µ) −→ (X, ν) in T-Mon. The previous construction yields T-algebras(X, a′) and (Y, b′), respectively. One then has

b′ · T ′f = b′ · pY · Tf · qX≤ b′ · pY · Tf · qX · e′X · a′ (a′ a e′X)≤ b′ · pY · µ · f · a′ (Tf · ν ≤ µ · f)= f · a′ (b′ · pY · µ = b′ · e′Y = 1Y ),

that is, b′ ·T ′f ≤ f ·a′. In the same way, one obtains a′ ·T ′(fa) ≤ fa · b′, which is equivalentto f ·a′ ≤ b′ ·T ′f , and we can conclude that f : (X, a′) −→ (Y, b′) is a morphism of T-algebras.Conversely, any T′-algebra ((X, ν), a′) makes the diagram (4.6.i) above commute. Thus,given a U -initial morphism j : (Y, µ) −→ (Z, ζ), and a morphism f : (Y, µ) −→ (X, ν), onecan define f := a′ · T ′f · (T ′j)a · e′Z :

(T ′Y, ωX)

T ′f ''

(T ′Z, ωZ)(T ′j)aoo (Z, ζ)

e′Zoo

f

(T ′X,ωX)a′// (X, ν)

The morphism f : (Z, ζ) −→ (X, ν) does indeed extend f along j, since by Lemmata 4.6.2and 4.6.1 one has

f · j = a′ · T ′f · (T ′j)a · e′Z · j = a′ · T ′f · e′Y = a′ · e′X · f = f .


This proves that (X, ν) is an Ini-injective object of T-Mon.If f : ((X, ν), a′) −→ ((Y, µ), b′) is a morphism of T′-algebras, then the T-monoid morphismfa : (Y, µ) −→ (X, ν) defined by

fa = a′ · pX · (Tf)a · µ

is right adjoint to f . Indeed, one readily verifies the inequalities

1X ≤ fa · f and f · fa ≤ 1Y .

The passages from Ini-injective Kleisli monoids to Eilenberg–Moore algebras and backdescribed above obviously define functors that commute with the forgetful functors toT-Mon and that are inverse to each other.

4.6.4 Corollary. If T is power-enriched, then there is an isomorphism between the categoryof RegMono-injective separated T-monoids (with left adjoint morphisms) and the categoryof T′-algebras (with T′ seen as a monad on T-Monsep):

RegMono-Inj(T-Monsep) ∼= (T-Monsep)T′ .

Moreover, the functors forming the isomorphism commute with the forgetful functors toT-Mon.

Proof. The proof that an Ini-injective object is an Eilenberg–Moore algebra in Theorem 4.6.3relies on the fact that the structure morphism ν : X −→ TX of a Kleisli monoid (X, ν)belongs to the class of U -initial morphisms. By definition, the structure morphisms ofseparated Kleisli monoids have monic underlying maps, so the same proof yields the statedisomorphism.

4.6.5 Corollary. Given an associative lax extension T of T to V-Rel, there is an isomor-phism

Ini-Inj((T,V)-Cat) ∼= (-Mon)′ .

In particular, the Ini-injective (T,V)-categories are precisely the ′-algebras (where is thediscrete presheaf monad associated to T).

Proof. By Proposition 3.2.1, the discrete presheaf monad associated to T is power-enriched.The result then follows directly from the isomorphisms of Theorems 4.6.3 and 3.2.2.

The following direct consequences of Theorem 4.6.3 and its corollaries allow us in particularto identify the injective objects of Ordsep and Top0.


4.6.6 Examples.

(1) Since SetP ∼= Sup and P-Mon ∼= Ord, complete lattices are both the Ini-injectiveobjects in Ord and the RegMono-injective objects in Ordsep (in this case the, regularmonomorphisms are the order-embeddings).

(2) Cocontinuous lattices can equivalently be seen as the Ini-injective objects in Topor the RegMono-injective objects in Top0 via the isomorphisms SetF ∼= Cntco, andF-Mon ∼= Top or F-Monsep ∼= Top0, see Exercise 4.B (here, regular monomorphismsare the topological embeddings).

(3) Completely distributive lattices are the Ini-injective objects of the category Cls ofclosure spaces (Exercise 4.E), and frames are the Ini-injective objects of the categoryClsfin of finitary closure spaces (Exercise 4.F).

(4) For a given quantale V, the Ini-injective objects of V-Cat are the V-actions in Sup(Exercise 4.G).

(5) Quantales are the Ini-injective objects of the category MultiOrd ∼= (L, 2)-Cat of multi-ordered sets and their morphisms (see V.1.4 and Exercise 4.H).

Exercises

4.A A distributive law for the down-set-filter monad. The down-set-filter monad FilDn is acomposite monad obtained from a distributive law of Dn over Fil (see 4.4). The monadDn is of Kock–Zoberlein type and Fil is of dual Kock–Zoberlein type, but FilDn is neither.Moreover, if (X, a) is a Dn-algebra, then the order on X is necessarily separated, so the2-isomorphisms of Example II.4.9.3 become

OrdDn ∼= Sup and OrdUp ∼= Inf .

Hence, OrdFil is 2-isomorphic to the category of ordered sets in which every down-directedset has an infima, with maps preserving down-directed infima.

4.B T0-spaces. Via the isomorphism F-Mon ∼= Top of Proposition 1.1.1, the categoryF-Monsep of separated F-monoids is isomorphic to the category Top0 of T0-spaces.

4.C Separated representable Kleisli monoids. Let T = (T,m, e) be a power-enrichedmonad with its Kleisli extension T to Rel. It follows from III.5.1 that T induces a monadT = (T,m, e) on Ord. Show:

(1) the functor T : Set −→ Set sends surjections to split epimorphisms;

(2) the full subcategory (OrdT)sep of OrdT spanned by all separated ordered sets is reflective;

(3) the composite functor (OrdT)sep → OrdT K−→ (T, 2)-Cat (see III.5.3) is strictly monadic;


(4) there is an isomorphism SetT ∼= (OrdT)sep that commutes with the underlying-setfunctors;

(5) the functor SetT −→ T-Mon of 4.2 factors asSetT ∼= (OrdT)sep −→ OrdT −→ (T, 2)-Cat ∼= T-Mon .

4.D Open sets and continuity in the Scott topology. The open sets of the Scott topology ona cocontinuous lattice X are those down-sets O ∈ DnX such that

∀A ∈ FilX (∧A ∈ O =⇒ A ∩O 6= ∅) .If Y is another cocontinuous lattice, then a map f : X −→ Y is continuous if and only if itpreserves down-directed infima.

4.E Completely distributive lattices as Eilenberg–Moore algebras. The Eilenberg–Moorecategory of the up-set monad U on Set is the category Dst of completely distributive latticesand maps that preserve all infima and suprema:

SetU ∼= Dst .

The Ini-injective objects of the category Cls of closure spaces are precisely the completelydistributive lattices (use Example 1.3.2(4)).

4.F Frames as Eilenberg–Moore algebras. The Eilenberg–Moore category of the finitary-up-set monad of Exercise 1.E is the category frames and frame homomorphisms:

SetUfin ∼= Frm .

The Ini-injective objects of the category Clsfin of finitary closure spaces are precisely frames(use Example 1.E).

4.G Categories associated to the V-powerset monad. The V-powerset monad (Exer-cise III.1.D) is power-enriched, and

PV-Mon ∼= V-Cat and SetPV ∼= SupV ,

(see Example II.4.3.1(3)). Hence, the Ini-injective objects of V-Cat are the V-actions in Sup.

4.H Quantales. There is a distributive law δ : LP −→ PL of the list monad L over thepowerset monad P given by

δ(A1, . . . , An) = (a1, . . . , an) | ai ∈ Ai .The resulting composite monad PL is power-enriched, and is induced by the monadic com-position of the monadic forgetful functors Qnt −→ Mon and Mon −→ Set (see Exercise II.2.Tand Example II.3.2.2(1)). Show that PL is isomorphic to the discrete presheaf monad ofthe list monad, so that

PL-Mon ∼= (L, 2)-Cat(see Exercise 3.C).


5 Domains as lax algebras and Kleisli monoids

It is shown in 4.4 that the cocontinuous lattices are the Eilenberg–Moore algebras of thefilter monad F on Set, whereas in Corollary 1.5.4 we saw that the category (F, 2)-Cat of laxalgebras of the Kleisli extension of the filter monad is isomorphic to the category Top of alltopological spaces. We saw in 4.4 that there is a monadic functor

Cntco ∼= SetF −→ F-Mon ∼= (F, 2)-Cat ∼= Top

which associates to each cocontinuous lattice its Scott topology (see Exercise 4.D). We shallsee that the lax algebra thus associated with each cocontinuous lattice is strict, meaningthat it satisfies the lax multiplicative Eilenberg–Moore strictly. Further, we shall showin Theorem 5.4.1 that the strict lax algebras of F constitute a class of topological spacesthat not only generalize the continuous lattices but in fact provide a close topologicalgeneralization of the broader class of continuous dcpos (see 5.2). The spaces in question—which we call the observable realization spaces—capture an essential domain-theoreticapproximation property in topological terms, and the continuous dcpos occur among thesespaces as precisely the sober observable realization spaces. In 5.9 we will characterize thosestrict lax algebras that are associated to continuous lattices.In Theorem 5.7.2 we shall also show that the observable realization spaces are Kleislimonoids of the ordered-filter monad Fil on Ord.

5.1 Modules and adjunctions. Recall from II.2.2 that the category Ord of ordered setsand monotone maps is manifested within the category Mod of ordered sets and modules astheir morphisms via functors

(−)∗ : Ord −→ Mod and (−)∗ : Ordop −→ Mod

which send a monotone map f : X → Y to the modules

f∗ = (≤Y ) · f : X −→ Y and f ∗ = f · (≤Y ) : Y −→ X ,

respectively, where f : Y −→7 X is the converse relation of f (given by y f x ⇐⇒ y =f(x)). Endowing the hom-sets Ord(X, Y ) and Mod(X, Y ) with the pointwise and inclusionorders, respectively, we find that for monotone maps f, g : X → Y ,

f∗ ≥ g∗ ⇐⇒ f ≤ g ⇐⇒ f ∗ ≤ g∗ ,

so that we have order-embeddings

(−)∗ : Ord(X, Y )op → Mod(X, Y ) and (−)∗ : Ord(X, Y )→ Mod(Y,X) .

We recall (see 1.4) that for monotone maps f : X → Y , g : Y → X, one has

f a g ⇐⇒ ∀x ∈ X, y ∈ Y (f(x) ≤ y ⇐⇒ x ≤ g(y)) ⇐⇒ f∗ = g∗ .

5. DOMAINS AS LAX ALGEBRAS AND KLEISLI MONOIDS 351

As a quantaloid, Mod is in particular an ordered category and hence a 2-category. For anymonotone map f : X → Y we have

1∗X ≤ f ∗ · f∗ and f∗ · f ∗ ≤ 1∗Y ,

so that f∗ is left adjoint to f ∗ in the 2-category Mod; in symbols, f∗ a f ∗.

5.2 Cocontinuous ordered sets. Section 4.4 defines a cocontinuous lattice X as acomplete lattice for which the infimum map ∧X : FilX −→ X has a right adjoint X ; if sucha right adjoint exists, then it is given by

x ∈ a ⇐⇒ ∀S ∈ FilX (∧S ≤ a =⇒ x ∈ S) . (5.2.i)

Considering an arbitrary ordered set X instead of a complete lattice, let us define a relation( ) : X −→7 X by

a x ⇐⇒ ∀S ∈ Fil X (∧S ≤ a =⇒ x ∈ S) ,

where Fil X is the set of all filters S in X for which a meet ∧S exists. Letting x := y ∈X | x y for each x ∈ X, we say that the ordered set X is cocontinuous if

x ∈ Fil X and ∧ x ∼= x (5.2.ii)

for each x ∈ X. For example, if X is a co-dcpo, that is, if X is separated and Fil X = FilX,then the infimum map ∧

X : FilX −→ X has a right adjoint X if and only if X iscocontinuous; if this is the case then the right adjoint is given by X x = x and we saythat X is a cocontinuous co-dcpo. The dual notion, that of a continuous dcpo, is more oftenemployed. The term dcpo was originally introduced as an acronym for directed-completepartial order.For any ordered set X, we may define the Scott topology on X as the collection of allScott-open sets, that is, of all down-sets U ∈ DnX such that

∀S ∈ Fil X (∧S ∈ U =⇒ ∃x ∈ S ∩ U) .

As in Exercise 4.D with regard to cocontinuous lattices, one may readily verify that afunction f : X → Y between ordered sets is continuous with respect to the Scott topologieson X and Y if and only if f preserves meets of down-directed subsets. Furthermore, theScott topology on an ordered set X is order-compatible, in the sense that the underlyingorder associated to the Scott topology of an ordered set X coincides with the order relationpossessed by X. In a topological space X endowed with its underlying order, we say that apoint x specializes a point y if x ≤ y.The nature of the Scott topology is related to the preservation of down-directed meetsby the neighborhood-filter map ν : X → FX of a topological space X endowed with itsunderlying order. For any down-directed subset S ⊆ X with a meet in X, the image ν(S)


is down-directed in FX by monotonicity, and directed meets in FX are given by union, soto say that ν preserves the meet ∧S means that

ν(∧S) = ⋃z∈S ν(z) .

5.2.1 Definition. Let X be a topological space, endowed with its underlying order, andlet S ⊆ X be a down-directed subset of X with a meet ∧S in X. We say that ∧S is atopological meet of S in X if the neighborhood-filter map ν : X → FX preserves this meet.

Given an ordered set X, the Scott topology on X is the largest order-compatible topologywhose neighborhood filter map ν preserves all directed meets, as follows:

5.2.2 Proposition. Let X be a topological space endowed with its underlying order. Thefollowing are equivalent:

(i) OX is coarser than the Scott topology on X; that is, every open set is Scott-open;

(ii) every directed meet in X is a topological meet.

Proof. This is Exercise 5.A.

We shall also require the following Lemma.

5.2.3 Lemma. Let X be a cocontinuous ordered set. Then for any y ∈ X, the set

y := x ∈ X | x y

is Scott-open.

Proof. Suppose that S ∈ Fil X and ∧S ∈ y. Observe that∧S =

∧∧ x | x ∈ S =

∧(⋃ x | x ∈ S) .

Since ⋃ x | x ∈ S is the union of a down-directed subset of FilX, it is an element ofFilX. The result follows.

One can provide an alternate characterization of the property of cocontinuity of an orderedset with reference to the Scott topology. For any topological space X with its underlyingorder and with neighborhood-filter map ν : X → FX, we may define a relation (≺) : X −→7X by

x ≺ y ⇐⇒ ↓ y ∈ ν(x) ,so in particular the set

y := x ∈ X | x ≺ yis open for all y ∈ X. Since ↓ y is the set of points that specialize y, the statement thatx ≺ y for points x, y ∈ X is in general stronger than the statement that x specializes y,


and we say that x observably specializes y if x ≺ y. We refer to (≺) as the observablespecialization relation associated to X.In the case that X is a cocontinuous ordered set under the Scott topology, we find that(≺) = ( ), as follows. For an ordered set X and a relation r : X −→7 X, we say that r isapproximating if for each x ∈ X, r(x) is a filter in X and x is a meet of r(x). For example,X is cocontinuous if and only if ( ) is approximating.

5.2.4 Lemma. For an ordered set X endowed with the Scott topology, if either ( ) or (≺)is approximating, then ( ) = (≺).

Proof. See Exercise 5.B.

5.2.5 Proposition. For an ordered set X endowed with the Scott topology, the followingare equivalent:

(i) X is cocontinuous.

(ii) the relation (≺) is approximating;

(iii) each x ∈ X is a topological meet of x := y ∈ X | x ≺ y.

Proof. This is Exercise 5.C.

5.3 Observable realization spaces. The third characterization of continuity in orderedsets in Proposition 5.2.5 affords an interesting generalization to arbitrary topological spaces:

5.3.1 Proposition. Let X be a topological space endowed with its underlying order. Thereis an embedding of ordered sets ι : FilX → FX given by S 7−→ ↑PX↓ y | y ∈ S, and thefollowing are equivalent:

(i) each x ∈ X is a topological meet of x;

(ii) there is a map n : X → FilX such that the neighborhood-filter map ν : X → FXfactors as

Xn // FilX ι // FX ;

(iii) there is a module c : FilX −→ X such that the convergence relation a : FX −→7 X ofX factors as

FXι∗// FilX

c// X ;

(iv) for all U ∈ OX and x ∈ U , there exists y ∈ U with x ≺ y.


If these conditions hold then the map n of (ii) and the relation c of (iii) are uniquelydetermined as

n = : X → FilX and c = a · ι =

∗ : FilX −→ X

with Sc−→ x ⇐⇒ ι(S) ⊇ ν(x) ⇐⇒ S ⊇ x, and : X → FilX is a fully faithful

monotone map.

Proof. It is readily verified that ι as given is well-defined, injective, monotone, and fullyfaithful.Suppose (iv) holds. Then for any x ∈ X we have that⋃

y∈ x ν(y) ⊇ ν(x) ,

and since x ⊆ ↑x, this inclusion is in fact an equality, from which it follows that x is ameet of x. Further, x is down-directed, since if y, z ∈ x then the sets

y = t ∈ X | t ≺ y and

z = t ∈ X | t ≺ z

are open neighborhoods of x and hence by (iv) there is u ∈ x with u ∈

y∩

z ⊆ ↓ y∩↓ zas needed. Thus, x is a topological meet of x, showing that (i) holds.Suppose (i) holds. Then for any x ∈ X

ν(x) = ⋃y∈ x ν(y)

and hence (iv) holds. We find also that (ii) holds, as follows. The preceding equationimplies that ν(x) ⊆ ι( x) since open sets are down-closed. This inclusion may be replacedby an equality, since for any y ∈ x we have that ν(x) 3 ↓ y, so we find that ν factorsthrough the map : X → FilX x 7−→ x as ν = ι · .If there is a map n as in (ii), then for any x ∈ X we find that

n(x) 3 y ⇐⇒ ι(n(x)) 3 ↓ y ⇐⇒ ν(x) 3 ↓ y ⇐⇒ x ≺ y

for each y ∈ X, whence n(x) = x and we find also that (iv) holds. Further, since ν andι are monotone and fully faithful and ι · n = ν, it follows that n is monotone and fullyfaithful. Letting c := n∗, we have that

a = ν∗ = (ι · n)∗ = n∗ · ι∗ = c · ι∗ ,

so (iii) holds.Suppose we have a relation c as in (iii). Since ι is fully faithful we have that ι∗ · ι∗ = 1∗FilXand hence

a · ι = a · ι∗ = c · ι∗ · ι∗ = c .

Further,

1X ≤ 1∗X ≤ ν∗ · ν∗ = a · ν∗ = c · ι∗ · ν∗ = a · ι∗ · ι∗ · ν∗ = ν∗ · ι∗ · ι∗ · ν∗ = ν∗ · ι · ι · ν∗ ,

so for each x ∈ X there is S ∈ FilX with ν(x) ⊇ ι(S) ⊇ ν(x), whence ν(x) = ι(S), andone verifies that S = x and hence ν(x) = ι( x). Therefore, (ii) holds with n := .


5.3.2 Definition. For a topological space with convergence relation a : FX −→7 X, we findthat the associated relation a · ι : FilX −→7 X of the preceding proposition is such thatS

a·ι−→ x if and only if S enters every open neighborhood of x, and if this is the case thenwe say that S observably realizes x. Note that S observably realizes x if and only if x is inthe closure of S.If the equivalent conditions of Proposition 5.3.1 hold for a topological space X, then we saythat X is an observable realization space.

5.3.3 Example. Every cocontinuous ordered set is an observable realization space whenendowed with the Scott topology. In fact, we shall show in 5.8 that the cocontinuousco-dcpos under the Scott topology are precisely the sober observable realization spaces. Fora cocontinuous co-dcpo, the map : X −→ FilX is given by x 7−→ x, and the observablerealization relation c = a · ι : FilX −→7 X is given by

Sa·ι−→ x ⇐⇒ ∧

S ≤ x ;

the associated filter convergence relation a = c · ι∗ is called the relation of Scott convergence.

Thus, when we express the domain-theoretic approximation property of continuity intopological rather than order-theoretic terms, relative to an arbitrary topology, we obtain acharacterization of the observable realization spaces (Proposition 5.3.1(i)). Moreover, wefind that these spaces can also be characterized via condition 5.3.1(iii) as those in whichfilter convergence reduces to a relation of convergence of directed sets that captures theessential topological character of the notion of convergence by directed meets supported bycocontinuous co-dcpos. Thus, these spaces provide our general notion of a domain, suchthat in 5.8 we shall see that the continuous dcpos are those domains that are sober.The following Lemma entails in particular that every observable realization space carries atopology finer than the Scott topology on its specialization order (that is, a topology inwhich every Scott-open subset is open). The proof is straightforward.

5.3.4 Lemma. Suppose X is a topological space with its underlying order such that (≺) isapproximating. Then OX is finer than the Scott topology on X; that is, every Scott-opensubset is open.

5.4 Observable realization spaces as lax algebras. We now observe that the ob-servable realization spaces may be characterized among arbitrary topological spaces asthose whose corresponding (F, 2)-category (with respect to the Kleisli extension F, see 1.4)satisfies the lax Eilenberg–Moore multiplicative law up to equality rather than simply aninequality, as follows:

5.4.1 Theorem. Let X be a topological space with associated (F, 2)-category (X, a) (wherea : FX −→7 X is the filter convergence relation associated to X). Then

a ·mX = a · F a ⇐⇒ X is an observable realization space.


Thus, the isomorphism (F, 2)-Cat ∼= Top of Corollary 1.5.4 restricts to an isomorphism

(F, 2)-Cat= ∼= ObsReSp

between the full subcategory (F, 2)-Cat= of (F, 2)-Cat, consisting of those lax algebras of Fthat satisfy the multiplicative law up to equality, and the full subcategory ObsReSp of Topwith objects all observable realization spaces.

Proof. Since F is a power-enriched monad, we know by Proposition 1.2.1 that mX : FFX →FX is a sup-map and hence has a right adjoint maX : FX → FFX given by

maX(x ) = ↑PFXAF | A ∈ x ,

for all x ∈ FX (where AF = y ∈ FX | A ∈ y). Also, with reference to (1.4.i), thereis a monotone map aτ : FX → FFX, given by aτ (y ) = ↑PFXa(A) | A ∈ y, such thatF a = (aτ )∗. Note also that a = ν∗, where ν : X → FX is the neighborhood filter map ofX. Letting OX(x) denote the set of open neighborhoods of a point x ∈ X, we may reasonas follows:

a ·mX = a · F a⇐⇒ a ·mX ≤ a · F a ((X, a) ∈ (F, 2)-Cat)⇐⇒ a · (mX)∗ ≤ a · F a⇐⇒ ν∗ · (mX)∗ ≤ ν∗ · F a⇐⇒ ν∗ · (maX)∗ ≤ ν∗ · F a (mX a maX)⇐⇒ ν∗ · (maX)∗ ≤ ν∗ · (aτ )∗

⇐⇒ (maX · ν)∗ ≤ (aτ · ν)∗

⇐⇒ maX · ν ≤ aτ · ν⇐⇒ ∀x ∈ X

(↑AF | A ∈ ν(x) ⊇ ↑a(A) | A ∈ ν(x)

)⇐⇒ ∀x ∈ X

(↑V F | V ∈ OX(x) ⊇ ↑a(U) | U ∈ OX(x)

)⇐⇒ ∀x ∈ X,U ∈ OX(x)

(∃V ∈ OX(x) (V F ⊆ a(U))

).

This last condition is equivalent to condition (iv) of Proposition 5.3.1, as follows. Considerany x ∈ X and U ∈ OX(x). For any V ∈ OX(x), we have that

V F ⊆ a(U) ⇐⇒ ↑V ∈ a(U)

and hence moreover

∃V ∈ OX(x) (V F ⊆ a(U)) ⇐⇒ ∃V ∈ OX(x), y ∈ U (↑V a−→ y)⇐⇒ ∃y ∈ U (x ≺ y) ,


with the last equivalence holding as follows. Firstly, if V ∈ OX(x), y ∈ U , and ↑V a−→ ythen any z ∈ V specializes y since ν(z) ⊇ ↑V ⊇ ν(y), so that x ∈ V ⊆ ↓ y and hencex ≺ y. Conversely, if x ≺ y then there is V ∈ OX with x ∈ V ⊆ ↓ y, so for any openneighborhood U of y, V ⊆ ↓ y ⊆ U , whence we have that ↑V ⊇ ν(y); equivalently,↑V a−→ y.

5.4.2 Remark. Related to the characterization of the observable realization spaces givenin Theorem 5.4.1, we already mentioned in Remark III.5.6.5 that (, 2)-Cat=, taken withrespect to the Barr extension, is isomorphic to the full subcategory of Top consisting ofexponentiable spaces (see Proposition III.5.6.6 and Theorem III.5.8.5). In fact, if the lastcondition in the first chain of equivalences in the proof of Theorem 5.4.1 is modified bysubstituting the set V = x ∈ V F | x is an ultrafilter in place of V F, we obtain thefollowing result.

5.4.3 Corollary. c©Every observable realization space is exponentiable in Top.

5.5 Observable specialization systems. We now show that observable realizationspaces may be axiomatized quite directly as a set equipped with a binary relation (≺). Wethen find that this axiomatization in turn leads to a description of these spaces as Kleislimonoids.

5.5.1 Definition. Let X be a set and (≺) : X −→7 X a relation. We say that (X, (≺)) isan observable specialization system if the following statements hold for all x, y, z ∈ X:

(1) x ≺ y ≺ z =⇒ x ≺ z;

(2) ∃u ∈ X (x ≺ u);

(3) x ≺ y, z =⇒ (∃u : x ≺ u ≺ y, z);

(4) x ≺ y & (∀u (z ≺ u =⇒ y ≺ u)) =⇒ x ≺ z.

If (1)–(3) hold, then we say that (X,≺) is an abstract basis.The relation (≺) induces an order on X, given by

x ≤ y ⇐⇒ x ⊇ y (5.5.i)

for x, y ∈ X, where x := y ∈ X | x ≺ y. With respect to this order, note that (4)requires exactly that (≺) : X −→7 X is a module, since one always has (≺) · (≤) ⊆ (≺).The category ObsSys has objects all observable specialization systems and morphisms allmaps f : (X,≺)→ (Y,≺) such that

∀x ∈ X y ∈ Y (f(x) ≺ y =⇒ ∃u ∈ X (x ≺ u & f(

u) ⊆

y)) ,

where

y := z ∈ Y | z ≺ y.


5.5.2 Theorem. There is an isomorphism

ObsReSp ∼= ObsSys

which commutes with the underlying-set functors and sends an observable realization spaceX to the pair (X,≺) consisting of the underlying set and observable specialization relation(≺) of X.Quite generally, if (X,≺) is an abstract basis, then B :=

y | y ∈ X is a base for a

topology on X, under which X is an observable realization space, and if (X,≺) is moreoveran observable specialization system, then (≺) coincides with the observable specializationrelation of this observable realization space.

Proof. If X is an observable realization space, then by Proposition 5.3.1, X : X −→ FilXis a fully faithful monotone map, so the underlying order (≤X) is the order induced by(≺X) (5.5.i), and it is an easy exercise to verify that (1)–(4) hold for (≺X).Next suppose (X,≺) is an abstract basis. It is readily verified that B as given is a base fora topology on X (meaning that for each x ∈ X, the set B ∈ B | x ∈ B is either emptyor down-directed) and that the order induced by (≺) (5.5.i) coincides with the associatedunderlying order (≤X) of the resulting topological space X. It then follows by (1) that(≺) ≤ (≤X), and in turn that (≺) ≤ (≺X) using (3). We may use this and (3) to deduce bymeans of Proposition 5.3.1 (iv) that X is an observable realization space.Supposing moreover that (X,≺) also satisfies (4), we now show that (≺X) ≤ (≺) as well.If x ≺X y then there is some u with x ∈

u ⊆ ↓X y, and by (3) there is some v withx ≺ v ≺ u, whence x ≺ v ≤X y, so by (4), x ≺ y.Noting also that for an observable realization space X the base B associated to (X,≺X) isa base for X, the result follows.

Thus, while the topological concept of a domain embodied by the observable realizationspaces is not the usual order-theoretic one, in which the specialization order is primitiveand the topology is derivative, the observable realization spaces are nevertheless determinedby a transitive relation, but one that is not necessarily reflexive (nor irreflexive). For thesespaces, observable specialization, rather than specialization, provides the primitive notion.

5.6 Ordered abstract bases and round filters. For an ordered set X, recall that amodule (≺) : X −→ X is an auxiliary relation onX if (≺) ≤ (≤X) (see also Example 2.5.5(6)).Observe that any auxiliary relation (≺) on X is necessarily transitive, since (≺) · (≺) ≤ (≺) · (≤X) = (≺).

5.6.1 Definition. For an auxiliary relation (≺) on an ordered set X, we say that (X,≺) isan ordered abstract basis if the underlying set X and the relation (≺) constitute an abstractbasis.


Ordered abstract bases provide a mild generalization of observable specialization systems,as follows:

5.6.2 Proposition. Let X be a set and (≺) : X −→7 X a relation. A pair (X,≺) is anobservable specialization system if and only if (X,≺) is an ordered abstract basis when X isendowed with the order induced by (≺)(see (5.5.i)).

The following alternative axioms lead us to a description of ordered abstract bases andobservable specialization systems as Kleisli monoids in 5.7:

5.6.3 Proposition. For an ordered set X and a relation (≺) : X −→7 X, the pair (X, (≺))is an ordered abstract basis if and only if the following statements hold, where x := y ∈X | x ≺ y for each x ∈ X:

(1) (≺) is an auxiliary relation on X;

(2) For each x ∈ X, x is down-directed;

(3) For any x, z ∈ X, if x ≺ z then there is some y ∈ X with x ≺ y ≺ z.

5.6.4 Definition. Given an ordered set X equipped with an auxiliary relation (≺), we saythat a filter S in X is a round filter in (X, (≺)) if for any y ∈ S there is some x ∈ S withx ≺ y. Observe that (X, (≺)) is an ordered abstract basis if and only if x is a round filterfor each x ∈ X.

5.7 Domains as Kleisli monoids of the ordered-filter monad. For an ordered setX, modules (≺) : X −→ X correspond bijectively to monotone maps n : X −→ Up X via

x ≺ y ⇐⇒ n(x) 3 y ,

and we have n(x) = x. In the case that (X,≺) is an ordered abstract basis we have thateach n(x) is a round filter, so we obtain a map n : X −→ FilX. In fact, we may employ thiscorrespondence to obtain an axiomatization of observable realization spaces and orderedabstract bases as Kleisli monoids of the ordered-filter monad Fil = (Fil , ↑,⋃) (see 4.4).

5.7.1 Lemma. For a monotone map n : X −→ Up X with corresponding module (≺) :X −→ X, the following are equivalent:

(i) n : X −→ Up X restricts to an Fil-monoid structure n′ : X −→ FilX;

(ii) (X,≺) is an ordered abstract basis.

Proof. (i) is equivalent to the statement that n restricts to a (monotone) map n′ : X −→ FilXwith

↑X x ≤ n′ and ⋃ ·Filn′ · n′ ≤ n′ .


The first inequality requires that ↑X ⊇ x for all x ∈ X, or equivalently, that (≺) ≤ (≤X);since (≺) is a module, this is equivalent to the requirement that (≺) be an auxiliary relationon X. Next observe that

y ∈ (⋃ ·Filn′ · n′)(x) ⇐⇒ ↑X y ∈ (Filn′ · n′)(x)⇐⇒ ∃u ∈ n(x) (↑X y ⊆ n(u)) ⇐⇒ ∃u (u ∈ n(x) & y ∈ n(u))

for all x, y ∈ X, so that the second inequation requires that

∀x, y (x ≺ y =⇒ ∃u (x ≺ u ≺ y)) .

5.7.2 Theorem. Let X be an ordered set. There is a bijection between Fil-monoids (X,n :X −→ FilX) and ordered abstract bases (X,≺), wherein we associate to each Kleisli monoid(X,n) the ordered abstract basis whose auxiliary relation (≺) is the module associated tothe composite X n−→ FilX → Up X under the bijection Ord(X,Up X) ∼= Mod(X,X) givenabove.Consequently, observable specialization systems (X,≺) correspond bijectively to Kleislimonoids (X,n) in which n : X −→ FilX is fully faithful, and hence there are isomorphisms

ObsReSp ∼= ObsSys ∼= Fil-Monff

which commute with the underlying-set functors, where Fil-Monff is the full subcategory ofFil-Mon consisting of those Kleisli monoids (X,n) with n fully faithful.

Proof. The preceding lemma yields the bijection between Kleisli monoids and orderedabstract bases, and an ordered abstract basis (X,≺) carries the induced order if and onlyif its mate n : X −→ FilX is fully faithful.Consider observable realization spaces X,Y with associated Kleisli monoid structures nX , nYand a function f : X −→ Y . If f is continuous, then f is monotone, so we may assumethat f is monotone. Then one may employ the characterization of morphisms given inTheorem 5.5.2 to verify that f is continuous if and only if

∀x ∈ X, y ∈ Y (f(x) ≺ y =⇒ ∃u ∈ X (x ≺ u & f(u) ≺ y)) ,

using the fact that (≺X) is interpolating and (≺X) ≤ (≤X). On the other hand, f :(X,nX) −→ (Y, nY ) is a morphism of Kleisli monoids if and only if Filf · nX ≤ nY · f ,equivalently, if and only if

∀x ∈ X, y ∈ Y (f(x) ≺ y =⇒ ∃v (x ≺ v & f(v) ≤ y)) .

This is entailed by continuity, since (≺Y ) ≤ (≤Y ), and entails continuity, since if f(x) ≺ ythen we may interpolate to obtain y′ with f(x) ≺ y′ ≺ y.


5.8 Continuous dcpos as sober domains. Herein we shall show that the continuousdcpos under the Scott topology are precisely the sober observable realization spaces.Moreover, we find that the observable realization spaces are closed under sobrificationand hence that the continuous dcpos occur as a reflective subcategory of ObsReSp. Thesobrification of an observable realization space X may be formed by topologizing the set ofround filters in X.For each point x in a topological space X, the open neighborhood filter OX(x) = OX ∩ x ofx is a filter in the frame OX. The filter p = OX(x) is completely prime, meaning that forany family (Ui ∈ OX)i∈I ,

p 3 ⋃i∈I Ui ⇐⇒ ∃i ∈ I (p 3 Ui) . (5.8.i)

Quite generally, given any frame O, we refer to the completely prime filters in O as thepoints of O, and we denote the set of all such points by ptO. The idea is that we may likenthe elements u ∈ O to open sets in a topological space and the points p ∈ ptO to points ina topological space, interpreting statements of the form p 3 u as statements of membershipof a point in an open set. We can make this concrete by defining u := p ∈ ptO | p 3 ufor each u ∈ O and observing that there is a resulting frame homomorphism

O −→ P ptO , u 7−→ u

whose image is thus a topology on ptO, so that the points of O are in fact the points of atopological space ptO which we call the spatialization of O. However, the frame O ptOneed not be isomorphic to O, and in fact the discipline of locale theory pursues a formulationof general topology in which frames, rather than the usual topological spaces, are taken asthe primitive objects of study (see e.g. [Johnstone, 1982]).In the case that O is the frame OX of open sets of a topological space X, the given framehomomorphism OX −→ O ptOX has a retraction given by U 7−→ x ∈ X | OX(x) ∈ Uand hence is actually an isomorphism of frames OX ∼= O ptOX. The original space Xis manifested within the spatialization ptOX via an initial continuous map OX : X −→ptOX—the open neighborhood filter map—and this map is an embedding if and onlyif X is T0. If this map is a bijection then, since it is initial, it is a homeomorphism(that is, an isomorphism in Top) X ∼= ptOX and we say that X is sober , in which casethe points of X may be viewed as order-theoretic features inherent in the frame OX ofopens. Regardless, the isomorphism of frames OX ∼= O ptOX induces an evident bijectionptOX ∼= ptO ptOX, and in fact this bijection coincides with the open neighborhood filtermap of the space ptOX, which is thus always sober and is called the sobrification of X.For any frame O, there is a bijection ptO ∼= Frm(O, 2) given by associating to a completelyprime filter p ∈ ptO its characteristic function. Thus, we may alternatively construct thespatialization of O by taking ptO := Frm(O, 2) under the topology given by analogy with theabove. In fact, it is trivially verified that the contravariant hom-functor Frm(−, 2) : Frmop −→Set sends each frame homomorphism h : O −→ N to a continuous map pth : ptN −→ ptOand hence factors through a functor pt : Frmop −→ Top. Composing this functor with


the functor O : Top −→ Frmop that sends a continuous map f : X −→ Y to the framehomomorphism OY −→ OX given by inverse image, we obtain a covariant functor

Top O// Frmop pt

// Top

which sends a space X to its sobrification. This functor factors through the full subcategorySob of Top consisting of sober spaces as Top R−→ Sob J

−→ Top, and the maps OX : X −→ptOX, which are isomorphisms for precisely the objects of Sob, serve as the components ofa natural transformation ρ : 1Top −→ JR. One may verify immediately that ρJR = JRρand then use this fact and the naturality of ρ to verify that ρ is the unit of an adjunctionR a J , so that Sob is a reflective subcategory of Top.For observable realization spaces, we have the following:5.8.1 Proposition. Let X be an observable realization space. Then the sobrification ptOXof X is an observable realization space and is homeomorphic to the space FilX of all roundfilters in X under a topology consisting of the sets U♦ = R ∈ FilX | ∃y (R 3 y ∈ U)with U ∈ OX, and there is a commutative diagram

X

OX

FilX∼= // ptOX

in Top whose bottom row is an isomorphism.

Proof. It is straightforward to verify that the map OX −→ P FilX given by U 7−→ U♦ is aframe homomorphism, so the image of this map is a topology O FilX on FilX. Moreover,since X is an observable realization space we have that for each U ∈ OX, x ∈ X,

x ∈ U ⇐⇒ ∃y (x ≺ y ∈ U) ⇐⇒ x ∈ U♦ , (5.8.ii)

so the given frame homomorphismOX −→ O FilX has a retraction U 7−→ x ∈ X | x ∈ Uand hence is an isomorphism of frames OX ∼= O FilX. Thus, there is a homeomorphismptOX ∼= ptO FilX, wherein each completely prime filter q ∈ ptO FilX is the imageq = U♦ | U ∈ p of a unique p ∈ ptOX. It is readily verified that R = y ∈ X | p 3

yis a round filter, and then for any U ∈ OX we find that

p 3 U ⇐⇒ p 3 ⋃y∈U y ⇐⇒ ∃y ∈ U (p 3

y) ⇐⇒ R ∈ U♦ ,

whence q = OFilX(R). This shows that the initial continuous map OFilX : FilX −→ptO FilX is surjective, and we shall show below that this map is injective and hence ahomeomorphism, so we obtain a composite homeomorphism FilX ∼= ptO FilX ∼= ptOXwhich, by (5.8.ii), commutes with the maps ,OX as in the above diagram.Indeed, FilX is T0, since its underlying order (≤) is the reverse inclusion order, as follows.We have that

R ∈ (

y)♦ ⇐⇒ ∃u ∈ X (R 3 u ≺ y) ⇐⇒ R 3 y


for any R ∈ FilX, y ∈ X. Thus, for R, S ∈ FilX, one readily verifies the followingimplications,

R ≤ S =⇒ ∀y ∈ X (R ∈ (

y)♦ ⇐= S ∈ (

y)♦) ⇐⇒ R ⊇ S =⇒ R ≤ S ,

thus obtaining the needed conclusion.Lastly, we show that FilX is an observable realization space via Proposition 5.3.1(iv).Suppose R ∈ U♦. Then

R ∈ U♦ = (⋃y∈U y)♦ = ⋃y∈U(

y)♦

and hence there is y ∈ U with R ∈ (

y)♦. We know that y ∈ FilX, and each S ∈ (

y)♦specializes y since S 3 y and hence S ⊇ y. Thus, we have that R ≺ y ∈ U♦.

Before characterizing the sober observable realization spaces as precisely the continuousdcpos under the Scott topology, we note the following well-known result concerning soberspaces:

5.8.2 Proposition. Let X be a sober space. Then, under the underlying order, X isa co-dcpo, and OX is coarser than the Scott topology on X (that is, every open set isScott-open).

Proof. It is straightforward to verify that the union of a directed subset of ptOX is acompletely prime filter on OX and hence that directed meets in ptOX are given by union(as is the case in TX), so that these directed meets are moreover preserved by the monotonemap ↑PX : ptOX −→ TX. Further, the top row of the following commutative diagram isan Ord-isomorphism

XOX

∼= //

ν

ptOX

↑PX

TX

and hence X is a separated ordered set with all directed meets such that these meets arepreserved by ν. Therefore, X is a co-dcpo, and by Proposition 5.2.2, OX is coarser thanthe Scott topology.

In observable realization spaces, these conditions are also sufficient for sobriety, and weobtain the following result:

5.8.3 Theorem. For any topological space X endowed with the underlying order, thefollowing are equivalent:

(i) X is a sober observable realization space;

(ii) X is a cocontinuous co-dcpo and carries the Scott topology.


Proof. First suppose (i). Then by the preceding Proposition, X is a co-dcpo and OXis coarser than the Scott topology, so by Lemma 5.3.4, OX coincides with the Scotttopology. Thus, since X is an observable realization space, the co-dcpo X is cocontinuous,by Corollary 5.2.5 and Proposition 5.3.1.Next, suppose (ii). Since X is a cocontinuous co-dcpo carrying the Scott topology, X is anobservable realization space. X is T0, since it carries a separated underlying order, so toshow that X is sober it suffices by Proposition 5.8.1 to show that the map : X −→ FilXis surjective, so let R ∈ FilX. Since X is a co-dcpo and carries the Scott topology, byProposition 5.2.2 there is a topological meet ∧R of R in X, so that OX(∧R) = ⋃

y∈ROX(y).Thus, for any z ∈ X,∧

R ≺ z ⇐⇒ OX(∧R) 3

z ⇐⇒ ∃y (R 3 y ∈

z) ⇐⇒ R 3 z ,

since R is a round filter, so

∧R = R.

Letting CntDcpoco be the category of cocontinuous co-dcpos and directed-meet-preservingmaps, we obtain the following:

5.8.4 Corollary. There is a reflective full embedding

CntDcpoco → ObsReSp

which commutes with the underlying-set functors and endows a cocontinuous co-dcpo withits Scott topology and whose image is the full subcategory of ObsReSp consisting of soberobservable realization spaces. A left adjoint to this embedding may be taken as a restrictionof the composite

Top ptO// Top S

// Ordof the sobrification functor ptO and the concrete functor S, which sends a topological spaceto its underlying order.

Proof. By the preceding Theorem and the remarks following the definition of the Scotttopology above, there is an embedding CntDcpoco → ObsReSp as described, and thisembedding restricts to an isomorphism CntDcpoco ∼= Sob∩ObsReSp with inverse a restrictionof S. As discussed above, the embedding Sob → Top has a left adjoint R : Top −→ Sob whichis a restriction of ptO, and since ObsReSp is closed under sobrification by Proposition 5.8.1,R restricts to a left adjoint of the inclusion Sob ∩ ObsReSp → ObsReSp, thus yielding theneeded reflector.

5.9 Cocontinuous lattices among lax algebras. A cocontinuous lattice is a completelattice that is cocontinuous (see 5.2). In 4.4 it was shown that the category of Cntco ofcocontinuous lattices and directed-meet-preserving sup-maps is isomorphic to the categorySetF of Eilenberg–Moore algebras of the filter monad F. By Theorem 5.8.3, when we endow acocontinuous lattice with its Scott topology, we obtain a (sober) observable realization space.


The resulting functor Cntco −→ ObsReSp is faithful, but not full, and given the isomorphismObsReSp ∼= (F, 2)-Cat= of Theorem 5.4.1, we may form an evident commutative diagram

Cntco //

∼=

ObsReSp //

∼=

Top∼=

SetF // (F, 2)-Cat= // (F, 2)-Cat .

The functors in the leftmost square allow us to consider cocontinuous lattices as spaces,F-algebras, and strict lax algebras of F, as follows.

5.9.1 Proposition. Let X be a cocontinuous lattice.

(1) The relation a : FX −→7 X of Scott convergence of filters (see Example 5.3.3) isdetermined by a map

l : FX → X , x 7−→ ∧z ∈ X |↓ z ∈ x = ∧A∈x

∨A ,

in the sense thata = l∗ = (≤X) · l .

(2) By (1), the neighborhood filter map ν : X → FX is related to l via l∗ = a = ν∗ andhence l a ν.

(3) The isomorphism Cntco ∼= SetF of 4.4 associates to X the F-algebra (X, l).

(4) The given functor Cntco → ObsReSp endows the cocontinuous lattice X with its Scotttopology, and the isomorphism ObsReSp ∼= (F, 2)-Cat= of Theorem 5.4.1 associates tothis space the strict lax algebra (X, l∗ : FX −→7 X).

Proof. The verification is straightforward.

We now characterize the image of the given composite Cntco ∼= SetF → (F, 2)-Cat=, thuscharacterizing the cocontinuous lattices among strict lax algebras.

5.9.2 Theorem. Among lax algebras (X, a : FX −→7 X) of F, those associated to cocontin-uous lattices in the above commutative diagram are precisely those that are

(1) strict,

(2) map-determined, in the sense that a = l∗ for some monotone map l : FX → X, and

(3) T0, in the sense that (≤X) is separated,

when we endow X with the underlying order (≤X) = a · eX .Thus, the category of cocontinuous lattices and directed-meet-preserving maps is evidentlyisomorphic to the full subcategory of (F, 2)-Cat= whose objects satisfy (2) and (3).


Proof. By the above, it is clear that if X is a cocontinuous lattice, then the associated laxalgebra satisfies all three conditions. Conversely, consider an arbitrary (X, a) ∈ ob(F, 2)-Catsatisfying (1)–(3). Since (X, a) is strict, the associated topological space is an observablerealization space by Theorem 5.4.1. Therefore, by Proposition 5.2.2, Lemma 5.3.4, andProposition 5.2.5, it suffices to show that X is a complete lattice and that the neighborhood-filter map ν : X −→ FX preserves directed meets.Since a is the convergence relation of the associated space, we have that l∗ = a = ν∗. Thus,

FX >l// X ,

νoo

so since FX is a complete lattice and ν : X → FX is a fully faithful functor, it follows thatX is a complete lattice and that ν preserves all meets.

5.9.3 Remark. Condition 5.9.2(2) means

x a x ⇐⇒ l(x ) ≤ x

for all x ∈ FX, x ∈ X. In other words, l chooses for every x ∈ FX a smallest convergencepoint which, in conjunction with (3), is uniquely determined. Conversely, if every filter xhas a least convergence point l(x ), we obtain a map l : FX −→ X which is automaticallymonotone and and satisfies a = l∗.

Exercises

5.A Topological meets and the Scott topology. For a topological space X endowed with itsunderlying order, the statement that every open subset of X is Scott-open is equivalent tothe statement that every directed meet in X is a topological meet (Proposition 5.2.2).

5.B Comparing and ≺. Let X be an ordered set endowed with the Scott topology. Ifeither ( ) or (≺) is approximating, then ( ) = (≺) (see Lemma 5.2.4).Hint. Note that we have (≺) ≤ ( ) in general. Employ Lemma 5.2.3.

5.C Characterizations of cocontinuity. For an ordered set X endowed with the Scotttopology, the following are equivalent:

(i) X is cocontinuous;

(ii) the relation (≺) is approximating;

(iii) each x ∈ X is a topological meet of x := y ∈ X | x ≺ y.

(See Proposition 5.2.5.)

5.D Closure properties of the class of observable realization spaces. Show that ObsReSp isclosed in Top under finite products, arbitrary coproducts, and retracts.


Notes on Chapter IV

The provision of a satisfactory notion of convergence has been one of the major concerns of topology fromits very beginnings. In his pioneering work [Frechet, 1905, 1906], Frechet not only introduced metric spaces(the designation is due to Hausdorff) but also defined sequential convergence in an abstract way. However,his concept of convergence turned out to be insufficient as it lacks in particular the ability to deal withiterated limits. A more general type of convergence based on directed sets rather than sequences wasintroduced by Moore and his student Smith [Moore, 1915; Moore and Smith, 1922], and in 1937, Birkhoffcharacterized T1-topological spaces in terms of this net convergence (the term net was coined by Kelley[1950]). At a first glance, the notion of net resembles closely the notion of sequence; however, alreadythe definition of subnet turns out to be more sophisticated than the corresponding notion of subsequence.Birkhoff also showed that net convergence can be equivalently substituted by a notion of convergence basedon certain systems of sets that he had introduced earlier [Birkhoff, 1935], called filter bases in today’slanguage. Approximately at the same time, Cartan [1937a,b] introduced the notion of filter convergence,and this idea became central in Bourbaki’s treatment of topology [Bourbaki, 1942]. The notions of filterand ultrafilter and their predecessors per se had emerged earlier in various mathematical contexts; in fact,the ideale Verdichtungsstelle mentioned by Riesz [1908] may be considered a precursor of the notion ofultrafilter, and the Polish School of measure theorists and functional analysts had studied the dual conceptof ideal, with Tarski [1930] in effect proving that every filter is contained in an ultrafilter.Grimeisen [1960, 1961] characterized topological spaces among pretopological spaces as precisely thosewhich have the “iterated limit propery”. Later on, Cook and Fischer [1967] presented four axioms thatcharacterize topological spaces in terms of filter convergence. Barr [1970] extended Manes’ equationalpresentation of compact Hausdorff spaces (see [Manes, 1969]) to characterize topological spaces in terms oftwo simple inequalities for an ultrafilter convergence relation that are amenable to algebraic manipulation.The fact that Barr’s two ultrafilter convergence axioms may be taken verbatim as filter convergence axiomsto already fully characterize topological spaces appeared only in [Seal, 2005]; a slightly weaker version ofthis result had been established earlier in [Pisani, 1999].The original definition of a topological space given by Hausdorff [1914] used neighborhood systems as theprimary structure. (His axioms included the “Hausdorff” separation axiom which, however, is independentof his other axioms.) The fact that this definition is equivalent to that of a monoid in the Kleisli category ofthe filter monad on Set (Proposition 1.1.1) was first observed in [Gahler, 1992]. Therein, Gahler introducedthe notion of a preordered monad that is at the origin of our power-enriched monads 1.2, and their associatedcategory of monadic topologies that, similarly to Kleisli monoids 1.3, are defined as monoids in the Kleislicategory of the given monad. Hohle’s notion of a topological space object (see [Hohle, 2001]) uses the sameidea in a framework different from the one used in this book, but his work targeted many of the categoriesalso of interest here. The presentation of closure spaces and the up-set monad as in Example 1.3.2(4)appeared in [Seal, 2009] which also defined the Kleisli and strata extensions of a power-enriched monadas described in 1.4 and 2.4. The strata extensions of a general lax monad first appeared in [Clementinoand Hofmann, 2004b]. Initial extensions of a functor and of a monad 2.1 were introduced in [Schubertand Seal, 2008] and [Colebunders, Lowen, and Rosiers, 2011], respectively. The Kleisli towers in 2.5 are aparticular case of the tower extension construction over a topological functor from [Zhang, 2000], whereZhang generalized the presentation of approach spaces as towers [Lowen, 1989].The construction of the discrete presheaf monad (as a power-enriched monad) from an arbitrary monadwith an associative lax extension to V-Rel as given in Section 3—which enables the surprising result that(T,V)-categories may always be considered as relational algebras (see Corollary 3.2.3)—was motivated bythe presentation of approach spaces as relational algebras in [Lowen and Vroegrijk, 2008]. The alternativedescription of the discrete presheaf monad in 3.3 is similar to that one of the monad in [Colebunders,Lowen, and Rosiers, 2011].


The identification of Eilenberg–Moore algebras for the filter monad on Set or Top0 as continuous lattices 4.4is due to Day [1975]. In 1981, Wyler noted that Cnt is also monadic over Sup and CompHaus. The fact thatcompletely distributive lattices are the Eilenberg–Moore algebras for the up-set monad on Set (Exercise 4.E)was noted in Pedicchio and Wood [1999]; the corresponding observation for frames (Exercise 4.F) overSet can be found in Johnstone [1982]. The description of PV -algebras as V-actions in Sup (Exercise 4.G)comes from [Pedicchio and Tholen, 1989] (following the more technical description of [Machner, 1985]);Pedicchio and Tholen [1989] also remarked that SetV is monadic over Ord and V-Cat. Further developmentsin the context of categories enriched in a quantaloid can be found in [Stubbe, 2007]. The description of thedistributive law of the list monad over the powerset monad and its category of Eilenberg–Moore algebras(Exercise 4.H) appeared in [Manes and Mulry, 2007]. Finally, the distributive law of Exercise 4.A can befound in [Schubert, 2006] and uses the distributive law of Dn over Up described in Marmolejo et al. [2002]that, in turn, finds its source in a distributive law for the frame monad over Ord mentioned by Linton ata meeting in 1984. The proof that the RegMono-injective T0-spaces are the continuous lattices endowedwith their Scott topology (Example 4.6.6(2)) goes back to Scott [1972], but the treatment we give of thesubject in Section 4.6—and in particular the proof of Theorem 4.6.3—is more directly inspired by Escardo[1998]. The power-enriched treatment was developed in [Seal, 2010] and [Seal, 2011].Section 5 is based largely on [Lucyshyn-Wright, 2009, 2011], in which Theorems 5.4.1 and 5.9.2 were firstproved. The proof of Lemma 5.2.3 is derived from the proof (for dcpos) that was contributed by Escardoto [Gierz et al., 2003]. The given relation of Scott convergence (5.3.3) was studied (with respect to the dualorder) by Erne [1981] (under the name of s3-convergence), considered there among two other generalizationsto arbitrary preorders of Scott’s original notion of convergence in continuous lattices [Scott, 1972].The class of spaces that we call the observable realization spaces appeared in papers by Banaschewski[1977]; Erne [1980, 1981, 1991, 1999]; Erne and Wilke [1983]; Ershov [1993]; Hoffmann [1979, 1981]; Lawson[1979, 1997], but were often identified via quite different and less elementary characterizations. Thesespaces were called C-spaces by Erne. The basic role of these spaces in domain theory was noted by Ershov[1993], who had earlier developed domain theory in topological terms. The domain-theoretic relevance ofC-spaces was also underscored in [Lucyshyn-Wright, 2009, 2011]. Lemma 5.3.4 is part of Exercise II.34 of[Gierz et al., 2003], contributed by Escardo and Heckmann.The axioms for an observable specialization system given in Definition 5.5.1 and the main result ofTheorem 5.5.2 were given by Hoffmann [1981] (building upon work in [Hoffmann, 1979]), whereas theaxioms (1)–(3) defining an abstract basis were given by Smyth earlier [1977/78] as a way of generatingcontinuous dcpos. Each abstract basis gives rise to an associated continuous dcpo, the set of round idealsof the basis (see for example [Abramsky and Jung, 1994]). Herein, following [Lucyshyn-Wright, 2009], weregard those abstract bases satisfying (4) as domains in their own right. These domains—the observablerealization spaces—need not be sober, and the space of round ideals (or, rather, round filters) associated tosuch a space is in fact the sobrification of that space and is a continuous dcpo under the Scott topology(see 5.8). With regard to ordered abstract bases (see Definition 5.6.1), note that auxiliary relations withthe given properties were studied by Hoffmann [1981].

Suggestions for further reading: [Adamek et al., 2003; Gutierres and Hofmann, 2007, 2012; Hofmann andTholen, 2006; Johnstone, 1979; Kowalsky, 1954a,b; McShane, 1952; Tukey, 1940; Wyler, 1995].

V Lax Algebras as SpacesMaria Manuel Clementino, Eva Colebunders, Walter Tholen

Looking at (T,V)-categories as geometric objects, in this chapter we explore fundamentaltopological properties like compactness and Hausdorff separation, along with low-separationproperties, regularity, normality, and extremal disconnectedness. We do so first taking amore traditional object-oriented view, before highlighting the central role of proper andopen maps in (T,V)-Cat. Each of these classes give a “topology” on (T,V)-Cat and, alongwith other classes, allows us to investigate relativized topological properties in an axiomaticcategorical setting. We also explore the categorical notion of connected object in (T, V )-Cat.

1 Hausdorff separation and compactness

Topological spaces have been described as lax algebras via ultrafilter convergence in III.2.2,that is, as sets X equipped with a relation a : βX −→7 X satisfying the two requirements ofa (, 2)-category. The relation a will actually be a map if, when we write x −→ x insteadof x a x,

∀x, y ∈ X, z ∈ TX (z −→ x & z −→ y =⇒ x = y), that is, a · a ≤ 1X ,

and∀x ∈ TX ∃x ∈ X (x −→ x), that is, 1TX ≤ a · a.

The first property identifies X as a Hausdorff space, and the second one as a compact space(see III.2.3). This observation leads us to consider the following notions in the generalcontext of a monad T = (T,m, e) with a lax extension T to V-Rel, where V = (V ,⊗, k) is aquantale.

1.1 Basic definitions and properties. As we are considering topologically-inspiredproperties of (T,V)-categories, we will most often refer to the objects of (T,V)-Cat as(T,V)-spaces, and to its morphisms as (T,V)-continuous maps. Objects and morphisms ofV-Cat are simply called V-spaces and V-continuous maps, respectively.

1.1.1 Definition. A (T,V)-space (X, a) is Hausdorff if

a · a ≤ 1X ,

369

370 CHAPTER V. LAX ALGEBRAS AS SPACES

and it is compact if1TX ≤ a · a .

The resulting full subcategories of (T,V)-Cat are denoted by

(T,V)-CatHaus and (T,V)-CatComp ,

respectively, and their intersection by (T,V)-CatCompHaus.

1.1.2 Proposition. Let (X, a) be a (T,V)-space.

(1) (X, a) is Hausdorff if and only if for all x, y ∈ X and z ∈ TX,

⊥ < a(z, x)⊗ a(z, y) =⇒ x = y and a(z, x)⊗ a(z, x) ≤ k ,

where the latter condition holds trivially when V is integral.

(2) (X, a) is compact if and only if for all x ∈ TX,

k ≤ ∨z∈X a(x , z)⊗ a(x , z) . (1.1.i)

Proof. The assertions follow immediately from

(a · a)(x, y) = ∨z∈TX a(z, x)⊗ a(z, y) , and

(a · a)(x , y ) = ∨z∈X a(x , z)⊗ a(y , z)

for all x, y ∈ X, x , y ∈ TX.

1.1.3 Remark. If the quantale V is superior , that is, if

k ≤ ∨i∈I ui ⊗ ui ⇐⇒ k ≤ ∨i∈I ui (1.1.ii)

for all families ui ∈ V , i ∈ I, then the compactness condition (1.1.i) simplifies to

k ≤ ∨z∈X a(x , z) , (1.1.iii)

for all x ∈ TX. Note that for V integral (so that k = >) one has u⊗u ≤ u⊗k = u, so thatthe implication =⇒ of (1.1.ii) holds trivially. If V is a frame with ⊗ = ∧, or when V = P+,the implication ⇐= of (1.1.ii) also holds, so that in this case the simplified compactnesscondition (1.1.iii) applies.An example of a non-superior quantale is given below in Remark 1.4.3(1).

1. HAUSDORFF SEPARATION AND COMPACTNESS 371

1.1.4 Examples.

(1) Let T = I be the identity monad (identically extended to V-Rel). Then, for x 6= y in aHausdorff (T,V)-space (X, a), we must have a(x, y) = ⊥. Consequently, if k = > inV, necessarily a = 1X , that is, (X, a) is discrete. However, quite trivially, (X, a) isalways compact. Briefly, if V is integral, then V-CatComp = V-Cat, while V-CatHaus isthe full coreflective subcategory of discrete V-categories in V-Cat.

(2) In Top ∼= (, 2)-Cat (see III.2.2), the Hausdorff separation property and the compact-ness property of 1.1.1 are equivalent to the usual notions that are expressed in termsof open sets, provided that the Axiom of Choice is granted: a topological space Xis Hausdorff if and only if for any distinct points x, y one finds disjoint open subsetsU 3 x, V 3 y, and X is compact if and only if any open cover of X (that is, any setU of open subsets with ⋃U = X) contains a finite subcover (see Propositions III.2.3.1and III.2.3.2). Briefly,

(, 2)-CatHaus ∼= Haus , (, 2)-CatComp ∼= Comp . c©

(3) In Top ∼= (F, 2)-Cat (see Corollary IV.1.5.4), since the filter PX on X converges toevery point in X, a Hausdorff (F, 2)-space can have at most one point: (F, 2)-CatHaus isequivalent to ∅, 1. Considering the filter X, one sees that a compact (F, 2)-spaceX must contain a point whose only neighborhood is X. Since every other filter on Xconverges to that point as well, this property characterizes compactness in (F, 2)-Cat.Spaces with this property are called supercompact since they may equivalently bedescribed by the property that every open cover of X contains at most one opensubset of X (see Exercise 1.A).

(4) If we replace the filter monad by the submonad Fp = (Fp,m, e), where FpX =FX \ PX is the set of proper filters on X, equipped with its Kleisli extensionFp, then the Hausdorff property assumes its usual meaning: (Fp, 2)-CatHaus ∼= Haus.However, (Fp, 2)-CatComp = (F, 2)-CatComp \ ∅. Consequently, compact Hausdorffobjects in (Fp, 2)-Cat can have at most one point (see Exercise 1.A).

(5) An object (X, a) in (,P+)-Cat ∼= App (see III.2.4.5) is Hausdorff precisely whena(z, x) < ∞ & a(z, y) < ∞ implies x = y, for all x, y ∈ X, z ∈ βX. Obviously,(X, a) is Hausdorff if and only if its pseudotopological modification (X, oa) of (X, a)is Hausdorff. (Here o : [0,∞]op −→ 2 = ⊥,> is the “optimist’s map” with (o(v) => ⇐⇒ v <∞) which induces a functor

(,P+)-Cat ∼= App −→ (, 2)-Gph ∼= PsTop ,

see Exercise II.1.I and Examples III.4.1.3; a pseudotopological space (X,−→) is calledHausdorff if z −→ x & z −→ y implies x = y, for all x, y ∈ X, x ∈ βX.) Theapproach space (X, a) is compact if and only if

infx∈X a(x , x) = 0


for all x ∈ βX, a property called 0-compact in approach theory (see [Lowen, 1988]).

1.1.5 Proposition.

(1) (T,V)-CatHaus is closed under non-empty mono-sources in (T,V)-Cat; it is closed underall mono-sources and, therefore, it is strongly epireflective in (T,V)-Cat if V is integralor T1 ∼= 1.

(2) (T,V)-CatComp is closed under those sinks gi : (Xi, ai) −→ (Y, b) in (T,V)-Cat for which(Tgi)i∈I is epic in Set.

Proof.

(1) For a mono-source (fi : (X, a) −→ (Yi, bi))i∈I with I 6= ∅ in (T,V)-Cat, one has1X = ∧

i∈I fi · fi (see Proposition III.1.2.2). Hence, if all (Yi, bi) are Hausdorff, froma · a ≤ (f i · bi · Tfi) · ((Tfi) · bi · fi) ≤ f i · bi · bi · fi ≤ f i · fi

one obtains a · a ≤ 1X , as desired. If I = ∅ then |X| ≤ 1 and the Hausdorffcriterion 1.1.2(1) is trivially satisfied when k = > or T1 ∼= 1. The additional claimfollows with Proposition II.5.10.1.

(2) Similarly, the hypothesis on the sink (gi)i∈I guarantees 1TY = ∨i∈I Tgi · (Tgi).

Consequently, with all (Xi, ai) compact, one obtainsb · b ≥ (Tgi · ai · gi ) · (gi · ai · (Tgi)) ≥ Tgi · ai · ai · (Tgi) ≥ Tgi · (Tgi)

and, hence, b · b ≥ 1TY .

1.1.6 Corollary.

(1)c© For a surjective (T,V)-continuous map g : (X, a) −→ (Y, b) between (T,V)-spaces, if(X, a) is compact, then (Y, b) is compact.

(2) If the Set-functor T preserves coproducts, (T,V)-CatComp is closed under epi-sinks in(T,V)-Cat. The same statement holds for finite coproducts with closure under finiteepi-sinks.

Proof.

(1) g is a split epimorphism in Set and is preserved by T .c©

(2) Decompose (Tgi)i∈I as

TXi//∐i TXi

d // T (∐iXi)Tg// TY .

c© Here the induced morphism g is epic and therefore preserved by T , and the canonicalcomparison morphism d is bijective by hypothesis. It actually suffices to know thatd is surjective to be able to conclude that (Tgi)i∈I is an epimorphism and to applyProposition 1.1.5.


1.1.7 Example. Compact spaces are closed under finite epi-sinks in Top ∼= (, 2)-Cat (sinceβ : Set −→ Set preserves finite coproducts), but not under countable epi-sinks (consider(n : 1 −→ N)n∈N with N discrete). Likewise, 0-compact spaces are closed under finiteepi-sinks in App ∼= (,P+)-Cat, but not under countable epi-sinks.

Finally, Hausdorffness and compactness are preserved by algebraic functors (see III.3.4).

1.1.8 Proposition. Let α : (S, S) −→ (T, T ) be a morphism of lax extensions. Then theassociated algebraic functor Aα : (T,V)-Cat −→ (S,V)-Cat preserves Hausdorffness andcompactness.

Proof. Preservation of Hausdorffness follows from (a·αX)·(a·αX) = a·αX ·αX ·a ≤ a·a ≤1X , and preservation of compactness from (a·αX)·(a·αX) = αX ·a·a·αX ≥ αX ·αX ≥ 1TX .

1.2 Tychonoff’s Theorem, Cech–Stone compactification. For T and V as in 1.1, a(T,V)-space (X, a) is by definition compact Hausdorff when a : TX −→7 X is a map in theordered category V-Rel. By Proposition III.1.2.1, a is then actually a Set-map TX −→ X,provided that V is integral and lean. In that case, the V-relational inequalities

a · Ta ≤ a · T a ≤ a ·mX and 1X ≤ a · eX

between Set-maps must actually be equalities; likewise the (T,V)-continuity conditionf · a ≤ b · Tf for f : (X, a) −→ (Y, b) must actually be an equality if (X, a), (Y, b) are bothcompact Hausdorff. This proves the following result.

1.2.1 Proposition. Let V be integral and lean. Then (T,V)-CatCompHaus is precisely thefull subcategory of SetT containing those T-algebras (X, a) with a · T a = a ·mX . In particular,if T is flat, then

(T,V)-CatCompHaus = SetT .

1.2.2 Examples.

(1) The category of 0-compact approach spaces with Hausdorff pseudotopological modi-fication (see Example 1.1.4(5)) is isomorphic to the category of compact Hausdorffspaces:

(,P+)-CatCompHaus ∼= (, 2)-CatCompHaus ∼= Set ∼= CompHaus .

(2) Example 1.1.4.(3) shows that flatness of T is essential for (T,V)-CatCompHaus = SetT:while SetF is the category of cocontinuous lattices (see IV.4.4), (F, 2)-CatCompHauscontains just singletons.

Proposition 1.2.1 entails a Tychonoff Theorem for compact Hausdorff objects in (T,V)-Cat,with the needed hypotheses on T and V granted, as follows. With (Xi, ai) in SetT for all i ∈ I,


the product X = ∏i∈I Xi in SetT carries the structure a : TX −→ X with πi · a = ai · Tπi

(where πi are the product projections for all i ∈ I), and using Proposition III.1.2.2 one obtainsthat a coincides with the product structure formed in (T,V)-Cat (see Proposition III.3.1.1):

a = (∧i∈I πi · πi) · a = ∧i∈I π

i · πi · a = ∧

i∈I πi · ai · Tπi .

With the help of the Adjoint Functor Theorem II.2.12.1 we can now go one step furtherand guarantee the existence of a Cech–Stone compactification for (T,V)-spaces.

1.2.3 Theorem. Let V be integral and lean, and let T be flat. Then (T,V)-CatCompHaus isreflective in (T,V)-CatHaus which, in turn, is strongly epireflective in (T,V)-Cat.

Proof. The embedding SetT → (T,V)-Cat preserves not only small products, but alsoequalizers. Indeed, the T-algebra structure of the equalizer j : E → X of f, g : (X, a) −→(Y, b) in SetT is the map a0 : TE −→ E with j · a0 = a · Tj; hence, a0 = j · a · Tj, whichis the structure of the equalizer formed in (T,V)-Cat. Consequently, SetT is closed undersmall limits in (T,V)-Cat and therefore in (T,V)-CatHaus (which is strongly epireflective in(T,V)-Cat, see Proposition 1.1.5).In order to build a solution set in SetT = (T,V)-CatCompHaus for (X, a) in (T,V)-CatHaus,consider any (T,V)-continuous f : (X, a) −→ (Y, b) with (Y, b) in SetT and form the leastT-subalgebra 〈M〉 of (Y, b) containing M = f(X), which may be constructed as theimage h(TM) of the T-homomorphism h : (TM,mM) −→ (Y, b) with h · eM = (M → Y ).Hence, f factors as (X −→ 〈M〉 → Y ), where the cardinality of 〈M〉 cannot exceed thecardinality of TX. Hence a solution set for (X, a) can be given by a representative systemc©

of non-isomorphic T-algebras (Y, b) whose cardinalities do not exceed that of TX.

Theorem 1.2.3 relies on the algebraic realization of (T,V)-CatCompHaus as SetT. The theme ofHausdorff-separation and compactness will be taken up again in Sections 3 and 4, throughan algebraic approach to proper maps.

1.3 Compactness for Kleisli-extended monads. The example Top ∼= (, 2)-Cat ∼=(F, 2)-Cat shows that the notion of compactness may change dramatically when changingthe parameters T or V , see Examples 1.1.4(2) and (3). Here, we wish to explore this changewhen (T,V)-Cat is presented relationally as ((T,V), 2)-Cat, as in Corollary IV.3.2.3.With that goal in mind, let us first look at relational T-algebras in general. Independentlyof the lax extension of T to Rel, for (X, a) in (T, 2)-Cat one has (writing x −→ x instead ofa(x , x) = >):

(X,−→) is Hausdorff ⇐⇒ ∀x, y ∈ X, z ∈ TX (z −→ x & z −→ y =⇒ x = y) ,(X,−→) is compact ⇐⇒ ∀x ∈ TX ∃z ∈ X (x −→ z) .

In the case where T = (T,m, e) is power-enriched via τ : P −→ T, so that τ : (P, P ) −→ (T, T )is a morphism of the respective Kleisli extensions, the algebraic functor

Aτ : (T, 2)-Cat // (P, 2)-Cat ∼= Ord


simply equips (X,−→) with the induced order considered in III.3.3:

x ≤ y ⇐⇒ τ(x) −→ y ⇐⇒ eX(x) −→ y .

1.3.1 Proposition. For a monad T power-enriched by τ : P −→ T and (X,−→) in(T, 2, T)-Cat, one has:

(1) (X,−→) is Hausdorff if and only if |X| ≤ 1.

(2) If (X,−→) is compact, X has a largest element, with the converse statement holdingwhen τ(X) is the largest element in TX.

Proof.

(1) The complete lattice TX has a least element p. By Proposition III.3.3.6 and Re-mark IV.1.4.4 one has

(p ≤ x & x −→ x) =⇒ (p −→ x)

for all x ∈ X, x ∈ TX. Since eX(x) −→ x, one has p −→ x for all x ∈ X, and (1)follows.

(2) For (X,−→) compact there is x0 with τ(X) −→ x0. Since τ(X) = ∨x∈X eX(x)

(see Exercise IV.1.B) one obtains eX(x) −→ x0 and, hence, x ≤ x0 for all x ∈ X.Conversely, assuming x0 a largest element in X, with Exercise IV.1.H one concludesτ(X) −→ x0 which, with τ(X) the largest element in TX, gives x −→ x0 for allx ∈ TX.

For T = F the filter monad, the least element p of FX is the filter PX, while the largestelement τ(X) is the filter X. Compactness in (F, 2)-Cat (that is, supercompactness,see 1.1.4(3)) may seem like a rare property at first sight. However, the following Propositionshows that in general there is a rich supply of such spaces.Recall from IV.4.1 and Theorem IV.1.5.3 that for a power-enriched monad T one has thefunctors

SetT // T-Mon // (T, 2)-Cat(X,α) // (X,αa) = (X, ν) // (X,−→) ,

where αa is right adjoint to the sup-map α : TX −→ X, and

x −→ x ⇐⇒ x ≤ ν(x) ⇐⇒ x ≤ αa(x) ⇐⇒ α(x ) ≤ x (1.3.i)

for all x ∈ X, x ∈ TX. In particular, setting x = α(x ) one obtains the following result.

1.3.2 Proposition. For a power-enriched monad T, every T-algebra is compact whenconsidered as a (T, 2)-space via (1.3.i). In fact, for (X,α) ∈ SetT and every x ∈ TX thereis a least point x ∈ X with x −→ x.


For T = F one obtains with IV.4.4:1.3.3 Corollary. Every continuous lattice endowed with its Scott topology is supercompact.

Let us now consider the discrete presheaf monad

= (T,V , T)

induced by a monad T and an associative lax extension T to V-Rel and apply Proposition 1.3.2to with its Kleisli extension (see Corollary IV.3.2.3):

Set // -Mon // (, 2)-Cat ∼= (T,V)-Cat(X,α) // (X,αa) // (X,−→) .

1.3.4 Corollary. For a monad T with an associative lax extension T to V-Rel, every(T,V)-algebra is compact as an object of (, 2)-Cat.

1.3.5 Example. Consider T = with its Barr extension to V = P+ = ([0,∞]op,+, 0),which carries a (T,V)-algebra structure

α : Π[0,∞] = (T,V)-URel(1, V ) −→ (T,V)-URel(1, 1) ∼= [0,∞]ϕ 7−→ ϕ ι = infx ∈[0,∞] ϕ(x ) + ξ(x ) ,

where ι : 1 −→7 V is given by the identity map on [0,∞] and ξ(x ) = supA∈x infu∈A u for allx ∈ β[0,∞]; see IV.3.3. (Note that the convergence map ξ : β[0,∞] −→ [0,∞] provides[0,∞] with the standard topology of [0,∞].) By Corollary 1.3.4, the space ([0,∞],−→)with

ϕ −→ v ⇐⇒ infx ∈β[0,∞](ϕ(x ) + ξ(x )) ≥ v

becomes compact in (, 2)-Cat ∼= App.

Finally, let us examine compactness in ((T,V), 2)-Cat beyond spaces that arise from-algebras. Recall from IV.3.2 the isomorphisms

(T,V)-Cat∼= // -Mon

∼= // (, 2)-Cat(X, a) // (X, a[) // (X,−→) ,

where a[ : X −→ ΠX is given by a[(x) = x · a = a(−, x), and

ϕ −→ x ⇐⇒ ϕ ≤ a(−, x)

for all ϕ ∈ ΠX.1.3.6 Proposition. For a monad T with a flat associative lax extension to V-Rel and a(T,V)-space (X, a), each of the following statements implies the next:

(i) (X,−→) is compact in ((T,V), 2)-Cat;


(ii) ∀x ∈ TX ∃x ∈ X (a(x , x) ≥ k);

(iii) (X, a) is compact in (T,V)-Cat.

Proof. For (i) =⇒ (ii), we observe that compactness of (X,−→) means

∀ϕ ∈ ΠX ∃x ∈ X (ϕ −→ x) .

Hence, given x ∈ TX we may exploit this property for ϕ = YX(x ), with Y : T −→ as inProposition IV.3.2.5. With ϕ −→ x ∈ X we then have

k = ϕ(x ) ≤ a(x , x) ,

as desired. (ii) =⇒ (iii) is immediate.

1.3.7 Remark. For T = I the identity monad identically extended to V-Rel, (I,V) = PVis the V-powerset monad and (, 2)-Cat ∼= V-Cat; see Corollary IV.3.2.3.For V = 2, so that V-Cat ∼= Ord and ϕ ∈ P2X corresponds to A ⊆ X, one has

A −→ x ⇐⇒ A ⊆ ↓x ,

so that compactness of (X,≤) in (, 2)-Cat means existence of a largest element in X(as already confirmed in Proposition 1.3.1(2)). When V = P+, so that V-Cat ∼= Met, forϕ ∈ ΠX one has

ϕ −→ x ⇐⇒ ϕ ≥ a(−, x)

for a metric space (X, a) considered as a (, 2)-space. With ϕ constantly 0 one sees:

(X,−→) compact ⇐⇒ ∃x ∈ X ∀y ∈ X (a(y, x) = 0)⇐⇒ (X,≤) compact,

where ≤ is the underlying order of (X, a), making (X,≤) an object of ((I, 2), 2)-Cat.

This invariance property of compactness is now shown to hold in full generality, not justfor T = I.As in Theorem IV.3.4.4, let ϕ : (V,⊗, k) −→ (W,⊗, l) be a homomorphism of integralquantales with a right adjoint ψ (in Ord); furthermore, for a monad T we assume ϕ to bestrictly compatible with the associative lax extensions T, T to V-Rel, W-Rel, respectively, sothat

T (ϕr) = ϕ(T r)

for all V-relations r. With V = (T,V), W = (T,W) there is then a morphism

ϕT = (T, ϕ) : V −→ W


of power-enriched monads whose algebraic functor commutes with the change-of-basefunctor induced by ψ:

(T,W)-Cat //

Bψ

(W , 2)-CatAϕT

(T,V)-Cat // (V , 2)-Cat

(X, a) //

_

(X,−→)_

(X,ψa) // (X, ) ,

where

ρ −→ x ⇐⇒ ρ ≤ a(−, x) ,θ x ⇐⇒ θ ≤ ψa(−, x) ⇐⇒ ϕθ ≤ a(−, x) ,

for all ρ ∈ ΠWX, θ ∈ ΠVX, x ∈ X.

1.3.8 Theorem. For (X, a) ∈ (T,W)-Cat,

(X,−→) compact in (W , 2)-Cat ⇐⇒ (X, ) compact in (V , 2)-Cat.

Proof. “ =⇒ ” follows from Proposition 1.1.8. For “ ⇐= ”, consider θ : TX −→7 1 withθ(x ) = > for all x ∈ TX. Since

θ · T1X ≥ θ and e1 · T θ ·mX ≥ e1 · T θ · eTX ≥ θ ,

θ is unitary when k = >. Compactness of (X, ) gives x0 ∈ X with θ −→ x0, so thatθ ≤ ψa(−, x0) implies ϕθ ≤ a(−, x0), with ϕθ the top element in ΠWX. Consequently,ρ −→ x0 for all ρ ∈ ΠWX.

1.3.9 Corollary. An approach space is compact as an object of ((,P+), 2)-Cat if andonly if its topological coreflection is supercompact.

Proof. Apply Theorem 1.3.8 with ϕ = ι : 2 −→ P+ and ψ = p the “pessimist’s map” (seeExercise II.1.I).

1.4 Examples involving monoids. For a monoid (H,µ, η) we consider the monad

H = (H × (−),m, e)

on Set, with mX = µ× 1X : H ×H ×X −→ H ×X and eX = 〈η, 1X〉 : X −→ H ×X (seeExercise II.3.B). The Barr extension H : Rel −→ Rel is a lax extension of the monad H, andit is given explicitly by

(α, x) Hr (β, y) ⇐⇒ α = β & x r y ,


for r : X −→7 Y , x ∈ X, y ∈ Y , and α, β ∈ H. For a relation a : H×X −→7 X, when we writex

α−→ y instead of (α, x) a y (that is, “x is related to y with weight α”), an (H, 2)-category(X, a) is characterized by the conditions

xη−→ x and (x α−→ y & y

β−→ z =⇒ xβ·α−→ z) ,

where β · α = µ(β, α) in H; a morphism f : (X, a) −→ (Y, b) must satisfy

xα−→ y =⇒ f(x) α−→ f(y)

for all x, y ∈ X, α ∈ H. For an (H, 2)-category (X, a), one has:

(X, a) is compact ⇐⇒ ∀x ∈ X,α ∈ H ∃y ∈ Y (x α−→ y) ,(X, a) is Hausdorff ⇐⇒ ∀x, y, z ∈ X,α ∈ H (x α−→ y & x

α−→ z =⇒ y = z) ,

and the conjunction of both properties makes a : H × X −→ X an action of H on X:x

α−→ y means y = α · x when we write the action multiplicatively. These observationsillustrate the assertions of Proposition 1.2.1 and Theorem 1.2.3.

1.4.1 Corollary. The category of compact Hausdorff (H, 2)-spaces and (H, 2)-continuousmaps is isomorphic to the category of H-actions and equivariant maps:

(H, 2)-CatCompHaus ∼= SetH

It is reflective in (H, 2)-CatHaus which, in turn, is strongly epireflective in (H, 2)-Cat.

Our arrow notation for the structure of an (H, 2)-category (X, a) emphasizes that X isactually the object set of a small category, denoted again by X, with hom-sets

X(x, y) = (x, α, y) | α ∈ H, x α−→ y in (X, a) ;

moreover, this category comes with a faithful functor

p : X −→ H , (x, α, y) 7−→ α ,

with H considered as a one-object category. (Hence, p identifies the objects while leavingthe morphisms intact.) Considering now more generally small categories X which comeequipped with an H-valued norm, that is, a functor n : X −→ H, we see that there is a fullembedding

E : (H, 2)-Cat → Cat/H , (X, a) 7−→ (X, p) .

1.4.2 Proposition. The functor E is reflective and identifies (H, 2)-categories as thosesmall categories over H whose norm is faithful.


Proof. The reflection of an object (X, n) in Cat/H into (H, 2)-Cat may be formed byX := ob X with (H, 2)-structure

xα−→ y ⇐⇒ ∃ϕ ∈ X(x, y) (n(ϕ) = α)

which makes n : (X, n) −→ (X, p) a morphism in Cat/H and, in fact, the reflection morphism,as one easily verifies. Furthermore, n as a morphism in Cat/H becomes an isomorphism ifand only if the functor n : X −→ H is faithful. Consequently, (H, 2)-Cat is equivalent to thefull subcategory of Cat/H whose objects have a faithful norm.

1.4.3 Remarks.

(1) The category (H, 2)-Cat may also be presented as V-Cat with V the powerset of H,ordered by inclusion, and equipped with the tensor product ⊗ defined by

A⊗B = α · β | α ∈ A, β ∈ B ,

that preserves unions (see Exercise II.1.M). Then η is the tensor unit, but not thetop element of V , hence V is not integral unless H = 1. Moreover, if, for instance, His a non-trivial group, V is not superior: let A = α, α−1, with α 6= η; then η 6⊆ A,although η ⊆ A⊗ A.It is easy to check that if, to each V-space (X, a), we assign the (H, 2)-space (X, a),with a((α, x), y) = > if and only if α ∈ a(x, y), every V-functor f : (X, a) −→ (Y, b)is an (H, 2)-functor f : (X, a) −→ (Y, b), and in fact this correspondence defines anisomorphism V-Cat −→ (H, 2)-Cat (see also Example III.3.5.2(2)). We note, however,that Hausdorff and compact V-spaces do not coincide with Hausdorff and compact(H, 2)-spaces.

(2) In the trivial case H = 1, we have H = I, so that (H, 2)-Cat reproduces 2-Cat ∼= Ord,also identified as the full subcategory of Cat given by small categories X for whichX −→ 1 is faithful.

We now turn to the list monad L = (L,m, e) induced by the right adjoint functor Mon −→ Set(see Example II.3.1.1(2)). The Barr extension L : Rel −→ Rel of the list monad is given by

(x1, . . . , xn) Lr (y1, . . . , ym) ⇐⇒ n = m & x1 r y1 & . . . & xn r yn

for all (x1, . . . , xn) ∈ LX, (y1, . . . , ym) ∈ LY , and relations r : X −→7 Y ; it is obviouslyan associative flat lax extension of L (by Corollary III.1.12.2 and Exercise III.1.Q). An(L, 2)-category is a multi-ordered set, that is, a set X with a relation a : LX −→7 X that,when we write

(x1, . . . , xn) ` y for (x1, . . . , xn) a y ,must satisfy (x) ` x and the transitivity condition

(x11, . . . , x

1n1) ` y1, . . . , (xm1 , . . . , xmnm) ` ym & (y1, . . . , ym) ` z =⇒ (x1

1, . . . , xmnm) ` z .


For an (L, 2)-continuous map f : (X, a) −→ (Y, b) one must have the monotonicity condition

(x1, . . . , xn) ` y =⇒ (f(x1), . . . , f(xn)) ` f(y) .

An (L, 2)-space (X, a) is compact precisely when for all integer n ≥ 0,

∀x1, . . . , xn ∈ X ∃z ∈ X ((x1, . . . , xn) ` z) ,

and (X, a) is Hausdorff if and only if for all n ≥ 0,

∀x1, . . . , xn, y, z ∈ X ((x1, . . . , xn) ` y & (x1, . . . , xn) ` z =⇒ y = z) .

In particular, compactness forces X 6= ∅ (from the case n = 0). The conjunction of bothproperties makes (x1, . . . , xn) ` y the n-ary extension of a binary monoid operation on X.1.4.4 Corollary. The category of compact Hausdorff (L, 2)-spaces and (L, 2)-continuousmaps is isomorphic to the category of monoids and their homomorphisms:

(L, 2)-CatCompHaus ∼= Mon .

It is reflective in (L, 2)-CatHaus which, in turn, is strongly epireflective in (L, 2)-Cat.

Proof. The result is a direct consequence of Proposition 1.2.1 and Theorem 1.2.3.

Exercises

1.A Supercompact topological spaces. Show that the compact objects in (Fp, 2)-Cat areprecisely the supercompact topological spaces (see 1.1.4(4)). Furthermore,

(Fp, 2)-CatHaus ∼= Haus and (F, 2)-CatCompHaus ∼= X ∈ Top | |X| ≤ 1 .

1.B Algebraic functors induced by monoid homomorphisms.

(1) For a homomorphism α : H −→ K of monoids one obtains a morphism α : H −→ Kof monads on Set which, in turn, induces an algebraic functor Aα : (K, 2)-Cat −→(H, 2)-Cat (see Exercise II.3.B and 1.4).

(2) Considering that the monad associated to the trivial monoid 1 is the identity monad Ion Set, show that for every monoid H there are algebraic functors A : (H, 2)-Cat −→ Ordand P : Ord −→ (H, 2)-Cat with AP = 1Ord.

1.C Induced algebraic functors that are not adjoint. Let H = (0, 1,+, 0) and K =(0, 1, ·, 1) be the monoids that make up the ring Z/2Z.

(1) Show that (H, 2)-Cat is isomorphic to the category of ordered sets (X,≤) that comewith an additional relation ≺ satisfying

(u ≤ x ≺ y ≤ z =⇒ u ≺ z) and (x ≺ y ≺ z =⇒ x ≤ z) .


(2) Show that (K, 2)-Cat may be described as the category of ordered sets (X,≤) thatcome with an additional relation ≺ satisfying

u ≤ x ≺ y ≺ z ≤ w =⇒ u ≺ w .

(3) Identify the compact and Hausdorff objects in (H, 2)-Cat and (K, 2)-Cat.

(4) Describe the functors A and P of 1.B for both H and K and show that in both casesA and P fail to be adjoint to each other.

1.D Superior but not integral. The powerset of the multiplicative monoid 0, 1 yields a4-element commutative and superior, but non-integral, quantale.

1.E Characterizing superior completely distributive quantales. If the ⊗-neutral element k ofV satisfies k = ∨

uk u, then the following conditions are equivalent:

(i) ∀ui ∈ V (i ∈ I) (k ≤ ∨i∈I ui =⇒ k ≤ ∨i∈I ui ⊗ ui);(ii) k ≤ ∨uk u⊗ u.

Conclude that an integral completely distributive quantale is superior if and only if (ii)holds.

2. LOW SEPARATION, REGULARITY AND NORMALITY 383

2 Low separation, regularity and normality

We continue to work with a monad T = (T,m, e) on Set, laxly extended by T to V-Rel, fora quantale V = (V ,⊗, k).

2.1 Order separation. Every (T,V)-category (X, a) comes with an underlying order,defined by

x ≤ y ⇐⇒ k ≤ a(eX(x), y)for all x, y ∈ X (see Proposition III.3.3.1). We recall that (X, a) is separated if its underlyingorder is separated; that is, if for all x, y ∈ X

x ≤ y & y ≤ x =⇒ x = y .

Hence, for T = I and V = 2 or P+, this notion returns the terminology used for orderedsets and metric spaces (see II.1.3 and Examples III.1.3.1). For topological spaces, whetherconsidered as objects of (, 2)-Cat or (F, 2)-Cat, separation means T0-separation:

x −→ y & y −→ x =⇒ x = y ;

equivalently, the map ν : X −→ FX which assigns to every point its neighborhood filter isinjective. An approach space (X, a), as an object of (,P+)-Cat, is separated if and only iffor all x, y ∈ X

a(x, y) = 0 = a(y, x) =⇒ x = y ;equivalently, in terms of its approach distance δ, if

δ(x, y) = 0 = δ(y, x) =⇒ x = y .

2.1.1 Proposition. Let (X, a) be a (T,V)-space.

(1) If (X, a) is Hausdorff, then (X, a) is separated.

(2) If (X, a) is separated, any (T,V)-continuous map f : (2,>) −→ (X, a) from a two-element indiscrete (T,V)-space is constant, and this property is equivalent to (X, a)being separated if a(a, i) = > for all a ∈ T2 \ e2(i), i ∈ 2 = 0, 1.

(3) The full subcategory (T,V)-Catsep of separated (T,V)-spaces is closed under mono-sources in (T,V)-Cat.

Proof.

(1) If (X, a) is Hausdorff, already k ≤ a(eX(x), y) implies

⊥ < k = k ⊗ k ≤ a(eX(x), x)⊗ a(eX(x), y)

and then x = y.


(2) For a (T,V)-functor f : (2,>) −→ (X, a) one has

k ≤ > ≤ a(Tf(e2(0)), f(1)) = a(eX(f(0)), f(1))

and, likewise, k ≤ a(eX(f(1)), f(0)), hence f(0) = f(1) if (X, a) is separated. Con-versely, if x ≤ y and y ≤ x in (X, a), the additional hypothesis makes f : (0 7−→ x, 1 7−→y) (T,V)-continuous, hence constant.

(3) This is a straightforward exercise.

Closure under mono-sources makes (T,V)-Catsep strongly epireflective in (T,V)-Cat. Wegive an easy ad-hoc description of the reflector, modeled after the situation in Top. For atopological space X, the quotient topology on the T0-reflection X/∼ with (x ∼ y ⇐⇒x = y) has the special property that it makes the projection p : X −→ X/∼ not onlyO-final but also O-initial, with respect to O : Top −→ Set. This property is crucial forestablishing the reflection for (T,V)-spaces.2.1.2 Theorem.c© For a (T,V)-space (X, a), the projection p : X −→ X/∼ of the separated-order reflection, given by

x ∼ y ⇐⇒ x ≤ y & y ≤ x ,

serves also as a reflection into (T,V)-Catsep when X/∼ is endowed with the (T,V)-categorystructure a = p · a · (Tp), making p both O-final and O-initial with respect to the functorO : (T,V)-Cat −→ Set.

Proof. By definition of ∼, one has p · p ≤ a · eX , which implies

a · eX/∼ = p · a · (Tp) · eX/∼≥ p · a · eX · p

≥ p · p = 1X/∼ ,

as well as

p · a · Tp = p · p · a · (Tp) · Tp≤ a · eX · a · T (a · eX)≤ a · eX · a≤ a · T a · eTX= a .

We can now concludec© a · T a ·mX/∼ = p · p · a · Tp · (Tp) · T a · TTp · (TTp) ·mX/∼

≤ p · a · T (p · a · Tp) ·mX · (Tp)

≤ p · a · T a ·mX · (Tp)

= p · a · (Tp) = a .


(Here we used the fact that Tp and TTp are surjective since p is surjective, the Axiomof Choice granted.) Consequently, (X/∼, a) is a (T,V)-space, and p is both O-initial andO-final. Quite trivially, (X/∼, a) is separated since p(x) ≤ p(y) implies

k ≤ a(eX/∼(p(x), p(y))≤ (p · a · eX/∼ · p)(x, y)≤ (p · a · Tp · eX)(x, y)= a(eX(x), y) ,

so x ≤ y.It now suffices to show that for any (T,V)-continuous map f : (X, a) −→ (Y, b) with (Y, b)separated one has (x ∼ y =⇒ f(x) = f(y)), since the induced map f : X/∼−→ Y withf · p = f is (T,V)-continuous by O-finality of p. Moreover, p(x) = p(y) implies

k ≤ a(eX(x), y) ≤ b(Tf · eX(x), f(y)) = b(eY (f(x)), f(y))

and, likewise, k ≤ b(eY (f(x)), f(y)), so f(x) = f(y) by separatedness of (Y, b).

2.1.3 Corollary. The separated-order reflection of a V-category carries its reflection intoV-Catsep; that is, the solid-arrow pullback diagram

V-Catsep //

a_

Ordsep

a_

V-Cat

OO

// Ord

OO

in CAT also commutes with its dotted-arrow left adjoints.

2.2 Between order separation and Hausdorff separation. For a (T,V)-space (X, a),we now consider an array of low separation and symmetry conditions, with the terminologyborrowed from the role model Top ∼= (, 2)-Cat, as follows:

(T0) (a · eX) ∧ (a · eX) ≤ 1X ;

(T1) a · eX ≤ 1X ;

(R0) (a · eX) ≤ a · eX ;

(R1) a · a ≤ a · eX .

The following scheme shows how these conditions are related with the Hausdorff separationcondition a · a ≤ 1X and with separatedness of (X, a):2.2.1 Proposition. The following implication hold for a (T,V)-space (X, a):

Hausdorff ⇐⇒ T1 & R1⇓ ⇓ ⇓

T1 ⇐⇒ T0 & R0⇓

separated.


Proof. Hausdorff =⇒ T1 & R1: Since eX ≤ a, one has a · eX ≤ a · a ≤ 1X ≤ a · eX .T1 & R1 =⇒ Hausdorff: One has a · a ≤ a · eX ≤ 1X .T1 =⇒ T0 & R0: One has (a · eX) ∧ (a · eX) ≤ a · eX ≤ 1X and (a · eX) ≤ 1X ≤ a · eX .T0 & R0 =⇒ T1: One has a · eX = (a · eX) ≤ (a · eX), so a · eX = (a · eX)∧ (a · eX) ≤ 1X .R1 =⇒ R0: One has (a · eX) = eX · a ≤ a · a ≤ a · eX .T0 =⇒ separated: If k ≤ a(eX(x), y)∧a(eX(y), x), then ⊥ < k ≤ ((a·eX)∧(a·eX))(x, y) ≤1X(x, y), hence x = y.

2.2.2 Examples.

(1) In Ord ∼= 2-Cat, T0-separation coincides with separation (see II.1.3), while R1 = R0means symmetry, that is, the order is an equivalence relation. Also in Met ∼= P+-Cat,R1 = R0 assumes the usual meaning of symmetry: a(x, y) = a(y, x) for all points inthe metric space (X, a). Like Hausdorffness, T1 means being discrete. When (X, a) issymmetric, even T0 means being discrete and is therefore considerably stronger thanbeing order-separated. But a two-point space X = u, v with a(u, v) =∞ and allother distances 0 is T0 but not T1.

(2) In Top ∼= (, 2)-Cat, or in Top ∼= (F, 2)-Cat, T0 means order-separated and T1 meansthat singleton sets are closed, that is, these conditions assume their usual meanings.Likewise for R0 (if y ∈ x, then x ∈ y) and R1 (if U ∩W 6= ∅ for all neighborhoodsU of x and W of y, then x = y). In App ∼= (,P+)-Cat one has the followingstraightforward characterizations:

(X, a) is T0 ⇐⇒ ∀x, y ∈ X (a(x, y) <∞ & a(y, x) <∞ =⇒ x = y) ;(X, a) is T1 ⇐⇒ ∀x, y ∈ X (a(x, y) <∞ =⇒ x = y);(X, a) is R0 ⇐⇒ ∀x, y ∈ X (a(y, x) = a(x, y));(X, a) is R1 ⇐⇒ ∀x, y ∈ X ∀z ∈ βX (δ(y, x) ≤ a(z, x) + a(z, y)) ,

where δ is the approach distance of (X, a).

2.2.3 Proposition.

(1) Let V be an integral quantale. The T0 and T1 properties are closed under mono-sourcesin (T,V)-Cat. Hence, the corresponding full subcategories are strongly epireflective in(T,V)-Cat.

(2) The R0 and R1 properties are closed under O-initial sources in (T,V)-Cat, whereO : (T,V)-Cat −→ Set. Hence, the corresponding full subcategories are both mono- andepireflective in (T,V)-Cat.


Proof. Let fi : (X, a) −→ (Yi, bi) (i ∈ I) be a source in (T,V)-Cat that is assumed to bemonic when all (Yi, bi) are assumed to be T0 or T1, and O-initial when all (Yi, bi) areassumed to be R0 or R1.Suppose that all (Yi, bi) are T0; then

a · eX ∧ (a · eX) ≤ ∧i,j∈I(f i · bi · Tfi · eX) ∧ (f j · bj · Tfj · eX)

≤ ∧i∈I(f i · bi · eYi · fi) ∧ (f i · eYi · bi · fi)

≤ ∧i∈I f i · (bi · eYi ∧ (bi · eYi)) · fi≤ ∧i∈I f i · fi = 1X ((Yi, bi) are T0).

The proof for T1 is similar.If all (Yi, bi) are R0, then

(a · eX) ≤ ∧i∈I(f i · bi · Tfi · eX)

= ∧i∈I f

i · (bi · eYi) · fi

≤ ∧i∈I f i · bi · eYi · fi ((Yi, bi) are R0)= (∧i∈I f i · bi · Tfi) · eX = a · eX .

Finally, if all (Yi, bi) are R1, then

a · a ≤ ∧i∈I(f i · bi · Tfi) ∧ ∧j∈I(f j · bj · Tfj)≤ ∧i∈I f i · bi · Tfi · (Tfi) · bi · fi≤ ∧i∈I f i · bi · bi · fi≤ ∧i∈I f i · bi · eYi · fi ((Yi, bi) are R1)= (∧i∈I f i · bi · Tfi) · eX = a · eX .

2.3 Regular spaces. Throughout this subsection we assume that

• V is commutative.

In order to formulate regularity and normality for (T,V)-spaces we will make essential useof the V-relation (see Theorem III.5.3.5)

(TX a // TX ) := (TX mX // TTX T a

// TX ) .

For every (T,V)-space (X, a), the V-relation a = T a ·mX is a V-graph structure on TX:

1TX ≤ T1X = T (eX) ·mX ≤ T a ·mX .


Therefore, both (X, a · a) and (X, a · a) are (T,V)-graphs. While the inequality a · a ≤ ais exactly the transitivity condition for a, the condition a · a ≤ a encodes an interestingseparation property of a.

2.3.1 Definition. A (T,V)-space (X, a) is regular if a · a ≤ a, that is a ·mX · (T a) ≤ a,or, in pointwise form,

T a(Y , x )⊗ a(mX(Y ), z) ≤ a(x , z) ,for all Y ∈ TTX, x ∈ TX and z ∈ X.

We denote the resulting full subcategory of (T,V)-Cat by (T,V)-Catreg.

2.3.2 Examples.

(1) If T = I is the identity monad (identically extended to V-Rel), then a · a = a · a,so that a · a ≤ a if and only if a = a. Indeed, if a = a then a · a ≤ a followsfrom transitivity. Conversely, if a · a ≤ a, then a ≤ a · a ≤ a. Hence, for V-spacesregularity means symmetry.

(2) A topological space X considered as a (, 2)-space (X, a) is regular if and only if it isregular in the usual sense, that is: if for x ∈ X and A ⊆ X closed with x 6∈ A, thereexist open sets U,W ⊆ X with x ∈ U , A ⊆ W and U ∩W = ∅. To prove this claimwe first recall that, as observed in Examples III.5.3.7, for x , y ∈ βX, with () = a,

y x ⇐⇒ ∀A ⊆ X,A closed(A ∈ y =⇒ A ∈ x )⇐⇒ ∀A ⊆ X,A open(A ∈ x =⇒ A ∈ y ) .

Furthermore, the condition a · a ≤ a means that

y x & y −→ x =⇒ x −→ x ,

for all x , y ∈ βX and x ∈ X. Therefore, if we write y // x whenever y x , wemay depict this property as follows:

y

x x =⇒ x // x .

If X is regular in the usual topological sense, then the closed neighborhoods of x ∈ Xform a neighborhood base for x. Therefore, if y −→ x, that is, if y contains theneighborhoods of x, and every closed subset A of X belonging to y also belongsto x , then clearly x −→ x. If X is not regular, that is, if there is x ∈ X witha neighborhood W that does not contain any closed neighborhood of x, consideran ultrafilter x containing all the closed neighborhoods of x and X \ W , and anultrafilter y containing all the neighborhoods of x. Then y x and y −→ y, thereforea · a(x , x) = >. However, by construction, one does not have x −→ x, so a · a ≤ afails.


(3) Let T = H as in 1.4. For an (H, 2)-space (X, a), the order () = a is defined by

(β, y) (α, x) ⇐⇒ Ha ·mX((β, y), (α, x)) = >

⇐⇒ ∃γ ∈ H (β = α · γ & yγ// x ) ,

for (α, x), (β, y) ∈ H ×X. Hence (X, a) is regular if and only if

yβ

α·β

x z =⇒ xα // z .

for all α, β ∈ H and all x, y, z ∈ X.

(4) Let T be the list monad L as in 1.4. If (X, a) is a multi-ordered set, then, for () = aand (x1, . . . , xn), (y1, . . . , ym) ∈ LX, one has (y1, . . . , ym) (x1, . . . , xn) if and only ifthere is an n-partition of m, that is, there exist 1 ≤ m1 < m2 < . . . < mn = m suchthat

(y1, . . . , ym1) ` x1, . . . , (ymn−1+1, . . . , ym) ` xn .Hence, (X, a) is regular if and only if for all (x1, . . . , xn), (y1, . . . , ym) ∈ LX, z ∈ X,one has

(y1, . . . , ym) (x1, . . . , xn) & (y1, . . . , ym) ` z =⇒ (x1, . . . , xn) ` z .

(5) Let T = be the ultrafilter monad and V = P+. An approach space, considered as a(,P+)-space (X, a), is regular if and only if for any x , y ∈ βX and x ∈ X,

a(x , x) ≤ a(y , x ) + a(y , x) ,

where a(y , x ) = infu ∈ [0,∞] | ∀A ∈ y (A(u) ∈ x ) (see Examples III.5.3.7).Analogously to the characterization for topological spaces one can write

yu

v

x x =⇒ x ≤u+v

// x .

2.3.3 Proposition. For a (,P+)-space (X, a) with approach distance δ, the followingconditions are equivalent:

(i) (X, a) is regular;

(ii) for every filter f on X, u ∈ [0,∞] and every B ⊆ X with B ∩F (u) 6= ∅ for all F ∈ f ,

δ(x,B) ≤ supA∈A δ(x,A) + u ,

with A = A ⊆ X | ∀F ∈ f (A ∩ F 6= ∅);


(iii) for every ultrafilter y on X, u ∈ [0,∞] and every B ⊆ X with B ∩ A(u) 6= ∅ for allA ∈ y ,

δ(x,B) ≤ a(y , x) + u .

Proof. (i) =⇒ (ii): Let f be a filter on X, u ∈ [0,∞] and B ⊆ X with B ∩ F (u) 6= ∅ forevery F ∈ f . Let x be an ultrafilter on X such that B ∈ x and F (u) ∈ x whenever F ∈ f .Since W ⊆ X | W (u) 6∈ x is an ideal disjoint from f , by Corollary II.1.13.5 there existsan ultrafilter y such that f ⊆ y and W (u) ∈ y whenever W ∈ x . Then, for every x ∈ X,

δ(x,B) ≤ a(x , x) ≤ a(y , x ) + a(y , x) ≤ u+ a(y , x) ≤ u+ supA∈A δ(x,A) .(ii) =⇒ (iii) is straightforward. For (iii) =⇒ (i), consider x , y ∈ βX, x ∈ X and u ∈ [0,∞]such that A(u) ∈ x whenever A ∈ y ; for every B ∈ x , one then has δ(x,B) ≤ a(y , x) + uand thereforea(x , x) = supB∈x δ(x,B) ≤ a(y , x) + infu | ∀A ∈ y (A(u) ∈ x ) = a(y , x) + a(y , x ) .

Here are some expected general assertions about regularity.2.3.4 Proposition.

(1) If V is lean and integral and T is flat, then every compact Hausdorff (T,V)-space isregular.

(2) (T,V)-Catreg is closed under O-initial sources (where O : (T,V)-Cat −→ Set is theforgetful functor), hence both epi- and mono-reflective in (T,V)-Cat.

Proof.

(1) Let (X, a) be a compact Hausdorff (T,V)-space. If V is lean and integral, a is a mapand a · Ta = a ·mX . Since T is flat,

a · a = a ·mX · (Ta) = a · Ta · (Ta) ≤ a · T (a · a) ≤ a .

(2) Let fi : (X, a) −→ (Yi, bi) be an O-initial source, that is a = ∧i∈I f

i · bi · Tfi, with

(Yi, bi) regular for every i ∈ I. Then, for every i ∈ I,

a ·mX · (T a) ≤ f i · bi · Tfi ·mX · (T (f i · bi · Tfi))

= f i · bi · Tfi ·mX · ((Tfi) · T bi · TTfi)

= f i · bi · Tfi ·mX · (TTfi) · (T bi) · Tfi= f i · bi ·mYi · TTfi · (TTfi) · (T bi) · Tfi≤ f i · bi ·mYi · (T bi) · Tfi ((Yi, bi) are regular)≤ f i · bi · Tfi

and therefore a ·mX · (T a) ≤ ∧i∈I f i · bi · Tfi = a.


2.3.5 Remarks.

(1) That a regular (T,V)-space does not need to be Hausdorff (or even separated) canbe seen already at the level of V-spaces: while Hausdorffness means discreteness,regularity means symmetry.

(2) As shown in Theorem III.5.3.5, if T is associative, for a (T,V)-space (X, a), a isnot only reflexive but also transitive, that is, (TX, a) is a V-space. Moreover, the(T,V)-space (X, a) is regular whenever the V-space (TX, a) is regular. Indeed, ifa = a, then using the equality a = a · a one gets

a · a = a · a · a ≤ a · a = a .

The converse statement is not true in general (see Exercise 2.F).

(3) A necessary condition for the (T,V)-space (X, a) to be regular is the regularity ofthe V-space (X, a · eX), provided that the lax extension T of T is symmetric, that isT (r) = (T r) for all r in V-Rel. The next result generalizes this remark.

2.3.6 Proposition. Let α : (S, S) −→ (T, T ) be a morphism of symmetric lax extensions.Then the algebraic functor Aα : (T,V)-Cat −→ (S,V)-Cat preserves regularity.

Proof. For (X, a) regular in (T,V)-Cat one has (with S = (S, n, d)):

a · αX · a · αX = a · αX · nX · (SαX) · (Sa)

= a ·mX · TαX · αSX · (SαX) · S(a) (α monad morphism)≤ a ·mX · TαX · (TαX) · αTX · S(a) (Tα · αS = αT · Sα)≤ a ·mX · T (a) · αX (α lax extension morphism)≤ a · a · αX≤ a · αX .

For (T,V) = (,P+) one concludes in particular that the underlying metric of a regularapproach space is symmetric: see 2.3.2(1).

2.4 Normal and extremally disconnected spaces. Throughout this subsection weassume that

• V is commutative;

• T is associative (see 2.3.5(2)).

Recall that a topological space is normal if for all A,B closed with A ∩B = ∅, there areopen sets U and W such that A ⊆ U , B ⊆ W and U ∩W = ∅. It was already shown in


Proposition III.5.6.2 that normality for a (, 2)-space (X, a) can be expressed by using theordered set (βX, a):

2.4.1 Proposition. For a topological space X presented as a (, 2)-space (X, a), the fol-lowing conditions are equivalent:

(i) X is a normal topological space.

(ii) a · a ≤ a · a.

The proposition leads us to the following definition.

2.4.2 Definition. A (T,V)-space (X, a) is normal if

a · a ≤ a · a , (2.4.i)

that is,a(z, x )⊗ a(z, y ) ≤

∨w∈TX

(a(x ,w)⊗ a(y ,w)) ,

for all x , y , z ∈ TX.

2.4.3 Proposition. For a (T,V)-space (X, a), the following conditions are equivalent:

(i) (X, a) is normal;

(ii) (TX, a) is a normal V-space;

(iii) (TX, a · a) is a V-space.

Proof. For (i)⇐⇒ (ii), we just notice that normality for (TX, a) means a · a ≤ a · a.(ii) =⇒ (iii): The structure a · a is obviously reflexive. If a · a ≤ a · a, then

a · a · a · a ≤ a · a · a · a ≤ a · a .

(iii) =⇒ (i): Transitivity of a · a and reflexivity of a and a give

a · a ≤ a · a · a · a ≤ a · a ,

that is, the (T,V)-space (X, a) is normal.

Reversing the inequality (2.4.i) has also an interesting topological meaning. It leads us toconsider extremally disconnected objects. Recall that a topological space X is extremallydisconnected if the closure of every open set in X is open.

2.4.4 Proposition. For a topological space X presented as a (, 2)-space (X, a), the fol-lowing conditions are equivalent:

(i) X is extremally disconnected;


(ii) for all open subsets U,W of X, if U ∩W = ∅ then U ∩W = ∅;

(iii) a · a ≤ a · a.

Proof. (i) =⇒ (ii): Let U,W be open subsets of X. If U ∩W = ∅, then U ∩W = ∅, and,since U is open, U ∩W = ∅.(ii) =⇒ (i): Let U be an open subset of X and W = X \U . Then U ∩W = ∅ and thereforeU ∩W = ∅. This implies that U is open since

U ⊆ X \W ⊆ X \W = U .

To show (ii) ⇐⇒ (iii), first we point out that, for any ultrafilters x , y on X and witha = (),

a · a(x , y ) = > ⇐⇒ ∃z ∈ βX (x z and y z)⇐⇒ ∃z ∈ βX (∀B ⊆ X,B closed B ∈ x ∪ y =⇒ B ∈ z)

a · a(x , y ) = > ⇐⇒ ∃w ∈ βX (w x and w y )⇐⇒ ∃w ∈ βX (∀A ⊆ X,A open A ∈ x ∪ y =⇒ A ∈ w) .

Assuming (ii), let x , y , z be ultrafilters on X with x z and y z. For any open subsetsU,W of X, if U ∈ x and W ∈ y , then U,W ∈ z and therefore U ∩W 6= ∅, which implieswith (ii) that U ∩W 6= ∅. The filter base

U ∩W | U,W open subsets of X,U ∈ x ,W ∈ y

is contained in an ultrafilter w . By construction, w x and w y , and thereforea · a(x , y ) = >.Conversely, assume that (ii) does not hold. Thus, there are disjoint open subsets U and Wof X with U ∩W 6= ∅. Let z be an ultrafilter containing U ∩W . Consider the ultrafiltersx and y with filter bases

Bx = A | A open and A ∈ z or A = U ,By = A | A open and A ∈ z or A = W .

Then x z and y z but there is no w with w x and w y , that is, (iii) fails.

2.4.5 Definition. A (T,V)-space (X, a) is extremally disconnected if

a · a ≤ a · a ;

that is,a(x , z)⊗ a(y , z) ≤

∨w∈TX

(a(w , x )⊗ a(w , y ))

for all x , y , z ∈ TX.


2.4.6 Remark. A V-space (X, a) is normal if and only if (X, a) is extremally disconnected.

Hence, using Proposition 2.4.3 we obtain:2.4.7 Corollary. For a (T,V)-space (X, a), the following conditions are equivalent.

(i) (X, a) is extremally disconnected;

(ii) (TX, a) is an extremally disconnected V-space;

(iii) (TX, a) is a normal V-space;

(iv) (TX, a · a) is a V-space.

2.4.8 Examples.

(1) Let T = I be the identity monad (identically extended to V-Rel). A V-space (X, a) isnormal if and only if

∀x, y, z ∈ X (a(x, y)⊗ a(x, z) ≤∨s∈X

a(y, s)⊗ a(z, s)) . (2.4.ii)

If we write x β−→ y whenever a(x, y) = β, diagrammatically this condition can berepresented as follows:

∀

xβ

γ

y z β ⊗ γ ≤ ∨s∈X βs ⊗ γs :

xβ

γ

y

βs

z

γss

Extremally disconnected V-spaces are described in a similar way, with the arrowsreversed.In case V = 2, an ordered set (X,≤), considered as a 2-space, is normal if and onlyif the order ≤ is confluent (see Exercise III.5.A). In particular, this shows a normal(T,V)-space does not need to be regular (see also Exercise 2.H).A regular V-space, that is a symmetric V-space, is trivially normal and extremallydisconnected.

(2) Let be the ultrafilter monad. By Proposition 2.4.3, in (, 2)-Cat a topological space(X, a) is normal if and only if the order on βX is confluent: for x , y , z ∈ βX, withx y , x z, there exists w ∈ βX with y w and z w :

x

y

z

w .


(3) Let T = H as in Example 2.3.2(3). Since () = a is transitive, normality of an(H, 2)-space is equivalent to the condition a · a ≤ a · a. We recall that

(β, y) (α, x) ⇐⇒ ∃γ ∈ H (β = α · γ & yγ// x ) .

According to this description, when (β, y) (α, x), we say that (β, y) is a multiple of(α, x) and that (α, x) is a divisor of (β, y). For any (α, x), (β, y) ∈ H ×X,

(α, x) (a · a) (β, y) ⇐⇒ ∃(γ, z) ∈ H ×X ((α, x) (γ, z) (β, y)) ,

that is, (α, x) and (β, y) have a common multiple, and

(α, x) (a · a) (β, y) ⇐⇒ ∃(δ, w) ∈ H ×X ((α, x) (δ, w) (β, y)) ,

that is, (α, x) and (β, y) have a common divisor. Hence (X, a) is normal if and only ifany pair of elements (α, x), (β, y) of H ×X with a common multiple has a commondivisor. That is, for each x, y, z ∈ X, α, β, α1, β1 ∈ H such that α · α1 = β · β1 andx

α1←− zβ1−→ y, there exists w ∈ X, δ, α2, β2 ∈ H with α = δ · α2, β = δ · β2 and

xα2−→ w

β2←− y:z

α1

~~

β1

x

α2

y

β2~~w

with δ · α2 · α1 = α · α1 = β · β1 = δ · β2 · β1.Reversing the arrows one obtains a description of extremally disconnected (H, 2)-spaces.

(4) If T = L is the list monad, then again we can use the condition a · a ≤ a · a to studynormality of a multi-ordered set (X, a). It is easy to check that (X, a) is normal ifand only if for all (x1, . . . , xn), (y1, . . . , ym), (z1, . . . , zl) ∈ LX the following conditionholds: if there exist partitions 1 ≤ i1 < · · · < in = l and 1 ≤ j1 < · · · < jm = l of lsuch that

((z1, . . . , zi1), . . . , (zin−1+1, . . . , zl)) ` (x1, . . . , xn) &((z1, . . . , zj1), . . . , (zjm−1+1, . . . , zl)) ` (y1, . . . , ym)

then there exist (w1, . . . , wk) ∈ LX and k-partitions 1 ≤ n1 < · · · < nk = n and1 ≤ m1 < · · · < mk = m, of n and m, such that

((x1, . . . , xn1), . . . , (xnk−1+1, . . . , xn)) ` (w1, . . . , wk) &((y1, . . . , ym1), . . . (ymk−1+1, . . . , ym)) ` (w1, . . . , wk)

(where ` abreviates La).


(5) In Met ∼= (I,P+)-Cat, a metric space (X, a) is normal if and only if, for every x, y, z ∈ X,

a(z, x) + a(z, y) ≥ infw∈X a(x,w) + a(y, w) .

Likewise in App ∼= (,P+)-Cat, an approach space (X, a) is normal if and only if forany x , y , z ∈ βX,

a(z, x ) + a(z, y ) ≥ infw∈βX a(x ,w) + a(y ,w) ,

where a(x , y ) = infu ∈ [0,∞] | ∀A ∈ x (A(u) ∈ y ).

Normal approach spaces will be investigated in 2.5 below. Here we give conditions on Tand V in general for compact Hausdorff (T,V)-spaces to be normal.

2.4.9 Proposition. If T is flat, then every T-algebra is a normal (T,V)-space.

Proof. First we remark that, when a : TX −→ X is a map, the V-space (TX, a = Ta ·mX)is completely determined by its underlying order (given by Bp : V-Cat −→ 2-Cat = Ord).Therefore, to show normality of the T-algebra (X, a) we have to check that () = a isconfluent. Let x , y , z ∈ TX with x y and x z; that is, there exist Y ,Z ∈ TTX suchthat mX(Y ) = mX(Z) = x and Ta(Y ) = y , Ta(Z) = z. For y = a(y ) and z = a(z), usingthe equality a · Ta = a ·mX we conclude that

y = a · Ta(Y ) = a ·mX(Y ) = a(x ) = a ·mX(Z) = a · Ta(Z) = z .

Now it is easy to conclude y eX(y), since, for W = eTX(y ), mX(W ) = y and Ta(W ) =Ta(eTX(y )) = eX(a(y )) = eX(y); an analogous argument shows that z eX(y).

Using Proposition 1.2.1 we may now conclude.

2.4.10 Corollary. If V is integral and lean and T is flat, then every compact Hausdorff(T,V)-space is normal.

2.5 Normal approach spaces. Normality of an approach space (X, a) implies a strongseparation property that can be expressed in terms of its approach distance. In preparationfor that we first show the next result.

2.5.1 Lemma. For subsets A, B of an approach space X and any real u > 0, the followingare equivalent:

(i) ∀C ⊆ X (A ∩ C(u) 6= ∅ or B ∩ (X \ C)(u) 6= ∅);

(ii) ∃x , y , z ∈ βX ∀C ∈ z (A ∩ C(u) ∈ x and B ∩ C(u) ∈ y ).


Proof. (i) =⇒ (ii): We must show the existence of an ultrafilter z such that both A∪C(u) |C ∈ z and B ∪ C(u) | C ∈ z generate proper filters on X. But failing that, for everyz ∈ βX we could choose D ∈ z with A ∩ D(u) = ∅ or B ∩ D(u) = ∅; moreover, bycompactness of βX, finitely many such sets D1, . . . , Dn could be found with ⋃

iDi = X.Then, with

C :=⋃Di | A ∩D(u)

i = ∅

one would trivially have A ∩ C(u) = ∅, and also B ∩ (X \ C)(u) = ∅ since X \ C ⊆ ⋃Di |B ∩D(u)

i = ∅, in contradiction to (i).(ii) =⇒ (i): For C ⊆ X one either has C ∈ z and then A∩C(u) ∈ x , or X \C ∈ z and thenB ∩ (X \ C)(u) ∈ z.

2.5.2 Theorem. For an approach space (X, a), each of the following statements impliesthe next:

(i) (X, a) is normal in (,P+)-Cat;

(ii) for all ultrafilters x , y , z on X and any real w > 0,

a(z, x ) < w & a(z, y ) < w =⇒ ∃w ∈ βX (a(x ,w) < 2w & a(y ,w) < 2w)

z<w

~~

<w

x

<2w

y

<2w

w ;

(iii) for all A,B ⊆ X and any real v > 0,

A(v) ∩ B(v) = ∅ =⇒ ∃u > 0 ∃C ⊆ X (A(u) ∩ C(u) = ∅ = B(u) ∩ (X \ C)(u)) ;

(iv) for all ultrafilters x , y , z on X,

a(z, x ) = 0 = a(z, y ) =⇒ ∃w ∈ βX (a(x ,w) = 0 = a(y ,w)) .

Proof. (i) =⇒ (ii) is an immediate consequence of the defining property of normality:

a(z, x ) + a(z, y ) ≥ infw∈βX(a(x ,w) + a(y ,w)) .

(ii) =⇒ (iii): Given v > 0, we claim that any u ≤ v4 has the property described by

(iii). Indeed, assuming failure of (iii), by Lemma 2.5.1 we would have A,B ⊆ X withA(v) ∩B(v) = ∅ and ultrafilters x , y , z on X such that

∀C ∈ z (A(u) ∩ C(u) ∈ x and B(u) ∩ C(u) ∈ y ) .


Since a(z, x ) = infw | ∀C ∈ z (C(w) ∈ x ), one has u ≥ a(z, x ) and, likewise, u ≥ a(z, y ).From (ii) (with w = 3

2u) one obtains w ∈ βX with a(x ,w) < 3u and a(y ,w) < 3u;therefore, since A(u) ∈ x and B(u) ∈ y ,

A(v) ∩B(v) ⊇ (A(u))(3u) ∩ (B(u))(3u) ∈ w ,

which contradicts A(v) ∩B(v) = ∅.(iii) =⇒ (iv): Consider x , y , z ∈ βX with a(z, x ) = 0 = a(z, y ). For any A ∈ x , B ∈ y andu > 0 one has

∀C ∈ z (A ∩ C(u) ∈ x and B ∩ C(u) ∈ y )

which, by Lemma 2.5.1, means equivalently

∀C ∈ z (A ∩ C(u) 6= ∅ or B ∩ (X \ C)(u) 6= ∅) ;

in particular∀C ∈ z (A(u) ∩ C(u) 6= ∅ or B(u) ∩ (X \ C)(u) 6= ∅) .

By hypothesis (iii) then, A(v) ∩B(v) 6= ∅ for all A ∈ x , B ∈ y , v > 0. With an ultrafilterw containing the filter base given by all these non-empty sets, one obtains

∀v > 0∀A ∈ x ∀B ∈ y (A(v) ∈ w and B(v) ∈ w) ,

that is, a(x ,w) = 0 = a(y ,w).

2.5.3 Remark. None of the three implications of Theorem 2.5.2 is reversible, as thefollowing three examples show. We point out that all the three examples are in particularmetric spaces, showing that the four conditions are distinct already at the level of metricspaces.

(1) Let X = x, y, z, w with a(z, x) = a(z, y) = a(z, w) = 1, a(x,w) = a(y, w) = 2,a(x′, x′) = 0 for any x′ ∈ X, and a(x′, y′) =∞ elsewhere. Then it is straightforward tocheck that (X, a) satisfies (ii) although a(z, y)+a(z, x) < infw′∈X(a(x,w′)+a(y, w′)) =4.

(2) Let X = x, y, z, w with a′(x′, y′) = a(x′, y′) except for a′(x,w) = a′(y, w) = 3. Then(ii) clearly fails. Since distinct points are at distance at least 1, for any A ⊆ X andany u < 1, one has A(u) = A, so (iii) holds.

(3) Let Y = x, y ∪ xn ; n ∈ N, with b(xn, x) = 1n

= b(xn, y), b(x′, x′) = 0 for allx′ ∈ X, and b(x′, y′) =∞ elsewhere. Then x(1) ∩ y(1) = x ∩ y = ∅ although,for any C ⊆ X, either C or X \ C has an infinite number of xn, and so, for anyu > 0, either x, y ∈ C(u) or x, y ∈ (X \ C)(u); that is, (iii) does not hold. Moreover,b(z, x) = 0 only if z = x and so (iv) holds.


Exercises

2.A Preservation of low separation by algebraic functors. Let α : (S, S) −→ (T, T ) be amorphism of lax extensions and Aα : (T,V)-Cat −→ (S,V)-Cat the induced algebraic functor.Prove that (X, a) is T0, T1, or R0 if and only if Aα(X, a) has the respective property.Furthermore, if (X, a) is R1 or Hausdorff, then Aα(X, a) has the respective property, withthe converse statement holding when αX is surjective. Exploit these statements for α = e,that is, when Aα = A: (T,V)-Cat −→ V-Cat.

2.B Low separation in (H, 2)-Cat. Present (H, 2)-Cat as V-Cat with V the powerset of H asin Remark 1.4.3(1) and show that each of T0, T1, R0 and R1 changes its meaning underthe change of presentation.

2.C Compact T1-spaces for power-enriched monads. When T is power-enriched, prove thatevery compact T1-space in (T, 2)-Cat has precisely one point.

2.D Strengthening R0. We say that a (T,V)-space is R0+ if a · eX ≤ a. Show that:

(1) every R0+ space is R0;

(2) every regular space is R0+.

2.E Separated reflections and Kleisli monoids. c©For a monad T on Set with an associativelax extension T to V-Rel, show that the separated reflection of Theorem 2.1.2 defines also aleft adjoint to the inclusion functor ((T,V)-CatT)sep −→ (T,V)-CatT, where T is considereda monad on (T,V)-Cat (see III.5.4). Conclude that if T is power-enriched, then T ′X asdefined in IV.4.2 is the separated reflection of TX.Hint. See Exercise IV.4.C.

2.F Regularity of (X, a) does not imply regularity of (TX, a). Let H be as in Exam-ple 2.3.2(3).

(1) Show that, when H = (N,×, 1) is the multiplicative monoid of natural numbers, the(H, 2)-space (N, a), with a((α, x), y) = > only when α · x = y, is regular, although theordered set (HN, a) is not regular (that is, symmetric).

(2) Show that, if H is a group, then an (H, 2)-space (X, a) is regular if and only if (HX, a)is regular.

2.G Discrete (T,V)-spaces. Show that, when T is flat, discrete (T,V)-spaces are bothnormal and extremally disconnected (see III.3.2).

2.H The extended real half-line. Show that, as a P+-space, the extended real half-line([0,∞],) is normal and extremally disconnected, but not regular.


2.I Normal Hausdorff spaces in (L, 2)-Cat. Let (X,`) be a normal Hausdorff space in(L, 2)-Cat. Then

x · y = z ⇐⇒ (x, y) ` zestablishes a partially-defined binary operation · on X such that, whenever x · y, y · z aredefined, then (x · y) · z, x · (y · z) are defined and equal, or x · y = x and y · z = z.

3. PROPER AND OPEN MAPS 401

3 Proper and open maps

The power of the notion of compact Hausdorff (T,V)-space arises from its equationaldescription as an Eilenberg–Moore algebra (see Proposition 1.2.1). In this subsection weconsider the equationally defined classes of proper and of open maps in (T,V)-Cat which inSection 4, inter alia, lead us to equationally defined modifications of the notions of compactand Hausdorff (T,V)-space, as defined in 1.1.Throughout the section, V = (V ,⊗, k) is a quantale and T = (T,m, e) is a Set-monad witha lax extension T to V-Rel.

3.1 Finitary stability properties. For (T,V)-spaces (X, a), (Y, b), (T,V)-continuity ofa map f : X −→ Y is equivalently expressed by the inequalities

f · a ≤ b · Tf and a · (Tf) ≤ f · b .

Considering equalities in either case leads us to the two key notions of this subsection.

3.1.1 Definition. A (T,V)-continuous map f : (X, a) −→ (Y, b) is proper if

b · Tf ≤ f · a ,

and f : (X, a) −→ (Y, b) is open if

f · b ≤ a · (Tf) ,

as in III.4.3.1.

Hence, f is proper if and only if

∀x ∈ TX, y ∈ Y (b(Tf(x ), y) ≤∨

z∈f−1ya(x , z)) , (3.1.i)

in which case the inequality is actually an equality; and f is open if and only if

∀x ∈ X ∀y ∈ TY (b(y , f(x)) ≤∨

z∈(Tf)−1 ya(z, x)) ,

in which case the inequality is again an equality.In order to emphasize their dependency on T and V, whenever needed we speak moreprecisely of (T,V)-proper maps and (T,V)-open maps.


3.1.2 Remarks.

(1) For V commutative, one has the dualization functor

(−)op : V-Cat −→ V-Cat , X = (X, a) 7−→ Xop = (X, a) ,

which maps morphisms identically. For T = I (identically extended to V-Rel) andf : (X, a) −→ (Y, b) one has

b · f ≤ f · a ⇐⇒ f · b ≤ a · f ,

that is:f proper ⇐⇒ f op open .

Hence, open is dual to proper.

(2) When V is integral and superior (see 1.1.3), the (T,V)-functor (X, a) −→ (1,>) isproper if and only if (X, a) is compact.

(3) When V is integral and T1 ∼= 1, then the (T,V)-functor (X, a) −→ (1,>) is open. (Seealso Exercise 3.C.)

3.1.3 Examples.

(1) In Ord ∼= 2-Cat, a monotone map f : (X,≤) −→ (Y,≤) is proper if and only if

f(x) ≤ y =⇒ ∃z ∈ f−1y (x ≤ z) , x ≤ z

f(x) ≤ y

and it is open if and only if

y ≤ f(x) =⇒ ∃z ∈ f−1y (z ≤ x) , z ≤ x

y ≤ f(x) .

In terms of the down- and up-closure operations (see II.1.7), one has

f proper ⇐⇒ ∀x ∈ X (↑Y f(x) ⊆ f(↑X x))⇐⇒ ∀A ⊆ X (↑Y f(A) ⊆ f(↑X A))⇐⇒ ∀y ∈ Y (f−1(↓Y y) ⊆ ↓X(f−1y))⇐⇒ ∀B ⊆ Y (f−1(↓Y B) ⊆ ↓X f−1(B)) ,

and all inclusions may equivalently be replaced by equalities. Openness of f ischaracterized by the order-dual conditions.


(2) If Ord is presented as (P, 2)-Cat (with the powerset monad laxly extended by P , seeExample III.1.6.2(1)), the meaning of proper map changes from the previous example.Indeed, in this presentation, an ordered set (X,≤) is considered as a (P, 2)-space(X, ) via

A y ⇐⇒ ∀x ∈ A (x ≤ y)for all A ⊆ X, y ∈ X, and with X A := x ∈ X | A x one obtains for a monotonemap f :

f is (P, 2)-proper ⇐⇒ ∀A ⊆ X ( Y f(A) ⊆ f( X A)) .Such maps must necessarily be surjective (A = ∅) and 2-proper (A = x), butnot conversely. However, the meaning of openness stays the same as in the previousexample:

f is (P, 2)-open ⇐⇒ f is 2-open .

(3) In Met ∼= P+-Cat, a non-expansive map f : (X, a) −→ (Y, b) is proper if and only if

b(f(x), y) = infa(x, z) | z ∈ X, f(z) = y

for all x ∈ X, y ∈ Y . Openness is characterized dually.

(4) For a monoid H and H as in 1.4, an (H, 2)-map f : X −→ Y is proper if

f(x) α−→ y =⇒ ∃z ∈ f−1y (x α−→ z) ,

and open ify

α−→ f(x) =⇒ ∃z ∈ f−1y (z α−→ x) ,(for all x ∈ X, y ∈ Y , α ∈ H).

(5) For the list monad L as in 1.4, an (L, 2)-map f : X −→ Y is proper if

(f(x1), . . . , f(xn)) ` y =⇒ ∃z ∈ f−1y ((x1, . . . , xn) ` z) ,

and open if

(y1, . . . , yn) ` f(x) =⇒ ∃zi ∈ f−1(yi) (i = 1, . . . , n) ((z1, . . . , zn) ` x) ,

(for all x, xi ∈ X, y, yi ∈ Y ).

The more important examples Top ∼= (, 2)-Cat and App ∼= (,P+)-Cat will be discussedin 3.4, after we have established the following stability properties.

3.1.4 Proposition.

(1) The classes of proper maps and of open maps in (T,V)-Cat are both closed undercomposition and contain all isomorphisms.


(2) For (T,V)-continuous maps f : (X, a) −→ (Y, b), m : (Y, b) −→ (Z, c), if m is injectiveand m · f proper or open, f is also proper or open, respectively.

(3)c© For (T,V)-continuous maps e : (X, a) −→ (Y, b), g : (Y, b) −→ (Z, c), if e is surjectiveand g · e proper or open, g is also proper or open, respectively.

(4) If V is cartesian closed, every pullback of a proper map is proper; if, in addition, Tsatisfies BC, every pullback of an open map is open.

Proof.

(1) Immediate.

(2) If m is injective, then m ·m = 1Y and (Tm) · Tm = 1TY (see Proposition III.1.2.2and Exercise III.1.P). Hence, if m · f is proper, then

b · Tf = b · (Tm) · Tm · Tf ≤ m · c · T (m · f) = m ·m · f · a = f · a

follows, and if m · f is open, then one obtains

f · b = f ·m ·m · b ≤ f ·m · c · Tm = c · (Tf) · (Tm) · Tm = c · (Tf) .

(3) If e is surjective, then (Te) · (Te) = 1TY , so that g · e proper impliesc©

c · Tg = c · Tg · Te · (Te) ≤ g · e · a · (Te) ≤ g · e · e · b ≤ g · b .

The case where g · e is open is treated similarly.

(4) Pullback stability of openness was shown in Proposition III.4.3.8. Keeping the notationused there one shows the same property for proper maps, but without requiring BCfor T , as follows:

b · Tq = (b ∧ b) · Tq≤ ((g · c · Tg) ∧ b) · Tq= (g · c · Tg · Tq) ∧ b · Tq= (g · c · Tf · Tp) ∧ b · Tq= (g · f · a · Tp) ∧ b · Tq (f proper)= (q · p · a · Tp) ∧ b · Tq= q · ((p · a · Tp) ∧ (q · b · Tq)) (V cartesian closed by III.4.3.7)= q · d .


3.2 First characterization theorems. In this subsection we show that openness of(T,V)-Cat is often described as openness in V-Cat and then explore to which extent thisreduction is possible also for proper maps. We then prove a first generalization of thecharacterization of proper maps in Top as the closed maps with compact fibres.

3.2.1 Remark. By Corollary III.1.4.4, every lax extension T of T satisfies

T (r · f) = T r · Tf and T (g · r) = (Tg) · T r

for all f : X −→ Y , r : Y −→7 Z, g : W −→ Z. We say that T is left-whiskering if

T (h · r) = Th · T r

for all r : Y −→7 Z, h : Z −→ W . This condition implies in particular T h = Th · T1Z , whichis also sufficient for the general case when T preserves composition and, a fortiori, when Tis associative (see Proposition III.1.9.4). Similarly one says that T is right-whiskering if

T (s · f ) = T s · (Tf)

for all f : X −→ Y , s : X −→7 Z, which implies T (f ) = T1X · (Tf) without loss ofinformation when T preserves composition. Note that a flat associative lax extension isalways left- and right-whiskering. Hence, the Barr extensions of to Rel and to V-Rel areleft- and right-whiskering. The Kleisli extension F to Rel of the filter monad is right- butnot left-whiskering: see Exercise 3.K. In III.1.9.6 we gave an example of a non-associativeflat lax extension which is left- and right-whiskering.

If m : T −→ T T is a natural transformation, in particular if T is associative, the problemof characterizing the proper and open maps may often be reduced to the V-Cat case, viathe composite functor

(T,V)-Cat M // (V-Cat)T GT // V-Cat(X, a) // (TX, T a ·mX ,mX) // (TX, T a ·mX)

of 2.3 and Theorem III.5.3.5 which, by abuse of notation, we denote by T again. Indeed,with T left-whiskering or right-whiskering one obtains the following criteria.

3.2.2 Proposition. Assume m : T −→ T T to be a natural transformation and f :(X, a) −→ (Y, b) to be (T,V)-continuous.

(1) If T is left-whiskering, then

f is (T,V)-proper =⇒ Tf is V-proper .

(2) If T right-whiskering, then

f is (T,V)-open ⇐⇒ Tf is V-open .


Proof.

(1) From b · Tf ≤ f · a, one obtains

T b ·mY · Tf ≤ T b · T T f ·mX (m natural)≤ T (b · T f) ·mX (T lax functor)= T (b · Tf) ·mX (b right unitary)= T (f · a) ·mX (f is (T,V)-proper)= Tf · T a ·mX (T left-whiskering.)

(2) From f · b ≤ a · (Tf), one derives

(Tf) · T b ·mY = T (f · b) ·mY≤ T (a · (Tf)) ·mY (f is (T,V)-open)= T a · (TTf) ·mY (T right-whiskering)= T a ·mX · (Tf) .

Conversely, from (Tf) · T b ·mY ≤ T a ·mX · (Tf) we can derive

f · b ≤ f · b · eTY · eTY≤ eX · T (f · b) · eTY≤ eX · (Tf) · T b ·mY≤ eX · T a ·mX · (Tf)

= (eX a) · (Tf) = a · (Tf) ,

since a is left unitary.

Lax extensions for which the implication of Proposition 3.2.2(1) becomes a logical equivalenceare, from a topological perspective, rare, but can be fully characterized, as follows.

3.2.3 Proposition. Assume m : T −→ T T to be a natural transformation and T left-whiskering. Consider the following assertions:

(i) every (T,V)-continuous map f : (X, a) −→ (Y, b) such that Tf is V-proper is (T,V)-proper;

(ii) every function f : X −→ Y gives a (T,V)-proper map f : (X, 1]X) −→ (Y, 1]Y );

(iii) e : 1 −→ T satisfies the Beck–Chevalley Condition.

Then (i)⇐⇒ (ii) ⇐= (iii), and all are equivalent when T is flat.


Proof. (i) =⇒ (ii): We first determine how T : (T,V)-Cat −→ V-Cat maps discrete (T,V)-spaces and prove T (X, 1]X) = (TX, T1X). Indeed, since m is a natural transformation,

1]X = T (eX · T1X) ·mX = (TeX) · T T1X ·mX = (TeX) ·mX · T1X = T1X .

Since T is left-whiskering, Tf · T1X = T f = T1Y · Tf , so that

Tf : (TX, T1X) −→ (TY, T1Y )

is V-proper, for every function f : X −→ Y . By hypothesis (i), the map f : (X, 1]X) −→ (Y, 1]Y )is (T,V)-proper.(ii) =⇒ (i): For f : (X, a) −→ (Y, b), let Tf : (TX, a) −→ (TY, b) be V-proper, andf : (X, 1]X) −→ (Y, 1]Y ) be (T,V)-proper. Since T is left-whiskering we obtain

b · Tf = (eY b) · Tf= eY · T b ·mY · Tf= eY · Tf · T a ·mX (Tf is V-proper)= eY · Tf · T1X · T a ·mX= eY · T1X · Tf · T a ·mX (T left-whiskering)= 1]Y · Tf · T a ·mX= f · 1]X · T a ·mX (f : (X, 1]X) −→ (Y, 1]Y ) is (T,V)-proper)= f · (eX a)= f · a .

(iii)⇐⇒ (ii): f · eX = eY · Tf implies f · eX · T1X = eY · Tf · T1X = eY · T1Y · Tf since Tleft-whiskering, and the reverse implication holds if T is flat.

3.2.4 Remarks.

(1) For a monoid H, consider the flat extension H of the associated monad H (see 1.4).An (H, 2)-functor f : (X, a) −→ (Y, b) is proper if it satisfies

f(x) α−→ y =⇒ ∃z ∈ f−1y (x α−→ z)

for all x ∈ X, y ∈ Y , α ∈ H. The lax extension satisfies the hypothesis of Proposi-tion 3.2.3, and the unit morphism of the monad satisfies BC. Proper (H, 2)-functorsf are therefore equivalently described as those (H, 2)-functors for which 1H × f isproper.

(2) For the ultrafilter monad , the unit fails to satisfy BC: see Proposition III.1.12.4;indeed, the naturality square for the map X −→ 1 with X infinite fails to be a


BC-square. Consequently, by Proposition 3.2.3, there are maps f : X −→ Y inTop ∼= (, 2)-Cat for which βf is proper, but f is not. We will see in Proposition 3.4.5that 3.2.3 gives the (T,V)-categorical reason for the existence of closed maps that arenot stable under pullback.

We can now give a complete characterization of (T,V)-proper maps in terms of the conditionthat Tf be V-proper which, as we will show in Proposition 3.4.5, generalizes the charac-terization of proper maps in Top = (, 2)-Cat as the closed maps with compact fibres, andsimilarly in App ∼= (,P+)-Cat. But in order to be able to talk about fibres of f , we shouldfirst clarify that very term. For each y ∈ Y , the assignment ∗ 7−→ y defines a (T,V)-functory : (1, 1]) −→ (Y, b), where 1] = e1 · T11 is the discrete structure on 1 = ∗ (see III.3.2);explicitly, for w ∈ T1,

1](w , ∗) = T11(w , e1(∗)) .

By fibre of f on y we mean the pullback (f−1y, a) −→ (1, 1]) of f along the (T,V)-functory : (1, 1]) −→ (Y, b). We note that (f−1y, a) −→ (X, a) is a monomorphism, but in general nota regular monomorphism, that is, a does not need to be the restriction of a : TX ×X −→ Vto T (f−1y)× f−1y:

a(x , x) = a(x , x) ∧ 1](T !(x ), ∗) (with ! : f−1y −→ 1)= a(x , x) ∧ T1X(T !(x ), e1(∗)) ,

for every x ∈ T (f−1y) and x ∈ f−1y. However, when T1 ∼= 1 and V is integral, frome−1

1 = e1 ≤ e1 · T1 one obtains 1] = >, and every (f−1y, a) becomes a subspace of (X, a).Cartesian closedness of V suffices to insure that proper (T,V)-functors have proper fibres,see Proposition 3.1.4.We can now prove a first characterization theorem.

3.2.5 Theorem. Let V be cartesian closed and T be taut, with T left-whiskering and m anatural transformation. Then a (T,V)-functor f : (X, a) −→ (Y, b) is proper if and only ifall of its fibres are proper, and the V-functor Tf : (TX, a) −→ (TY, b) is proper.

Proof. If f is proper, then the fibres of f are proper by Proposition 3.1.4 and Tf is properby Proposition 3.2.2.Conversely, assume that all fibres of f are proper in (T,V)-Cat and Tf is proper in V-Cat.Since

b = b · eTY ·mY ≤ eY · T b ·mY = eY · b ,


for all x ∈ TX, y ∈ Y one obtains:

b · Tf(x , y) = b(Tf(x ), y)≤ b(Tf(x ), eY (y))= ∨

z∈(Tf)−1(eY (y)) a(x , z) (Tf proper)= ∨

z∈(Tf)−1(eY (y))(T a ·mX)(x , z)= ∨

z∈(Tf)−1(eY (y))∨

X∈m−1X x T a(X , z)⊗ k .

Since tautness of T guarantees that the following diagram is a pullback

T (f−1y) T ! //

T1Ty

TXTf

// TY ,

every z ∈ (Tf)−1(eY (y)) = (Tf)−1(Ty(e1(∗))) satisfies z ∈ T (f−1y) and T !(z) = e1(∗).Using propriety of (f−1y, a) −→ (1, 1]) one gets:

∨z∈(Tf)−1(eY (y))

∨X∈m−1

X x T a(X , z)⊗ k ≤ ∨z∈(T !)−1(e1(∗))∨

X∈m−1X x T a(X , z)⊗ ∨x∈f−1y a(z, x)

≤ ∨z∈(T !)−1(e1(∗))∨

X∈m−1X x

∨x∈f−1y T a(X , z)⊗ a(z, x)

≤ ∨X∈m−1X x

∨x∈f−1y a(mX(X ), x)

≤ ∨x∈f−1y a(x , x)= (f · a)(x , y) .

Hence, f is proper.

Next we show that propriety of fibres trivializes whenever the unit e of the monad T satisfiesBC—a rather restrictive condition, as we have seen in III.1.12 and Exercise III.1.Q.

3.2.6 Proposition. If V is integral and e : T −→ 1 is a natural transformation, then any(T,V)-functor has proper fibres.

Proof. For a (T,V)-functor f : (X, a) −→ (Y, b) and y ∈ Y , we must show that the diagram

T (f−1y) T ! //

_a

T1_ 1]

f−1y! // 1


commutes; for that, it suffices to consider x ∈ T (f−1y) with 1](T !(x ), ∗) = T1(T !(x ), e1(∗)) >⊥ and show a(x , ∗) = >. From the commutativity of the diagram

T (f−1y) T !//

_e

T1 T1//

_e1

T1_ e1

f−1y ! // 1 1 // 1

we first obtain

⊥ < e1 · T1 · T !(x , ∗) = e1 · T !(x , ∗) = ! · e(x , ∗) =∨x∈f−1y

e(x , x) = k ,

and then! · a(x , ∗) ≥ ! · e(x , x) = k = > .

3.2.7 Corollary. Under the hypotheses of Theorem 3.2.5 and Proposition 3.2.6, a (T,V)-functor f : (X, a) −→ (Y, b) is proper if and only if the V-functor Tf is proper.

3.3 Notions of closure. Our next aim is to characterize (T,V)-proper and (T,V)-openmaps in terms of suitable notions of closure. Since the functor T of the monad T preservesinjections (see Exercise III.1.P), for a subset A ⊆ X we may assume TA ⊆ TX; withA ⊆ TX, v ∈ V we define:

A[v] := x ∈ X | ∨x ∈A a(x , x) ≥ v ,A(v) := TA[v] ,

A := ⋃v>⊥A

(v) = x ∈ X | ∃x ∈ TA (a(x , x) > ⊥) ,

and call A(v) and A the v-closure and grand closure of A, respectively.

3.3.1 Proposition. The grand closure defines a hereditaryM-closure operator on (T,V)-Cat,with M the class of embeddings, and so does the v-closure, for any fixed v ≤ k in V . Ingeneral, neither operator is idempotent.

Proof. See Exercise 3.A.

We can now analyse to which extent these closures help us characterize propriety andopenness of maps.

3.3.2 Proposition. The following statements on a (T,V)-continuous map f : (X, a) −→(Y, b) satisfy the implications

(i) =⇒ (iii) =⇒ (v)⇓ ⇓

(ii) =⇒ (iv);


furthermore, the vertical implications are equivalences if V is completely distributive, andone has (ii)⇐⇒ (iv) if T = I is the identity monad, and (iii)⇐⇒ (v) if V = 2.

(i) b · Tf ≤ f · a, that is, f is proper;

(ii) ∀A ⊆ TX ∀u v in V (Tf(A)[v] ⊆ f(A[u]));

(iii) ∀A ⊆ X (b · Tf · TiA· !TA ≤ f · a · TiA· !TA);

(iv) ∀A ⊆ X ∀u v in V (f(A)(v) ⊆ f(A(u)));

(v) ∀A ⊆ X (f(A) ⊆ f(A));

where iA : A → X is the inclusion and !TA : TA −→ 1.

Proof. The implications (i) =⇒ (iii) and (ii) =⇒ (iv) are trivial while the remaininggenerally valid implications become obvious once transcribed in elementwise terms, asfollows:

(i) ∀x ∈ TX, y ∈ Y (b(Tf(x ), y) ≤ ∨x∈f−1y a(x , x));

(ii) ∀A ⊆ TX, u v in V , y ∈ Y (v ≤ ∨x ∈A b(Tf(x ), y) =⇒ ∃x ∈ f−1y (u ≤∨

z∈A a(z, x)));

(iii) ∀A ⊆ X, y ∈ Y (∨x ∈TA b(Tf(x ), y) ≤ ∨x∈f−1y

∨z∈TA a(z, x));

(iv) ∀A ⊆ X, u v in V , y ∈ Y (v ≤ ∨x ∈TA b(Tf(x ), y) =⇒ ∃x ∈ f−1y (u ≤∨

z∈TA a(z, x)));

(v) ∀A ⊆ X, y ∈ Y (∃x ∈ TA (b(Tf(x ), y) > ⊥) =⇒ ∃x ∈ f−1y, z ∈ TA (a(z, x) >⊥)).

Here, for the reformulation of (iii) and (iv), note that T (f(A)) = Tf(TA) since T preservesthe surjection A −→ f(A).To see (ii) =⇒ (i) if V is completely distributive, put A := x and v := b(Tf(x ), y), andfor (iv) =⇒ (iii) put v := ∨

x ∈TA b(Tf(x ), y). One trivially has (iv) =⇒ (ii) when T is theidentity functor, and the elementwise formulations also show (v) =⇒ (iii) for V = 2 where∨i vi = > equivalently means vi > ⊥ for some i.

3.3.3 Remarks.

(1) With the reverse inequalities in Proposition 3.3.2(i) and (iii) holding by (T,V)-continuity of f , these inequalities may also be replaced by equalities. The sameis true for the inclusion in (v), and when V is completely distributive, statements (ii)and (iv) may respectively be replaced by:

(ii′) ∀A ⊆ TX, v ∈ V (Tf(A)[v] = ⋂uv f(A[u]));


(iv′) ∀A ⊆ X, v ∈ V (f(A)(v) = ⋂uv f(A(u))).

Briefly, all five conditions considered in 3.3.2 are equationally defined.

(2) While conditions (i)–(iv) are equivalent in V-Cat for V completely distributive (andT = I), condition (v) is in general considerably weaker: in Met = P+-Cat, everysurjective map (X, a) −→ (Y, b) with a, b finite satisfies condition (v)! By contrast,in Top = (, 2)-Cat the grand closure describes the ordinary Kuratowski-closure, sothat each of (iii)⇐⇒ (iv)⇐⇒ (v) describes closed continuous maps (continuous mapswhich preserve closed subsets), while each of (i)⇐⇒ (ii) describes stably-closed maps,as we will show in Proposition 3.4.5 and Theorem 3.4.6.

(3) For an embedding f : (X, a) → (Y, b) in (T,V)-Cat one has (i) ⇐⇒ (iii) ⇐⇒ (v)in 3.3.2; these equivalent conditions are satisfied precisely when the set X is closed in(Y, b) with respect to the grand closure, that is, when X

Y = X: see Exercise 3.A.

While Proposition 3.3.2 compares propriety with the behavior of closures under takingimages, we now similarly compare openness with the behavior under taking inverse images.In order to do so, we need that

T (f−1B) = (Tf)−1(TB)

for all B ⊆ Y and f : X −→ Y , that is, that T is taut (see III.4.3.5).3.3.4 Proposition. Let the functor T of the monad T be taut. The following statementson a (T,V)-continuous map f : (X, a) −→ (Y, b) satisfy the implications

(i) =⇒ (iii) =⇒ (v)⇑ ⇑

(ii) =⇒ (iv);

furthermore, the vertical implications are equivalences if V is completely distributive, andone has (ii)⇐⇒ (iv) if T = I is the identity monad, and (iii)⇐⇒ (v) if V = 2.

(i) f · b ≤ a · (Tf), that is, f is open;

(ii) ∀B ⊆ TY, v ∈ V (⋂uv f−1(B[u]) ⊆ (Tf)−1(B)[v]);

(iii) ∀B ⊆ Y (f · b · TiB· !TB ≤ a · (Tf) · TiB· !TB);

(iv) ∀B ⊆ Y, v ∈ V (⋂uv f−1(B(u)) ⊆ f−1(B)(v));

(v) ∀B ⊆ Y (f−1(B) ⊆ f−1(B)).

Proof. One proceeds schematically as in the proof for Proposition 3.3.2, by transcribingthe five statements in elementwise terms:

(i) ∀x ∈ X, y ∈ TY (b(y , f(x)) ≤ ∨z∈(Tf)−1 y a(z, x));


(ii) ∀B ⊆ TY, v ∈ V , x ∈ X ((∀u v (u ≤ ∨y∈B b(y , f(x))) =⇒ v ≤ ∨z∈(Tf)−1(B) a(z, x));

(iii) ∀B ⊆ Y, x ∈ X (∨y∈TB b(y , f(x)) ≤ ∨z∈(Tf)−1(TB) a(z, x));

(iv) ∀B ⊆ Y, v ∈ V , x ∈ X ((∀u v (u ≤ ∨y∈TB b(y , f(x)))) =⇒ v ≤ ∨z∈(Tf)−1(TB) a(z, x));

(v) ∀B ⊆ Y, x ∈ X (∃y ∈ TB (b(y , f(x)) > ⊥) =⇒ ∃z ∈ (Tf)−1(TB) (a(z, x) > ⊥)).

3.3.5 Remarks.

(1) Because of the (T,V)-continuity of f , the inequalities in (i), (iii) and the inclusion inProposition 3.3.4(v) may equivalently be replaced by equalities; likewise in (ii) and(iv) if V is completely distributive.

(2) As in Proposition 3.3.2, conditions (i)–(iv) of 3.3.4 are equivalent in V-Cat for Vcompletely distributive (and T = I), but condition (v) is generally weaker: again, anysurjective map (X, a) −→ (Y, b) in Met = P+-Cat with a, b finite satisfies condition (v).However, in Top = (, 2)-Cat all five conditions are equivalent and describe open mapsin the usual sense, that is, continuous maps which preserve open subsets. Indeed, sucha map f : X −→ Y satisfies condition (i) which reads as:

∀x ∈ X, y ∈ βY (y −→ f(x) =⇒ ∃z ∈ βX (f [z] = y & z −→ x)) , z // x

y // f(x) .

(One simply takes for z an ultrafilter on X containing the filter base f−1B | B ∈ y.)Since (i) implies (v), it suffices to show that (v) makes f open in the usual sense. Butfor A ⊆ X open one obtains

f−1(Y \ f(A)) = f−1(Y \ f(A)) ⊆ X \ A = X \ A

and then Y \ f(A) ⊆ Y \ f(A), so that f(A) is open.

(3) As in 3.3.2, one has (i)⇐⇒ (iii)⇐⇒ (v) in 3.3.4 for an embedding f : (X, a) → (Y, b),where now (v) reads as:

∀B ⊆ Y (B ∩X = B ∩X) .

Letting Top = (β, 2)-Cat again guide our terminology in the general context, we considerproperties 3.3.2(iii) and 3.3.4(iii) and define:

3.3.6 Definition. A (T,V)-continuous map f : (X, a) −→ (Y, b) is closed if

b · Tf · TiA· !TA ≤ f · a · TiA· !TA


for all A ⊆ X, and f : (X, a) −→ (Y, b) is inversely closed if

f · b · TiB· !TB ≤ a · (Tf) · TiB· !TB

for all B ⊆ Y .

Emphasizing the dependency on the parameters, we often add the prefixes (T,V) or V(when T = I). One trivially has:

f proper =⇒ f closed, f open =⇒ f inversely closed,

with the reversed implications holding for T = I (see Propositions 3.3.2 and 3.3.4). WithPropositions 3.2.2 and 3.2.3 we obtain:

3.3.7 Corollary. Assume m : T T −→ T to be a natural transformation, and let f :(X, a) −→ (Y, b) be (T,V)-continuous.

(1) If T is left-whiskering, then

f is proper =⇒ Tf : (TX, a) −→ (TY, b) is closed,

with the reverse implication holding when e : 1 −→ T satisfies BC.

(2) If T is right-whiskering, then

f is open ⇐⇒ Tf : (TX, a) −→ (TY, b) is inversely closed.

It is essential to consider Tf on the right-hand sides, not just f , as the following exampleshows.

3.3.8 Example. For a monoid H and the flat extension on the associated monad H =(H × (−), e,m), m is natural and e satisfies BC. However, it is easy to check that theidentity map f : (0, 1, a) −→ (0, 1, b), where a((α, 0), i) = > and a((α, 1), 1) = > only ifα = 1, and b((α, i), j) = > for every i, j ∈ 0, 1, α ∈ H, is closed and inversely closed butneither proper nor open.Nevertheless, in Proposition 3.4.5 below we will show that, in Top = (, 2)-Cat andApp = (,P+)-Cat, the condition that Tf be (inversely) closed may be replaced by thecondition that f be (inversely) closed.

3.4 The Kuratowski–Mrowka Theorem. We are now ready to characterize propermaps in terms of closure properties. The Kuratowski–Mrowka Theorem is the object versionof that characterization and facilitates the proof in the general case. For its proof we relyin turn on Proposition III.4.9.1 which asserts that, under the hypotheses that T preserve


disjointness, T be flat and e finitely strict, for any set X and x ∈ TX, one can define a(T,V)-space structure c = cx on Z = X + 1 by

c(z, z) =

k if z = eZ(z),> if z = x and z ∈ 1,⊥ otherwise,

for all z ∈ Z, z ∈ TZ. Whenever all (X + 1, c) are (T,V)-spaces, in particular under thehypotheses above, we say that (T,V)-Cat has enough KM-test spaces.3.4.1 Theorem. Let V be cartesian closed and let (T,V)-Cat have enough KM-test spaces.Then the following assertions for a (T,V)-space (X, a) are equivalent:

(i) (X, a) −→ (1,>) is proper;

(ii) the product projection X × Y −→ Y is closed for any (T,V)-space (Y, b).

When, in addition, V is integral and superior, we may extend the list by:

(iii) (X, a) is compact.

Proof. (i) =⇒ (ii) follows from Proposition 3.1.4(3) since X × Y −→ Y is a pullback ofX −→ 1. (i)⇐⇒ (iii) re-states Remark 3.1.2(2). For (ii) =⇒ (i) we must show∨

x∈Xa(x , x) = >

for all x ∈ TX. Hence we exploit the hypothesis for (Y, b) = (Z, cx ) and obtain, with∆ = (x, x) | x ∈ X ⊆ X × Z and q : (X × Z, d) −→ (Z, c) denoting the second productprojection,

c · Tq · Ti∆· !T∆ ≤ q · d · Ti∆· !T∆ . (3.4.i)With the natural bijection δ : X −→ ∆ every z ∈ T∆ is of the form z = Tδ(y ) with y ∈ TX,and Tp(z) = y = Tq(z) ∈ TX ⊆ TZ with p : X × Z −→ X the first projection. Hence, anexploitation of the elementwise form of (3.4.i) gives (with ∗ ∈ 1 ⊆ Z)

> = c(x , ∗) ≤ ∨x∈X ∨y∈TX d(Tδ(y ), (x, ∗))

= ∨x∈X

∨y∈TX a(y , x) ∧ c(y , ∗)

= ∨x∈X a(x , x) ,

since for every y 6= x one has c(y , ∗) = ⊥.

With Theorem 3.2.5 we conclude from Theorem 3.4.1 the desired characterization of propermaps, calling a map stably closed if all of its pullbacks are closed.3.4.2 Corollary. Let V be cartesian closed, T be taut with T∅ = ∅, T flat and left-whiskering, e finitely strict and m a natural transformation. Then the following assertionson a (T,V)-continuous map f : (X, a) −→ (Y, b) are equivalent:

(i) f is proper;


(ii) f is stably closed, and Tf : (TX, a) −→ (TY, b) is closed;

(iii) f−1y −→ 1 is proper for all y ∈ Y , and Tf is closed.

When, in addition, V is integral and superior, we may extend the list by:

(iv) f−1y is compact for all y ∈ Y , and Tf is closed.

Proof. (i) =⇒ (ii): From Propositions 3.1.4(4) and 3.2.2(1).(ii) =⇒ (iii): As a pullback of f , each fibre of f is stably closed and therefore proper, byTheorem 3.4.1.(iii) =⇒ (i): From Theorem 3.2.5.(iii)⇐⇒ (iv): From Remark 3.1.2(2).

3.4.3 Remark. The condition that Tf be closed can neither be removed from 3.4.2(ii)nor be substituted in (iii) by the requirement that f be closed. Indeed, going back toExample 3.3.8 one easily checks that f is stably closed but not proper, and Exercise 3.Dgives an example of a closed (T,V)-continuous map with compact fibres which is not proper.

When T = is the ultrafilter monad, with its Barr extension to V-Rel (see Corollary IV.2.4.5),for a (,V)-space (X, a) and x , y ∈ βX one has, by definition,

a(x , y ) =∨

X∈m−1X x

βa(X , y ) =∨

X∈m−1X x

∧A∈X ,B∈y

∨z∈A,y∈B

a(z, y) .

3.4.4 Lemma. If V is a chain (and, thus completely distributive), then

a(x , y ) =∨u ∈ V | ∀A ∈ x (A(u) ∈ y ) . (3.4.ii)

Proof. For “≤”, consider any X ∈ ββX with mX(X ) = x . It suffices to show that everyu ∧

A∈X ,B∈y∨

z∈A,y∈B a(z, y) has the property that A(u) ∈ y for all A ∈ x . But if forA ∈ x we assume A(u) 6∈ y , so that B := X \ A(u) ∈ y , considering

A := A = z ∈ βX | A ∈ z ∈ X (since A ∈ x )

we would concludeu

∨z∈A,y∈B

a(z, y)

and therefore A(u) ∩B 6= ∅, a contradiction.For “≥”, consider v ∨u ∈ V | ∀A ∈ x (A(u) ∈ y ) in V. For all A ∈ x , B ∈ y , theultrafilter y contains A(v) ∩B 6= ∅, so that v ≤ ∨z∈A a(z, y) for some y ∈ B, and

v ≤∧

B∈y

∨z∈A,y∈B

a(z, y)


follows for every A ∈ x . Now,

F = A ⊆ βX | A ⊆ A for some A ∈ x

is a filter on βX, and

J := B ⊆ βX | v >∧B∈y

∨z∈B,y∈B

a(z, y)

is an ideal on βX that is disjoint from F . In order to establish closure of J under binaryunion, we use the fact that the order of V is total, as follows: if B and C belong to J , thenv >

∨z∈B,y∈B a(z, y) and v >

∨z∈C,y∈C a(z, y) for some B,C ∈ y , hence

v >(∨

z∈B,y∈B a(z, y))∨(∨

z∈C,y∈C a(z, y))

≥ ∨z∈B∪C,y∈B∩C a(z, y) ,

and thenv >

∧D∈y

∨z∈B∪C,y∈D

a(z, y)

since B ∩ C belongs to y . The filter F must be contained in an ultrafilter X which doesnot meet the ideal J ; by definition of F one has x = mX(X ), and by definition of J

v ≤∧A∈X ,B∈y

∨z∈A,y∈B

a(z, y) ≤ a(x , y )

follows.

Since the structure a can be recovered from a as

a(x , x) = a(x , eX(x)) ,

the equality (3.4.ii) shows that v-closures on subsets of X encode completely the structurea.

3.4.5 Proposition. Let V be a chain and let f : (X, a) −→ (Y, b) be a (,V)-continuousmap. Then the following hold.

(1) f is closed⇐⇒ βf is closed⇐⇒ βf is proper.

(2) f is inversely closed⇐⇒ βf is inversely closed⇐⇒ f is open.

Proof. To see (1), we observe that with 3.3.7(1) it suffices to show that f is closed if andonly if βf is proper. First assuming f to be closed, we must verify

b(f [x ], y ) ≤∨a(x , z) | z ∈ βX, f [z] = y

for all x ∈ βX, y ∈ βY , where

b(f [x ], y ) =∨v ∈ V | ∀B ∈ f [x ] (B(v) ∈ y ) .


Since f is closed, for any v ∈ V contributing to the join on the right, one has f(A(u)) ∈ yfor all A ∈ x and every u v. Consequently, with an ultrafilter zu on X containing thefilter base

A(u) ∩ f−1(B) | A ∈ x , B ∈ y ,

one has f [zu] = y and A(u) ∈ zu for all A ∈ x , that is, u ≤ a(x , zu). Therefore v =∨uv u ≤ (βf · a)(x , y ), as desired.

Conversely, assuming βf to be proper we consider y ∈ f(A)(v) for A ⊆ X, v ∈ V , so that

v ≤∨

y3f(A) b(y , y) =∨

x 3Ab(f [x ], y) .

Sinceb(f [x ], y) = b(f [x ], eY (y)) =

∨a(x , z) | z ∈ βX, f [z] = eY (y)

with a(x , z) as in (3.4.ii), for all u v one obtains x u ∈ βX with f [x u] = eY (y) andA(u) ∈ x u, hence y ∈ f(A(u)).The proof of (2) is similar.

Finally we get a characterization theorem for (,V)-spaces that has as particular instancesthe well-known characterizations of proper maps in Top ∼= (, 2)-Cat and App ∼= (,P+)-Cat.

3.4.6 Theorem. Let V be totally ordered, integral and superior. The following assertionsare equivalent, for f : (X, a) −→ (Y, b) (,V)-continuous:

(i) f is proper;

(ii) f is stably closed;

(iii) f is closed and, for every y ∈ Y , the (,V)-functor f−1(y) −→ 1 is proper;

(iv) f is closed with compact fibres.

3.4.7 Examples.

(1) Proper maps in Top, introduced as the stably-closed maps by Bourbaki [1989] andcharacterized as the closed maps with compact fibres, have a description dual to openmaps (see Remark 3.3.5(2)) in (, 2)-Cat:

∀x ∈ βX, y ∈ Y (f [x ] −→ y =⇒ ∃ ∈ X (f(z) = y & x −→ z)) , x // z

f [x ] // y .

(2) When Top is presented as (Fp, 2)-Cat as in 1.1.4(4), then (Fp, 2)-proper maps arecalled superproper and characterized by the same property as in (1), except that x isnow allowed to be any proper filter. As compact spaces need not be supercompact,superproper is considerably stronger than proper.


(3) Closed maps f : (X, a) −→ (Y, b) in (,P+)-Cat ∼= App as introduced in 3.3.6 arecharacterized in terms of approach distances by

∀A ⊆ X, y ∈ Y (infx∈f−1y δX(x,A) ≤ δY (y, f(A)))

(see Proposition 3.3.2), and this is the description used in approach space theory.Hence, proper maps are also here characterized as the “classically” stably-closed maps,or the closed maps with 0-compact fibres.

(4) By Proposition 3.4.5, open maps f : (X, a) −→ (Y, b) in (,P+)-Cat ∼= App areequivalently described as the inversely closed maps which, in terms of approachdistances, leads to the classical description of open maps in App:

∀B ⊆ Y, x ∈ X (δX(x, f−1(B)) ≤ δY (f(x), B) .

3.5 Products of proper maps. Our goal is to prove that the product∏i∈I gi : ∏i∈I Xi −→∏i∈I Yi of proper maps gi : (Xi, ai) −→ (Yi, bi) is proper, subject to a condition on V that

entails its cartesian closedness and therefore makes proper maps stable under pullback in(T,V)-Cat (see Proposition 3.1.4). (Throughout this subsection, I is assumed to be small, aset.)Hence, we first point out that for any class P of morphisms in a small-complete category Xthat contains all isomorphisms and is stable under pullback, one has the following result.

3.5.1 Proposition. P is closed under small products if and only if P is stable under smallmultiple pullback in X, that is: for any small multiple pullback diagram

P

pi

f

##

Xifi // Y (i ∈ I)

(3.5.i)

in X, if fi ∈ P for all i ∈ I, then f ∈ P.

Proof. The multiple pullback f in (3.5.i) is a pullback of ∏i fi along δY : Y −→ Y I , as in

Pf

//

pi

Y

δY

1

∏iXi

∏ifi//

Y I

Xifi // Y .


Conversely, the product ∏i gi of morphisms gi : Xi −→ Yi is a multiple pullback of the familyof pullbacks hi : Pi −→

∏i Yi = Y of gi along the projection Y −→ Yi, as in

∏iXi

∏igi//

πi

∏i Yi

1

πi

Pihi //

Y

Xigi // Yi .

Hence, to verify closure of P under products, it suffices to prove stability under multiplepullbacks. Let us call V-Rel widely modular if for every non-empty family ri : Z −→7 Xi

(i ∈ I) of V-relations and every multiple pullback diagram (3.5.i) in Set, one has

f ·∧

i∈Ipi · ri =

∧i∈I

fi · ri (3.5.ii)

in V-Rel.

3.5.2 Proposition. If V-Rel is widely modular, the class of proper maps is stable undermultiple pullback in (T,V)-Cat.

Proof. The multiple pullback of fi : (Xi, ai) −→ (Y, b) in (T,V)-Cat is formed by providing thelimit P of the multiple-pullback diagram (3.5.i) in Set with the structure d = ∧

i∈I pi ·ai ·Tpi.

For I = ∅, the map f = 1Y is trivially proper. Otherwise, we may invoke the widemodularity of V and obtain

f · d = f ·∧ipi · ai · Tpi =

∧ifi · ai · Tpi =

∧ib · Tfi · Tpi = b · Tf

when all fi are proper. Hence, f is proper.

We must now analyze the status of the hypothesis on V-Rel in Proposition 3.5.2. We firstnote that this hypothesis constitutes no restriction if all fi are injective.

3.5.3 Remark. If all maps fi : Xi −→ Y are injective, then (3.5.ii) holds for all V-relationsri : Z −→ Xi (i ∈ I 6= ∅). Indeed, in that case we may assume that all maps of (3.5.i) areinclusion maps, with P = ⋂

i∈I Xi ⊆ Y , and for all z ∈ Z, y ∈ Y one has

(f · ∧i pi · ri)(z, y) =

∧i ri(z, y) if y ∈ P = ⋂

iXi,

⊥ otherwise= (∧i fi · ri)(z, y) .

Here is what unrestricted wide modularity of V-Rel means for V .

3.5.4 Proposition.c© V-Rel is widely modular if and only if V is completely distributive.


Proof. Given fi : Xi −→ Y and ri : Z −→7 Xi (i ∈ I 6= ∅) as in (3.5.ii), for all z ∈ Z, y ∈ Yone has

(f · ∧i∈I pi · ri)(z, y) = ∨w∈f−1y

∧i∈I ri(z, pi(w)) = ∨

(wi)∈∏if−1i y

∧i∈i ri(z, wi) and

(∧i∈I fi · ri)(z, y) = ∧i∈i

∨wi∈f−1

i y ri(z, wi) .

Complete distributivity of V makes the right-hand sides equal (see II.1.11). c©

Conversely, let us assume V-Rel to be widely modular and consider subsets Ai ⊆ V , i ∈ I.For I = ∅ one trivially has ∨

(ai)∈∏iAi

∧i∈I

ai =∧

i∈I

∨Ai , (3.5.iii)

and for I 6= ∅ one obtains this identity by considering the unique maps fi : Ai −→ 1 andthe V-relations ri : 1 −→7 Ai with ri(∗, ai) = ai for ai ∈ Ai, ∗ ∈ 1.

3.5.5 Remarks.(1) If V satisfies (3.5.iii) (for all Ai ⊆ V , i ∈ I), so does its complete sublattice 2 = ⊥,>.

But validity of (3.5.iii) for 2 is, in Zermelo–Fraenkel set theory, equivalent to theAxiom of Choice [Herrlich, 2006]. Hence, in ZF one can state Proposition 3.5.4 moreprecisely as:

V-Rel is widely modular ⇐⇒ AC holds & V is completely distributive .

(2) An injective proper (T,V)-continuous map f : (X, a) −→ (Y, b) is necessarily O-initial (for O : (T,V)-Cat −→ Set, see Exercises 3.A and 3.E), that is, an embedding.Proper embeddings are more commonly characterized as the closed embeddings (seeRemark 3.3.5(3)), and a multiple pullback of embeddings is better known as anintersection (see Theorem II.5.3.2).

3.5.6 Theorem.

(1) Closed embeddings in (T,V)-Cat are stable under intersections.

(2) If V is cartesian closed, then every product of closed embeddings is a closed embeddingin (T,V)-Cat.

(3) If V is completely distributive, then every product of proper (T,V)-continuous maps isproper.

Proof. Combine 3.3.3(3) with 3.5.1–3.5.5, observing also that complete distributivity impliesthat V , as a lattice, is a frame that is cartesian closed.

3.5.7 Corollary. c©Products of proper maps in Top and in App are proper.

Proof. Apply Theorem 3.5.6(2), noting that every chain is completely distributive; inparticular, 2 and P+ are so.


3.6 Coproducts of open maps. Throughout this subsection we assume that

• m : T −→ T T is a natural transformation;

• T is right-whiskering.

By Remarks III.4.3.4, these hypotheses guarantee that for a coproduct

ti : (Xi, ai) −→∐

i∈I(Xi, ai) = (X, a) (i ∈ I)

in (T,V)-Cat one hasa =

∨i∈I

ti · ai · (Tti) ,

with each ti open. These facts give us stability of openness under coproducts, as follows.

3.6.1 Proposition.

(1) If all fi : (Xi, ai) −→ (Y, b) in (T,V)-Cat are open, then the induced f : ∐i∈I(Xi, ai) −→(Y, b) is open.

(2) If all fi : (Xi, ai) −→ (Yi, bi) in (T,V)-Cat are open, then∐i∈I

fi :∐

i∈I(Xi, ai) −→

∐i∈I

(Yi, bi)

is open.

(3) Open embeddings are closed under set-theoretic union.

Proof.

(1) Since V-Rel is a quantaloid one has

f · b = (∨i∈I ti · ti ) · f · b= ∨

i∈I ti · ti · f · b= ∨

i∈I ti · f i · b= ∨

i∈I ti · ai · (Tfi) (fi open)= (∨i∈I ti · ai · (Tti)) · (Tf) .

(2) One applies (1) to the open maps

(Xi, ai)fi // (Yi, bi) //

∐i∈I(Yi, bi) .

(3) If in (1) all fi are inclusion maps, then the image of the open map f is precisely theunion ⋃i∈I Xi which (when provided with the O-initial structure of (Y, b)) is open inY .


The inclusion map A → X of a subset of a (T,V)-space (X, a) provided with the O-initialstructure is open if and only if

∀x ∈ X, x ∈ TX (a(x , x) > ⊥ & x ∈ A =⇒ x ∈ TA) , (3.6.i)

where we assume T (A → X) = (TA → TX) (see Exercise III.1.P). The open subsets of Xdefined in this way form a topology on X if T is taut. Denoting by Ω(X, a) the resultingtopological space, we obtain:

3.6.2 Corollary. If T is taut, then there is a functor

Ω : (T,V)-Cat −→ Top

that preserves the underlying sets and sends (T,V)-open maps to open maps. Ω also preservesand reflects coproducts.

Proof. Since T is taut, open embeddings in (T,V)-Cat are stable under pullbacks byProposition III.4.3.8, so they are stable under finite intersection. Since open embeddingsare also closed under unions (see Proposition 3.6.1), the set Ω(X, a) is in fact a topologicalspace with its open sets A defined by (3.6.i). Stability of open embeddings under pullbackalso shows that a (T,V)-continuous map f : (X, a) −→ (Y, b) gives a continuous mapΩf = f : Ω(X, a) −→ Ω(Y, b). As for the preservation of open subsets by Ωf when f is openin (T,V)-Cat, we note that for A ⊆ X open the composite morphism

(A // f(A) // (Y, b)) = (A // (X, a) f// (Y, b))

is open, so that f(A) is open in (Y, b), by Proposition 3.1.4(2).Since Ω preserves open maps, it preserves in particular open embeddings (see Exercise 3.E(4))and therefore the open injections of a coproduct in (T,V)-Cat. Preservation of coproductsfollows, and reflection thereof likewise.

3.6.3 Examples.

(1) For (T,V) = (I, 2), the functor Ω describes the full coreflective embedding Ord → Topwhich provides an ordered set with its Alexandroff topology, that is, its open sets arethe down-closed sets (see Examples III.3.5.2). For (T,V) = (, 2), since (3.6.i) describesopen sets in terms of ultrafilter convergence, Ω is the isomorphism (T,V)-Cat ∼= Topof Theorem III.2.2.5 expressed in terms of open sets.

(2) For (T,V) = (I,P+), Ω provides a metric space (X, a) with a rather crude topology:A ⊆ X is open if and only if

∀x, y ∈ X (a(x, y) <∞ & y ∈ A =⇒ x ∈ A) .

In particular, this choice of (T,V) shows that openness of Ωf does not imply opennessof f in general. In fact, if the metric a is finite, then ΩX is indiscrete. Hence,


openness of Ωf for f : (X, a) −→ (Y, b) in Met with a, b finite and X 6= ∅ justmeans that f is surjective which, in general, does not guarantee openness in Met (seeProposition 2.3.3(3)).

(3) A set C ⊆ X is open in (X, a) ∈ App ∼= (,P+)-Cat precisely when

∀B ⊆ X, x ∈ C (δ(x,C ∩B) ≤ δ(x,B)) ,

where δ is the associated approach distance of (X, a) (see 3.4.7(4)). Choosing B = yfor y ∈ X, we see that openness of C in X in App implies openness of C in X in Met,where X carries the metric a · eX (see III.3.5.2(2)). Consequently, for an approachspace X sent by the algebraic functor Ae : App −→ Met (and coreflector of Met → App)to a space with a finite metric, we obtain from (2) that the topological space ΩX isindiscrete.

While Examples (2) and (3) indicate the limitations of the functor Ω, we will use Ω as anessential tool in 5.3 below, especially when V = 2.

3.6.4 Remark. Taking as its closed sets those subsets A ⊆ X for which A → (X, a) isproper, one obtains a functor

Γ : (T,V)-Cat −→ Topthat sends (T,V)-proper maps to closed maps in Top, provided that T : Set −→ Set preservesfinite coproducts: see Exercise 3.F.

3.7 Preservation of space properties. For a (T,V)-continuous map f : (X, a) −→ (Y, b),we briefly discuss cases when the codomain inherits a special property from the domain,and vice versa. We remind the reader that injectivity of f is always inherited by Tf , andthe same is true for surjectivity if the Axiom of Choice is assumed.c©

3.7.1 Proposition.

(1) If f is injective and proper and (Y, b) is compact, then (X, a) is compact.

(2) If Tf is surjective and (X, a) is compact, then (Y, b) is compact.

Proof. (1) follows from Proposition 1.1.5. For (2), note simply

1TY = Tf · (Tf) ≤ Tf · a · a · (Tf) ≤ b · f · f · b ≤ b · b .

3.7.2 Proposition.

(1) If f is injective and (Y, b) Hausdorff, then (X, a) is Hausdorff.

(2) If Tf is surjective, f proper and (X, a) Hausdorff, then (Y, b) is Hausdorff.


Proof. (1) follows from Proposition 1.1.5. For (2), we have

1Y ≥ f · f ≥ f · a · a · f = b · Tf · (Tf) · b = b · b .

For preservation of normality and extremal disconnectedness, we first consider the caseT = I.

3.7.3 Proposition. Let (X, a), (Y, b) be V-spaces and f : (X, a) −→ (Y, b) a proper V-continuous map.

(1) If f is injective and (Y, b) normal, then (X, a) is normal.

(2) If f is surjective and (X, a) normal, then (Y, b) is normal.

Proof. (1) follows from

a · a ≤ f · b · f · f · b · f ≤ f · b · b · f≤ f · b · b · f = a · f · f · a = a · a .

(2) is proved analogously, using the equality f · f = 1Y .

3.7.4 Corollary. Let (X, a), (Y, b) be V-spaces and f : (X, a) −→ (Y, b) an open V-continuous map.

(1) If f is injective and (Y, b) extremally disconnected, then (X, a) is extremally discon-nected.

(2) If f is surjective and (X, a) extremally disconnected, then (Y, b) is extremally discon-nected.

Proof. Apply Proposition 3.7.3 to the V-functor f : (X, a) −→ (Y, b).

3.7.5 Theorem. Let m : T −→ T T be a natural transformation, T right-whiskering, andf : (X, a) −→ (Y, b) a (T,V)-continuous map.

(1) (a) If f is proper, Tf is injective and (Y, b) is normal, then (X, a) is normal.(b) If f is proper, Tf is surjective and (X, a) is normal, then (Y, b) is normal.

(2) (a) If f is open, Tf is injective and (Y, b) is extremally disconnected, then (X, a) isextremally disconnected.

(b) If f is open, Tf is surjective and (X, a) is extremally disconnected, then (Y, b) isextremally disconnected.


Proof. We observe that the claims of (1) follow from Proposition 3.7.3 applied to theV-continuous map Tf : (TX, a) −→ (TY, b), using commutativity of the diagram below:

TXTf

//

_mX

TY_ mY

TTXTTf

//

_T a

TTY_T b

TXTf

// TY .

(2) is proved analogously.

Exercises

3.A Closure operators.

(1) Show that, for v ≤ k in V , the v-closure and the grand closure provide hereditary butgenerally non-idempotent M-closure operators on (T,V)-Cat, with M the class ofembeddings. (For non-idempotency of the grand closure, see Exercise 3.B.)

(2) Show that proper and open embeddings can be characterized using the grand closureas stated in Remarks 3.3.3(3) and 3.3.5(3).

3.B Idempotency of the grand closure.

(1) For a monad T on Set with an associative lax extension to V-Rel, a (T,V)-space (X, a),and A ⊆ X with

TA ⊆ TA = y ∈ TX | ∃x ∈ TA (a(x , y ) > ⊥) ,

show A = A. Conclude that the grand closure is idempotent whenever TA ⊆ TA forall A ⊆ X.

(2) In App ∼= (,P+)-Cat, consider ([0,∞], a) with structure

a(u, v) = v ξ(u) ,

where ξ(u) = supC∈u infu∈C u for all u ∈ β[0,∞], v ∈ [0,∞] (see Exercise III.5.J).Show for A = 0:

(a) A = [0,∞),(b) βA 6⊆ βA,(c) A = [0,∞].


Hint. Consider an ultrafilter y on [0,∞] containing [0,∞) and all [v,∞], v < ∞.Then y ∈ βA, but y 6∈ βA.

3.C Openness of product projections. Suppose that the functor T satisfies BC.

(1) Prove that, if V is integral and T1 ∼= 1, then product projections are open in (T,V)-Cat.

(2) Show that both conditions are essential in the previous statement.

3.D Failure of the classical Kuratowski–Mrowka Theorem. Let H = N \ 0 be themultiplicative monoid of positive integers, and H the flat lax extension to Rel of theassociated monad H (see 1.4). Consider the (H, 2)-spaces (X, a), (2,>), with X = N∪∞,2 = 0, 1, and the only a-relations that hold are:

(α,∞) a∞ , (α, 0) a∞ , (α, 0) a 0 , (α, 0) a α , (α, n) a (α · n)

for α, n ∈ H ⊆ X. Show that the (H, 2)-continuous map f : (X, a) −→ (2,>), defined byf(0) = f(∞) = 0 and f(n) = 1 for n ≥ 1, is closed, has compact fibres, but is not proper.

3.E Injective and surjective proper and open maps. For f : (X, a) −→ (Y, b) in (T,V)-Catand O the underlying-set functor:

(1) f proper, f injective =⇒ f is O-initial,

(2) f proper, Tf surjective =⇒ f is O-final,

(3) f open, f surjective =⇒ f is O-final,

(4) f open, Tf injective =⇒ f is O-initial.

Furthermore, the O-final structure in (2) and (3) is described by b = f · a · (Tf).

3.F The functor Γ. For a (T,V)-space (X, a), declare A ⊆ X to be closed if A → (X, a) isproper (where A carries the O-initial (T,V)-structure). If T : Set −→ Set preserves finiteproducts, this defines the object part of a functor

Γ : (T,V)-Cat −→ Top

that sends (T,V)-proper maps to closed continuous maps. Describe this functor for T = Ior T = and V = 2 or V = P+.

3.G Closed subspaces of normal spaces. For T with an associative and left-whiskering laxextension to V-Rel, prove that closed subspaces of normal (T,V)-spaces are normal.

3.H Quasi-proper maps. A (T,V)-continuous map f : (X, a) −→ (Y, b) is quasi-proper ifb · Tf ≤ b · eY · f · a.

(1) Every quasi-proper map f : (X, a) −→ (Y, b) satisfies b · Tf = b · eY · f · a.


(2) Every proper map is quasi-proper.

(3) If T is associative, then every left adjoint map in (T,V)-Cat (see Exercise III.3.F) isquasi-proper.

(4) In Ord ∼= (I, 2)-Cat, the embedding 0, 1 → 0, 1, 2 is left adjoint but not proper,and 1, 2 → 0, 1, 2 is proper but not left adjoint.

(5) A monotone map f : X −→ Y of ordered sets is quasi-proper as a morphism in(P, 2)-Cat (see Example III.1.6.4(1)) if and only if f is left adjoint as a morphism in(I, 2)-Cat.

3.I Closure under composition. Show that the classes of closed maps, of inversely closedmaps and of quasi-proper maps are all closed under composition.

3.J Openness and near openness with respect to the filter monad. Show that in (F, 2)-Catand (Fp, 2)-Cat the notions of open map coincide with the usual topological notion. Nearlyopen maps in (Fp, 2)-Cat (see Definition III.4.3.1) are also open while in (F, 2)-Cat everymap is nearly open.

3.K Left- but not right-whiskering, and conversely.

(1) Prove that the lax extension P of the power-set monad to Rel (see Examples III.1.4.2)is left-, but not right-whiskering, and that the lax extension P behaves conversely.

(2) The Kleisli extension F of the filter monad is right- but not left-whiskering.

4. TOPOLOGIES ON A CATEGORY 429

4 Topologies on a category

In this subsection we give an axiomatic approach to considering objects in a category asspaces where the category comes equipped with a class of “proper maps”. This class, calledthe topology of the category, determines notions of compactness and separation and allowsus to exhibit their interrelations at both the object and morphism levels.

4.1 Topology, fibrewise topology, derived topology. The finitary stability propertiesof the classes of proper and of open maps in (T,V)-Cat studied in 3.1 lead us to consideringclasses of morphisms P in an arbitrary finitely complete category X and to call P a topologyon X if

(1) P contains all isomorphisms,

(2) P is closed under composition,

(3) P is stable under pullback.

For another morphism class E in X we call P an E-topology if in addition

(4) p · e ∈ P with e ∈ E implies p ∈ P .

Throughout this subsection we require that the class E itself is an E-topology. The presenceof such a class E constitutes no restriction of generality since every topology is an (Iso X)-topology. Note also that any pullback-stable class E that belongs to a factorization system(E ,M) is an E-topology (see Proposition II.5.1.1).

4.1.1 Examples. Let X = (T,V)-Cat and E be the class of epimorphisms (that is, surjective(T,V)-continuous maps). Then E is an E-topology, with respect to which we consider thefollowing classes P .

(1) The class P = Prop(T,V) of proper maps is an E-topology if V is cartesian closed: seeProposition 3.1.4.

(2) The class Open(T,V) of open maps is an E-topology if V is cartesian closed and Tsatisfies BC: see 3.1.4.

(3) For the forgetful functor O : (T,V)-Cat −→ Set, the class IniO of O-initial morphismsis an E-topology, and so are the classes Mono((T,V)-Cat) and RegMono((T,V)-Cat) =IniO ∩Mono((T,V)-Cat).

The class of closed continuous maps in Top satisfies properties (1), (2), (4) of an E-topology(E = Epi Top), but not (3) (for example, R −→ 1 is closed, but its pullback R ×R −→ R


along R −→ 1 is not). However, it contains a greatest pullback-stable subclass, the class ofall proper maps.In general, we call a class P in X an E-pretopology if it satisfies properties (1), (2), (4). Theclass P is hereditary if every pullback of a morphism in P along an embedding lies in P .

4.1.2 Proposition. An E-pretopology P on X contains a largest E-topology P∗ ⊆ P. Amorphism f lies in P∗ precisely when every pullback of f lies in P. If P is stable underpullback along split monomorphisms, in particular if P is hereditary, then f lies in P∗ ifand only if f × 1Z lies in P for every object Z in X.

Proof. For the “if” part of the last claim, we consider a pullback diagram

Pf ′

//

g′

Z

g

Xf

// Y

=

Pf ′

//

〈g′,f ′〉

Z

〈g,1Z〉

X × Z f×1Z //

Y × Z

Xf

// Y

(4.1.i)

and its decomposition into two pullback diagrams. The hypotheses yield

f ∈ P =⇒ f × 1Z ∈ P =⇒ f ′ ∈ P

since 〈g, 1Z〉 is split mono.

4.1.3 Corollary. The class Clo(T,V) of closed maps in (T,V)-Cat is a hereditary E-pretopology, with E the class of epimorphisms. One has

f ∈ Clo(T,V)∗ ⇐⇒ ∀(Z, c) ∈ (T,V)-Cat (f × 1Z is closed) ,

and if T = and V cartesian closed, then Clo(,V)∗ = Prop(,V).

Proof. For closure of Clo(T,V) under composition, see Exercise 3.I. If the composite map

(Z, c) e // (X, a) f// (Y, b)

is closed with e surjective, then for all A ⊆ X and C = e−1(A), one has

b · Tf · TiA· !TA = b · Tf · Te · TiC · !TC (Te is surjective)= f · e · c · TiC · !TC≤ f · b · Te · TiC · !TC= f · b · TiA· !TC


and therefore f is closed. For a pullback diagram

(W = f−1(Z), d) f ′//

_

(Z, c) _

(X, a) f// (Y, b)

(4.1.ii)

with f closed and c = iZ · b · TiZ , one has d = iW · a · TiW , and for all A ⊆ W one obtains

c · Tf ′ · TiA· !TA = iZ · b · Tf · TiW · TiA· !TA= iZ · f · a · TiW · TiA· !TA= f ′ · iW · a · TiW · TiA· !TA ((4.1.ii) is a BC-square)= f ′ · d · TiA· !TA ,

so that f ′ is closed. Consequently, Clo(T,V) is a hereditary E-pretopology. The remainingassertions follow from Proposition 4.1.2 and Theorem 3.4.6.

There are two ways of creating “new topologies from old” that are of particular importanceto us now. First, given an object Z, we can “slice at Z” any topology P of X, by consideringthe class

PZ := D−1Z (P)

where DZ : X/Z −→ X is the “domain functor”. The following result is easy to prove.

4.1.4 Proposition. If P is an E-topology of X, then PZ is an EZ-topology of X/Z, forevery object Z in X. The statement still holds if “E-topology” is replaced everywhere by“E-pretopology”.

We call PZ the fibrewise topology of P at Z (or the fibrewise pretopology accordingly).Less trivially, for any class P , one may consider the derived class

P ′ = f : X −→ Y | δf = 〈1X , 1X〉 : X −→ X ×Y X lies in P .

4.1.5 Proposition. For a topology P of X, the derived class P ′ is also a topology of X thatmoreover contains all monomorphisms and satisfies the cancellation condition

g · f ∈ P ′ =⇒ f ∈ P ′ .

If P is an E-topology, then P ′ is an (E ∩ P)-topology.

We call P ′ the derived topology of P .


Proof. A morphism f is a monomorphism if and only if δf is an isomorphism. Hence, P ′contains all monomorphisms. For f : X −→ Y and g : Y −→ Z in X, let h = g · f . There is aunique morphism t : X ×Y X −→ X ×Z X with h1 · t = f1, h2 · t = f2, where (f1, f2) and(h1, h2) form the kernel pairs of f and h, respectively. The right square of

Xδf//

1X

X ×Y Xf ·f1

//

t

Y

δg

Xδh // X ×Z X

f×f// Y ×Z Y

(4.1.iii)

is a pullback diagram since t is the equalizer of f · h1 and f · h2. Consequently, one has theimplications

f, g ∈ P ′ =⇒ δf , δg ∈ P =⇒ δf , t ∈ P =⇒ δh = t · δf ∈ P =⇒ h ∈ P ′ .

Since t is monic, the left square of (4.1.iii) is also a pullback diagram, and one concludes

g · f ∈ P ′ =⇒ δh ∈ P =⇒ δf ∈ P =⇒ f ∈ P ′ .

The pullback diagram of (f, g) can be factorized as the following outer pullback diagram:

Pδf ′//

g′

P ×Z P////

g′′

Pf ′//

g′

Z

g

Xδf// X ×Y X

//// X

f// Y .

Since the two rightmost diagrams form a pullback diagram, the left square is also one, so

f ∈ P =⇒ f ′ ∈ P .

Finally, when P is an E-topology, set h = g · f ∈ P ′ with f ∈ E ∩ P . From (4.1.iii) one hasδg · f = (f × f) · δh, with δh ∈ P and f × f = (f × 1) · (1× f) ∈ P (as the composite oftwo pullbacks of f). Since f ∈ E , one has δg ∈ P , that is, g ∈ P ′.

4.1.6 Remark. One has:

(Iso X)′ = Mono X and (SplitMono X)′ = mor X .

In particular, for any class P containing Iso X,

(P ′)′ = mor X ,

and for the forgetful functor O : (T,V)-Cat −→ Set,

(IniO)′ = mor(T,V)-Cat .


4.2 P-compactness, P-Hausdorffness. Let P be a topology of the finitely completecategory X. An object X is P-compact if

(X −→ 1) ∈ P

(with 1 a terminal object in X), and X is P-Hausdorff if

(X −→ 1) ∈ P ′ ,

that is, if (δX : X −→ X ×X) ∈ P .A morphism f : X −→ Y is P-proper if f is PY -compact in X/Y (see Proposition 4.1.2), fis P-Hausdorff if f is PY -Hausdorff in X/Y , and f is P-perfect if f is both P-proper andP-Hausdorff.Since 1Y is a terminal object in X/Y and f is the unique morphism f −→ 1Y in X/Y , onesees immediately that

f is P-proper ⇐⇒ f ∈ P ,

f is P-Hausdorff ⇐⇒ f ∈ P ′ ⇐⇒ (δf : X −→ X ×Y X) ∈ P .

We examine these notions first in terms of the principal topologies for (T,V)-Cat as consideredin Examples 4.1.1 and compare them with those introduced in 1.1.

4.2.1 Proposition. Let V be a cartesian closed quantale, and (X, a) a (T,V)-space.

(1) (X, a) is Prop(T,V)-compact if and only if

∀x ∈ TX (> ≤∨

z∈Xa(x , z)) .

(2) Suppose that V is integral; if (X, a) is compact then (X, a) is Prop(T,V)-compact,with the converse statement holding when V is superior.

(3) (X, a) is Prop(T,V)-Hausdorff if and only if

∀x, y ∈ X ∀z ∈ TX (⊥ < a(z, x) ∧ a(z, y) =⇒ x = y) .

(4) Suppose that V is integral; if (X, a) is Prop(T,V)-Hausdorff then (X, a) is Hausdorff,with the converse statement holding if the map o : V −→ 2 with (o(v) = ⊥ ⇐⇒ v = ⊥)is a lax homomorphism of quantales, that is, if V satisfies (u ⊗ v = ⊥ =⇒ u =⊥ or v = ⊥).

(5) A (T,V)-continuous map f : (X, a) −→ (Y, b) is Prop(T,V)-Hausdorff if and only if

∀x, y ∈ X ∀z ∈ TX (f(x) = f(y) & (⊥ < a(z, x) ∧ a(z, y) =⇒ x = y)) .


Proof. To see (1), observe that (!X : (X, a) −→ (1,>)) ∈ Prop(T,V) means > · T !X ≤ !X · a;this, when stated elementwise, reads as claimed.For (3), consider the structure b = (p · a · Tp) ∧ (q · a · Tq) of X ×X (with projectionsp, q); then (δX : (X, a) −→ (X ×X, b)) ∈ Prop(T,V) means b · TδX ≤ δX · a. Since for allx, y ∈ X, z ∈ TX,

(b · TδX)(z, (x, y)) = a(Tp · TδX(z), x) ∧ a(Tq · TδX(z), y) = a(z, x) ∧ a(z, y) ,

(δX · a)(z, (x, y)) = ∨z∈δ−1

X (x,y) a(z, z) =

a(z, x) if x = y,

⊥ otherwise,

the criterion follows.Finally, (2) and (4) follow from Proposition 1.1.2, and (5) is proved as (3).

4.2.2 Examples.

(1) Since 2 and P+ satisfy all additional hypotheses used in Proposition 4.2.1, in allexamples presented in 1.1.4 the notions of Prop(T,V)-compactness and Prop(T,V)-Hausdorffness for objects are equivalent to the notions of compactness and Hausdorff-ness of 1.1.

(2) For V integral and T = I (identically extended to V-Rel), a V-functor f : (X, a) −→ (Y, b)is Prop(T,V)-Hausdorff if every fibre of f (as a subspace of (X, a)) is discrete.

(3) In Top ∼= (, 2)-Cat, f : X −→ Y is Prop(, 2)-Hausdorff if and only if any two distinctpoints in the same fibre of f may be separated by disjoint open neighborhoods in X.Such maps are usually called separated in the literature on fibred topology (see [James,1989]). In App ∼= (,P+)-Cat, a map f is Prop(,P+)-Hausdorff if and only if, for allx, y ∈ X, z ∈ βX with f(x) = f(y), a(z, x) <∞ and a(z, y) <∞, one has x = y.

4.2.3 Proposition. Suppose that V is cartesian closed and that T satisfies BC.

(1) A (T,V)-space (X, a) is Open(T,V)-compact if and only if

∀a ∈ T1, x ∈ X (> ≤∨

x ∈(T !X)−1aa(x , x)) ,

where !X : X −→ 1.

(2) If V is integral and T1 ∼= 1, then every (T,V)-space is Open(T,V)-compact.

(3) An Open(T,V)-continuous map f : (X, a) −→ (Y, b) is Open(T,V)-Hausdorff if andonly if

∀x ∈ X ∀w ∈ T (X ×Y X) (⊥ < a(Tp(w), x) ∧ a(Tq(w), x) =⇒ w ∈ T∆X) ,

where p, q : X ×Y X −→ X are the projections, and w ∈ T∆X means w = Tδf (x ) forsome (uniquely determined) x ∈ TX.


Proof.

(1) The given criterion expresses !X · > ≤ a · (T !X) in pointwise form.

(2) If T1 ∼= 1, then (T !X)(eX(x)) is the only element in T1.

(3) Formulating δf · b ≤ a · (Tδf ) with b = (p · a · Tp)∧ (q · a · Tq) (p, q : X ×Y X −→ Xprojections) in pointwise form, one obtains

a(Tp(w), x) ∧ a(Tq(w), x) =

a(x , x) if ∃x ∈ TX (Tδf (x ) = w),⊥ otherwise,

for all w ∈ T (X ×Y X), x ∈ X. Since in the first case (that is, when w ∈ T∆X) theequality holds trivially, one obtains the criterion as stated.

4.2.4 Examples.

(1) For V integral and T = I, a V-functor is Open(I,V)-Hausdorff if and only if its fibres arediscrete. (Hence, Open(I,V)-Hausdorffness is equivalent to Prop(I,V)-Hausdorffness,see Example 4.2.2(2).)

(2) In Top ∼= (, 2)-Cat, a continuous map f : X −→ Y is Open(, 2)-Hausdorff if andonly if the diagonal ∆X ⊆ X ×Y X is open, and this means equivalently that forevery point in X there is a neighborhood U of x such that f |U : U −→ X is injective;such maps are called locally injective. When applied to X −→ 1, this yields that X isOpen(, 2)-Hausdorff if and only if X is discrete.

(3) An Open(,P+)-Hausdorff morphism f : (X, a) −→ (Y, b) has the property that forevery ultrafilter w on X ×Y X that converges to (x, x) in the induced pretopology(see III.3.6 and Example III.4.1.3), so that p[w ] −→ x and q[w ] −→ x, one has∆X ∈ w . Consequently, the neighborhood filter of any diagonal point (x, x) in thepseudotopological space X ×Y X contains ∆X . Hence, Open(,P+)-Hausdorffnessis characterized in App by local injectivity, with “local” referring to the inducedpretopology. In particular, Open(,P+)-Hausdorff approach spaces are discrete.

4.2.5 Remark. Only indiscrete (T,V)-spaces are (IniO)-compact, but every (T,V)-spaceis (IniO)-Hausdorff.

4.3 A categorical characterization theorem. The following easy-to-prove theoremcollects important characteristic properties of P-compact objects in any finitely completecategory X with an E-topology P :

4.3.1 Theorem. The following assertions for an object X are equivalent:

(i) X is P-compact;


(ii) every morphism f : X −→ Y with Y P-Hausdorff is P-proper;

(iii) there is a P-proper morphism f : X −→ Y such that Y is P-compact;

(iv) the projection X × Y −→ Y is P-proper for all objects Y ;

(v) X × Y is P-compact for every P-compact object Y ;

(vi) for every morphism e : X −→ Y in E, the object Y is P-compact.

Proof. (i) =⇒ (ii): In the graph factorization

X × Yp

X

〈1X ,f〉>>

f// Y .

〈1X , f〉 is in P as a pullback of δY , and p is in P as a pullback of X −→ 1.(ii) =⇒ (iii): Consider Y = 1.

(iii) =⇒ (i): (X // 1 ) = (X f// Y // 1).

(i) =⇒ (iv): p is a pullback of X −→ 1.(iv) =⇒ (v): (X × Y // 1) = (X × Y // Y // 1).(v) =⇒ (i): Consider Y = 1.(i) =⇒ (vi): Apply the E-topology property to !X = !Y · e.(vi) =⇒ (i): Consider e = 1X .

4.3.2 Corollary. Let the composite morphism g ·f be P-proper, with g P-Hausdorff. Thenf is P-proper.

Proof. Using Proposition 4.1.4, apply Theorem 4.3.1(i) =⇒ (ii) to the morphism f : g·f −→ gin X/ cod(g).

Next we apply Theorem 4.3.1 with P ′ in lieu of P. With Proposition 4.1.5 we obtain thefollowing characterization of P-separated objects.

4.3.3 Corollary. The following assertions for an object X are equivalent:

(i) X is P-Hausdorff;

(ii) every morphism f : X −→ Y is P-Hausdorff;

(iii) there is a P-Hausdorff morphism f : X −→ Y such that Y is P-Hausdorff;

(iv) the projection X × Y −→ Y is P-Hausdorff for all objects Y ;


(v) X × Y is P-Hausdorff for every P-Hausdorff object Y ;

(vi) for every P-proper morphism e : X −→ Y in E, the object Y is P-Hausdorff;

(vii) for every equalizer diagram Eu // Z

//// X , the morphism u is P-proper.

Proof. We just observe that P ′-compact translates to P-Hausdorff for objects, and P ′-properto P-Hausdorff for morphisms. Furthermore, since P ′ contains all monomorphisms, allobjects and morphisms are P ′-Hausdorff. Hence, the equivalence of (i)–(vi) follows fromTheorem 4.3.1. For the equivalence (i)⇐⇒ (vii), we just note that equalizers as in (vii) areprecisely the pullbacks of δX : X −→ X ×X.

4.3.4 Corollary. For a P-perfect morphism f : X −→ Y , one has

Y is P-compact and P-Hausdorff =⇒ X is P-compact P-Hausdorff ,

with the converse implication holding when f lies in E.

Proof. Use (iii) =⇒ (i) =⇒ (vi) of both Theorem 4.3.1 and Corollary 4.3.3.

4.3.5 Corollary. The full subcategory of P-compact objects is closed under finite productsin X, and the full subcategories of P-Hausdorff objects and of P-compact P-Hausdorffobjects are both closed under finite limits in X.

Proof. Closure under finite products for all three subcategories in question follows from(i)⇐⇒ (v) of 4.3.1 and 4.3.3. Closure under equalizers for the two subcategories mentionedfollows from (i)⇐⇒ (iii)⇐⇒ (vii) of 4.3.3 and (i)⇐⇒ (iii) of 4.3.1.

4.3.6 Remarks.

(1) P-Hausdorffness generally fails to be closed under infinite products, also in conjunctionwith P-compactness, as the example X = Top and P = Open(, 2) shows: a countablepower of a two-point discrete space is no longer discrete.

(2) Theorem 4.3.1 and its corollaries may, of course, be readily applied to the topologiesP = Prop(T,V) and P = Open(T,V), provided that V is cartesian closed and (forP = Open(T,V)) that T satisfies BC. We do not formulate explicitly the emergingfinitary stability properties in these two important cases. However, in what followswe highlight the most important infinite stability properties available for these twogeneral choices of P .

4.3.7 Proposition. For V cartesian closed, the full subcategory of Prop(T,V)-Hausdorffspaces is closed under mono-sources in (T,V)-Cat, hence is strongly epireflective in (T,V)-Cat.


Proof. Let (fi : (X, a) −→ (Yi, bi))i∈I be a point-separating source in (T,V)-Cat, with all(Yi, bi) Prop(T,V)-Hausdorff. Hence, for x 6= y in X there is j ∈ I with fj(x) 6= fj(y).Consequently, for all z ∈ TX, one has

a(z, x) ∧ a(z, y) ≤ bj(Tfj(z), fj(x)) ∧ bj(Tfj(z), fj(y)) = ⊥ .

Hence, by Proposition 4.2.1(3), the space (X, a) is Prop(T,V)-Hausdorff.

With Theorem 3.5.6 we obtain, of course, a Tychonoff Theorem for Prop(T,V)-compactness,and then with Proposition 4.2.1 also for compactness.

4.3.8 Corollary. Let V be completely distributive. Then every product of Prop(T,V)-compact (T,V)-spaces is Prop(T,V)-compact. If V is also integral and superior, then(T,V)-CatComp is closed under products in (T,V)-Cat, and likewise for (T,V)-CatHaus and(T,V)-CatCompHaus.

4.4 P-dense maps, P-open maps. Given an E-topology P on a finitely complete cate-gory X, we wish to develop an internal notion of P-open map in X. A crucial step towardsthis goal is the introduction of P-dense maps which, in the case of the principal exampleP = Prop(T,V), will be described via suitable closure operators.

4.4.1 Definition. A morphism f in X is P-dense if in any factorization f = p · h withp ∈ P one has p ∈ E . The class of all P-dense maps in X is denoted by P d.

4.4.2 Proposition. Consider a composite morphism g · f in X.

(1) If g · f ∈ P d, then g ∈ P d.

(2) If f ∈ P d and g ∈ E, then g · f ∈ P d.

(3) If X has pushouts of morphisms in E and these belong to E again, then f ∈ E andg ∈ P d implies g · f ∈ P d.

(4) E ⊆ P d if and only if P ∩ SplitEpi X ⊆ E.

(5) If (E ,M) is a factorization system, then f ∈ P d if and only if in any factorizationf = p · h with p ∈ P ∩M, p is an isomorphism.

Proof. (1) is trivial. For (2) and (3), consider the diagrams

·p

· s //

h

55

f))

·g′

@@

p′

·

·g

@@

·

f ′

p

))·

h

@@

f

· t // ·

·

h′@@

g

55

(4.4.i)


with p ∈ P , the square being a pullback in the left diagram and a pushout in the right one,and with s, t induced morphisms. From f ∈ P d, p′ ∈ P, one has p′ ∈ E , and from g ∈ Efollows g′ ∈ E and, hence, p · g′ ∈ E so finally p ∈ E (since E is an E-topology). Similarly,from f ∈ E one obtains f ′ ∈ E and then t ∈ P (since P is an E-topology), which impliest ∈ E when g ∈ P d and, hence, p ∈ E .To see (4), remark that if E ⊆ P d, then every identity morphism is P-dense. Hence, ifp · s = 1 with p ∈ P, we can conclude p ∈ E . Conversely, P ∩ SplitEpi X ⊆ E implies thatidentity morphisms lie in P d which, with (2), implies E ⊆ P d.(5) follows easily from E ∩M = Iso X and the fact that, in an (E ,M)-factorization p = m · eof p ∈ P , m is also in P .

By consideration of the factorizationf = (X −→ f(X) → Y ) ,

the P-dense maps in Top with P = proper maps are easily identified as the dense maps,that is, as those f with f(X) = Y . Using the grand closure introduced in 3.3 we mayproceed in the same way in (T,V)-Cat but must take into account that this closure may failto be idempotent (see Exercise 3.A). Hence we consider the idempotent hull of the grandclosure,

A∞ =

⋂B | A ⊆ B ⊆ X,B = B

for any A ⊆ X, whose inclusion map into X is (T,V)-proper by Remark 3.3.3(3). Obviously,A = A ⇐⇒ A = A

∞.

4.4.3 Lemma. An embedding A → (X, a) is Prop(T,V)-dense if and only if A∞ = X.

Proof. The embedding A → X factors through A∞→ X. If A → X is Prop(T,V)-dense,

then A∞ = X follows. Conversely, assuming A∞ = X and considering any (T,V)-proper

B → X with A ⊆ B, we have B = B and, hence, A = A∞ ⊆ B = B.

4.4.4 Corollary. A (T,V)-continuous map f : (X, a) −→ (Y, b) is Prop(T,V)-dense if andonly if f(X)∞ = Y .

Guided by Proposition 3.3.4, we can now introduce P-open maps, as follows.4.4.5 Definition. A morphism f : X −→ Y in X is P-open if, for every pullback f ′ of f ,pulling back along f ′ preserves P-density, that is: if for all stacked pullback diagrams

· //

d′

·d

· f ′//

·

· f

// ·d ∈ P d implies d′ ∈ P d. We let P o denote the class of P-open maps in X.


4.4.6 Proposition. For an E-topology P on X, P o is also an E-topology on X.

Proof. P o is clearly a topology. Since any pullback of f · e with e ∈ E is of the form f ′ · e′with e′ ∈ E , it suffices to show that P-density pulls back along f if it does so along f · e.But for the pullback diagrams

· e′ //

d′′

· f ′//

d′

·d

· e// ·

f// ·

d ∈ P d implies d′′ ∈ P d, which gives e · d′′ = d′ · e′ ∈ P d by Proposition 4.4.2(2) andtherefore d′ ∈ P d.

Our goal must now be to describe the Prop(T,V)-open maps, for a cartesian closed andintegral quantale V and a lax extension T of the Set-monad T = (T,m, e). For conveniencewe also assume T to satisfy the Beck–Chevalley condition, so that not only Prop(T,V) butalso Open(T,V) is an E-topology on (T,V)-Cat (see Examples 4.1.1). Since the class P o isfully determined by the class P d which, in turn, is fully determined by the grand closurewhen P = Prop(T,V) (see Remarks 3.3.3), Prop(T,V)-open maps retrieve only very littleinformation of the quantale V , unless V = 2. To make this statement more precise, let usintroduce the following auxiliary notion.

4.4.7 Definition. A (T,V)-continuous map f : (X, a) −→ (Y, b) is called 2-open if, for allx ∈ X, y ∈ TY ,

b(y , f(x)) > ⊥ =⇒ ∃x ∈ TX (Tf(x ) = y & a(x , x) > ⊥) .

The following remarks explain this terminology.

4.4.8 Remarks.

(1) Since V is integral, the quantale homomorphism ι : 2 −→ V has (as a monotone map)a left adjoint o (with o(v) = > ⇐⇒ v > ⊥) which is a quantale homomorphism ifand only if V satisfies

u⊗ v = ⊥ =⇒ u = ⊥ or v = ⊥ (4.4.ii)

(see Example III.3.5.2(1)).

(2) The lax extension T : V-Rel −→ V-Rel may be restricted to Rel = 2-Rel; one simplyconsiders the composite lax functor

T : (Rel ι // V-Rel T // V-Rel o // Rel)r // ιr // T (ιr) // oT (ιr) ,


that is,x (T r) y ⇐⇒ T (ιr)(x , y ) > ⊥ .

Although T is a lax extension of T (in the sense of III.1.4.1) which makes e : 1Rel −→ T

oplax, T generally fails to make the monad multiplication m oplax. Still, with respectto T , we can form the category (T, 2)-Gph of (T, 2)-graphs as in III.4.1 and obtain thefunctor

(T,V)-Cat −→ (T, 2)-Gph , (X, a) 7−→ (X, oa) .For example, for the Barr extension of the ultrafilter monad to P+-Rel, one obtainsthe “induced-pseudotopology” functor

App −→ PsTop ,

which we encountered in III.3.6.

(3) Defining a morphism of (T, 2)-graphs to be open as in the case of (T,V)-categories,one can now state that, for a (T,V)-functor f : (X, a) −→ (Y, b),

f is 2-open ⇐⇒ f is open in (T, 2)-Gph .

(4) If V = 2, then T = T , and the notion of 2-openness returns precisely the notion ofopenness in (T, 2)-Cat.

4.4.9 Proposition. Let V be cartesian closed and integral, satisfying (4.4.ii), and let Tsatisfy BC. Then, for the following statements on a (T,V)-continuous map f of (T,V)-categories, one has (i) =⇒ (ii) =⇒ (iii), with (i)⇐⇒ (ii) holding if V = 2, and (ii)⇐⇒ (iii)if T = I:

(i) f ∈ Open(T,V);

(ii) f is 2-open;

(iii) f is Prop(T,V)-open.

Proof. The claims about (i), (ii) are obvious (see Remarks 4.4.8). Now we consider a 2-open(T,V)-functor f : (X, a) −→ (Y, b) and first show (using the grand closure of 3.3)

f−1(N) ⊆ f−1(N)

for all N ⊆ Y . Indeed, x ∈ f−1(N) implies b(y , f(x)) > ⊥ for some y ∈ TN ⊆ TY , and2-openness gives some x ∈ TX with Tf(x ) = y and a(x , x) > ⊥. Moreover, since T istaut, x ∈ (Tf)−1(TN) = Tf−1(N), which implies x ∈ f−1(N).By ordinal recursion one concludes

f−1(N∞) ⊆ f−1(N)∞


for all N ⊆ Y , which implies that pulling back along f preserves Prop(T,V)-density (seeCorollary 4.4.4). This property holds in fact for every pullback f ′ of the 2-open map f ,since 2-openness is stable under pullback, thanks to T satisfying BC, as one easily shows asin Proposition 3.1.4(4). Consequently, f is Prop(T,V)-open.Let now T = I, identically extended to V-Rel. Condition (4.4.ii) guarantees that the grandclosure is idempotent in this case. Assuming now f : (X, a) −→ (Y, b) in V-Cat to beProp(I,V)-open, we consider x ∈ X, y ∈ Y with b(y, f(x)) > ⊥ and let f ′ be the restrictionf−1Z −→ Z of f , with the subspace Z = y, f(x) of Y . Trivially, as a subspace of Z, yis dense in Z, so that f−1y is dense in f−1Z. Hence, x ∈ f−1Z = f−1y gives z ∈ f−1y witha(z, x) > ⊥, which shows that f is 2-open.

4.4.10 Examples.

(1) A 2-open map may fail to be open, even when T = I. Indeed, for R with its Euclideanmetric d, all surjective non-expansive maps f : R −→ R are 2-open in Met = P+-Cat,but among them only those with

d(y, f(x)) = infz∈f−1y d(z, x)

are also open. Already, f(x) = 12x fails to be open in P+-Cat.

(2) For T = and V = 2, so that (T,V)-Cat ∼= Top, all three conditions of Proposition 4.4.9are equivalent. The only critical implication left to be shown is (iii) =⇒ (ii). Wealready know that (i) (or, equivalently, (ii)) describes open maps in Top, in theusual sense that images of open sets are open: see Remarks 3.3.5. Hence, assuming(iii), consider an open set A ⊆ X and let N = Y \ f(A). Since the density of Nin N pulls back along the restriction f ′ : f−1(N) −→ N of f , one has that f−1(N)is dense in f−1(N), that is: f−1(N) = f−1(N) ⊆ X \ A = X \ A and thereforeY \ f(A) ⊆ Y \ f(A), so that f(A) is open in Y .

(3) Consider the monad H with a monoid H as in 1.4. A 2-open morphism f : X −→ Y in(H, 2)-Cat is (in the notation of 1.4) easily characterized by

yα−→ f(x) =⇒ ∃z ∈ f−1y (z α−→ x) ,

whereas a Prop(H, 2)-open map is described by

yα−→ f(x) =⇒ ∃z ∈ f−1y, β ∈ H (z β−→ x) ,

as one shows similarly to the proof of 4.4.9 (iii) =⇒ (ii). However the two notions areequivalent when f is an embedding.

(4) For the list monad L (see 1.4), a Prop(L, 2)-open map may fail to be open as well: seeExercise 4.E. However, Prop(L, 2)-open embeddings are open in (L, 2)-Cat.


Returning to the general setting of an E-topology P in a category X, with the “new” E-topology P o at hand, one may now consider the derived topology (P o)′ (see Proposition 4.1.5)and explore the notions of P o-compactness and P o-Hausdorffness. While we must leave it tothe reader to pursue this program in general, let us take a glimpse at X = Top ∼= (, 2)-Catwith P = proper maps = Prop(, 2) again, so that P o = open maps = Open(, 2).In this case, every object is P o-compact, while P o-Hausdorffness means discreteness (seeExample 4.2.4(2)). Furthermore, for a continuous map f : X −→ Y one has:

f is P o-Hausdorff ⇐⇒ f is locally injective (see Example 4.2.4(2)),f is P o-perfect ⇐⇒ f is P o-proper and P o-Hausdorff

⇐⇒ f is open and locally injective⇐⇒ f is a local homeomorphism,

where f is a local homeomorphism if every point in X has an open neighborhood U suchthat f(U) is open and the restriction U −→ f(U) of f is a homeomorphism.

4.5 P-Tychonoff and locally P-compact Hausdorff objects. We continue to workin a finitely complete category X equipped with an E-topology; furthermore, we now assumethat E belongs to a proper (E ,M)-factorization system of X, with E stable under pullback.(In this case, E is automatically an E-topology.)We consider the classes

(P ∩ P ′) · M = p ·m | m ∈M, p is P-perfect ,

(P ∩ P ′) · (M∩P o) = p ·m | m ∈M is P-open, p is P-perfectand note that they contain all isomorphisms and are stable under pullback, but are notnecessarily closed under composition.For an object X and the unique morphism !X : X −→ 1, one has

!X ∈ (P ∩ P ′) · M ⇐⇒ ∃m : X −→ K (m ∈M, K is P-compact & P-Hausdorff) ,!X ∈ (P ∩ P ′) · (M∩P o) ⇐⇒ ∃m : X −→ K (m ∈M, K is P-compact & P-Hausdorff) .

Guided by the role model Top with P = proper maps and E = surjective maps, wherefor a space X,

X is Tychonoff ⇐⇒ X subspace of a compact Hausdorff space,X is locally compact Hausdorff ⇐⇒ X open subspace of a compact Hausdorff space,

in our abstract category X we say that a morphism

f is P-Tychonoff ⇐⇒ f ∈ (P ∩ P ′) · M ,

f is locally P-perfect ⇐⇒ f ∈ (P ∩ P ′) · (M∩P o) ,


and an object

X is P-Tychonoff ⇐⇒ !X is P-Tychonoff,X is locally P-compact Hausdorff ⇐⇒ !X is locally P-perfect.

Although the two classes of morphisms under consideration generally fail to be topologies,one may establish characteristic properties similar to those of Theorem 4.3.1, as follows.

4.5.1 Proposition. The following assertions for an object X are equivalent:

(i) X is P-Tychonoff;

(ii) every morphism f : X −→ Y is P-Tychonoff;

(iii) there is a P-Tychonoff morphism f : X −→ Y with Y P-compact P-Hausdorff;

(iv) the projection X × Y −→ Y is P-Tychonoff for all objects Y ;

(v) X × Y is P-Tychonoff for every P-Tychonoff object Y .

Proof. (i) =⇒ (ii): With m : X −→ K in M and K P-compact P-Hausdorff, we considerthe diagram

Xf

yy

m

%%

〈f,m〉

Y Y ×Kp1oo

p2// K .

Since p2 · 〈f,m〉 = m ∈M also 〈f,m〉 ∈ M (see Proposition II.5.1.1(3)), and p1 ∈ P ∩ P ′by Theorem 4.3.1 and Corollary 4.3.3.(ii) =⇒ (iii), (iv) =⇒ (i), (v) =⇒ (i): Choose Y = 1.(iii) =⇒ (i): By hypothesis, f = p · m with m ∈ M and p : Z −→ Y in P ∩ P ′. ByTheorem 4.3.1 and Corollary 4.3.3, p transfers the needed properties from Y to Z.(i) =⇒ (iv): X × Y −→ Y is a pullback of X −→ 1.(i) =⇒ (v): If m : X −→ K, n : Y −→ L are in M, and K, L are P-compact P-Hausdorff,then m× n = (m× 1L) · (1X × n) is in M, and K × L is P-compact P-Hausdorff.

4.5.2 Corollary. If the composite morphism g · f is P-Tychonoff, f is also P-Tychonoff.

Proof. One argues as in the proof of Corollary 4.3.2.

In order to establish analogous properties for locally P-compact Hausdorff objects we needan additional hypothesis, as follows. We say that X has the P-open-closed interchangeproperty if every composite morphism (X m // Y

n // Z ) with m ∈M∩P and n ∈M∩P o

may be rewritten as (X n′ //Wm′ // Z ) with n′ ∈M∩P o and m′ ∈M∩P . In the role


model Top, if X ⊆ Y is a closed subspace and Y ⊆ Z an open subspace, one may chooseW = X as the closure of X in Z.

4.5.3 Proposition. If X has the P-open-closed interchange property, the following condi-tions are equivalent for an object X:

(i) X is locally P-compact Hausdorff;

(ii) every morphism f : X −→ Y with Y P-Hausdorff is locally P-perfect;

(iii) there is a locally P-perfect morphism f : X −→ Y with Y P-compact P-Hausdorff;

(iv) the projection X × Y −→ Y is locally P-perfect for all objects Y ;

(v) X × Y is locally P-compact Hausdorff for every locally P-compact Hausdorff object Y .

Proof. (i) =⇒ (ii): Revisiting the proof of 4.5.1 (i) =⇒ (ii) one decomposes 〈f,m〉 ∈ M as

X〈f,1X〉

// Y ×X 1Y ×m // Y ×K .

Then 〈f, 1X〉 ∈ M (since E ⊆ Epi X) and 〈f, 1X〉 ∈ P as a pullback of δY : Y −→ Y × Y(since Y is P-Hausdorff); furthermore, 1Y ×m ∈ M∩ P o as a pullback of m ∈ M∩ P o.With the P-open-closed interchange property andM⊆ Mono X, 〈f,m〉 is locally P-perfect,and so is f = p1 · 〈f,m〉 because p1 ∈ P ∩ P ′.All other steps can be taken as in the proof of Proposition 4.5.3.

4.5.4 Corollary. Let X have the P-open-closed interchange property. If the compositemorphism g · f is locally P-perfect and g is P-perfect, then f is locally P-perfect.

Proof. Exploit Proposition 4.5.3(i) =⇒ (ii) in X/ cod(g).

Let us now consider X = (T,V)-Cat and P = Prop(T,V) with V cartesian closed andintegral. Since every V-space is P-compact and P-Hausdorffness means discreteness, wenote that for T = I, being P-Tychonoff or locally P-compact Hausdorff also amounts tobeing discrete. For a general monad T we remark that, trivially, subspaces of P-Tychonoffspaces and P-open subspaces of locally P-compact Hausdorff spaces maintain the respectiveproperties.

4.5.5 Proposition. For V lean and superior and T flat, every Prop(T,V)-Tychonoff spaceis regular and Hausdorff in (T,V)-Cat.

Proof. A P-Tychonoff space X is embeddable into a P-compact P-Hausdorff space K which,by Proposition 4.2.1, is a compact Hausdorff object in (T,V)-Cat and therefore regular byProposition 2.3.4. Regularity and Hausdorffness are both inherited by subspaces.


Let us now restrict our attention to P-Tychonoff maps among P-Hausdorff spaces; these aresimply restrictions of proper maps in (T,V)-CatHaus, and one may exploit Proposition 4.5.1and Corollary 4.5.2 in this case. To be able to apply Proposition 4.5.3 to (T,V)-Cat, wemust secure the P-open-closed interchange property:4.5.6 Proposition. If the grand closure is idempotent, then (T,V)-Cat has the Prop(T,V)-open-closed interchange property.

Proof. For a closed subspace X of a P-open subspace Y of Z, consider the closure W = XZ

of X in Z. Then W is closed in Z by the idempotency hypothesis, and since hereditarinessof the grand closure makes the diagram

X

// _

Y _

W // Z

a pullback, P-openess of Y in Z gives the same property for X in W .

4.5.7 Remark. The idempotency hypothesis is certainly restrictive, as the case T = ,V = P+ shows (see Exercise 3.B). Trying to strengthen Proposition 4.5.6 one may be temptedto consider the idempotent hull of the grand closure. However, already in the generalcase, the idempotent hull of the grand closure in (T,V)-Cat can be hereditary only if thegrand closure is idempotent. Indeed, assume that for A ⊆ X we have x ∈ A

X, but x 6∈ AX .

For the subspace Y := A ∪ x one trivially has AY ⊆ AX ∩ Y and therefore x 6∈ A

Y .Consequently, AY = A is closed in Y . Denoting the idempotent hull by A = A

∞ andassuming its hereditariness one obtains A = AY = AX ∩ Y ⊇ A

X∩ Y 3 x, a contradiction.

4.5.8 Examples.(1) In Top = (, 2)-Cat with P = proper maps, P-Tychonoff spaces are characterized

as the completely regular Hausdorff spaces (or Tychonoff spaces), that is, as T1-spacesX with the property that for every closed set A ⊆ X and every x ∈ X \ A there is acontinuous map f : X −→ [0, 1] with f(x) = 0 and f(y) = 1 for all y ∈ A. Such spacesare easily seen to be embeddable into powers of the unit interval and, hence, into acompact Hausdorff space. Locally P-compact Hausdorff spaces are locally compactHausdorff spaces, that is, Hausdorff spaces that are locally compact (see III.5.7);in the presence of Hausdorff separation, these are the spaces in which every pointhas a compact neighborhood. Such spaces are Tychonoff spaces and, in fact, openlyembeddable into a compact Hausdorff space (see Engelking [1989]).

(2) For a monoid H and H as in 1.4 and P = proper maps as described in Exam-ple 3.1.3(4), an (H, 2)-space (X,−→) is P-Tychonoff if and only if it is embeddableinto an H-action Y , that is,

xα−→ y in X ⇐⇒ α · x = y in Y


for all x, y ∈ X, α ∈ H (with α ·x denoting the action of H on Y ). Since P-open mapsare open (see Example 4.4.10(3)), locally P-compact Hausdorff spaces in (H, 2)-Cathave the additional property that the embedding X → Y can be chosen to be open,that is, if α · x ∈ X for x ∈ Y , then x ∈ X. Finally, we note that (H, 2)-Cat has theP-open-closed interchange property since the grand closure is idempotent.

(3) In (L, 2)-Cat with P = Prop(L, 2), P-Tychonoff spaces are those (X,`) that areembeddable into monoids Y , that is,

(x1, . . . , xn) ` y in X ⇐⇒ x1 · x2 · . . . · xn = y in Y .

If the embedding X → Y is open, one has the additional property that

∀x, y ∈ Y (x · y ∈ X =⇒ x, y ∈ X) ,

and this property makes X → Y P-open (see 4.4.10(4)) and therefore characterizeslocally P-compact Hausdorff spaces in (L, 2)-Cat. In addition, since the grand closureis idempotent in (L, 2)-Cat, Proposition 4.5.3 is applicable in this category as well.

Exercises

4.A P-discrete objects. For an E-topology P on a finitely complete category X, call amorphism f : X −→ Y locally P-injective if δf : X −→ X ×Y X is P-open, and call an objectX P-discrete if !X is locally P-injective. Show the equivalence of the following statementson X:

(i) X is P-discrete;

(ii) every morphism f : X −→ Y is locally P-injective;

(iii) there is a locally P-injective morphism f : X −→ Y with Y P-discrete;

(iv) the projection X × Y −→ Y is locally P-injective for all objects Y ;

(v) X × Y is P-discrete for every P-discrete object Y ;

(vi) Y is P-discrete for every P-open morphism f : X −→ Y in E .

4.B P o-compact objects.

(1) Let X be extensive, and let the class E be closed under coproducts (so that ∐i∈I pi :∐i∈I Xi −→

∐i∈I Yi lies in E whenever all pi ∈ E). Show that P d is closed under

coproducts, for any E-topology P .


(2) For an E-topology P , assume P d to be closed under coproducts and let X be an objectsuch that, for all objects U , the morphism

eU : ∐x:1−→X U −→ X × U ,

whose x-th restriction to U is 〈x, 1U〉 : U −→ X × U , lies in E . Show that X isP o-compact. Conclude that every topological space is open map-compact.

4.C The left adjoint left-inverse topology. Recall that f : (X, a) −→ (Y, b) is left adjoint tog : (Y, b) −→ (X, a) in (T,V)-Cat if g · a = b · Tf ; f is left adjoint left-inverse to g if, inaddition, f · g = 1Y . Let L be the class of all left adjoint left-inverse maps. Show:

(1) L is a topology on (T,V)-Cat. For T = I, V = 2, a monotone map f : X −→ Y with Yseparated is left adjoint left-inverse if and only if f is proper and left adjoint.

(2) A (T,V)-space (X, a) is L-compact if and only if there is a point x0 ∈ X witha(x , x0) = > for all x ∈ TX. In particular, for T = and V = 2, a topological spaceis L-compact if and only if it contains a point whose only neighborhood is the spaceitself.

(3) If V is cartesian closed and T satisfies BC, then L is a Q-topology on (T,V)-Cat,where Q is the class of open surjections in (T,V)-Cat.

4.D The exponentiable topology. Show that in a finitely complete category X, the classExp X of exponentiable morphisms of X forms a topology. The Exp X-compact objects arethe exponentiable objects. For X = Top, prove that a space X is Exp X-Hausdorff if andonly if every point in X has a Hausdorff neighborhood.

4.E P-open versus open. Show that in (H, 2)-Cat and (L, 2)-Cat there are proper maps-open maps which fail to be open.

4.F Nearly open maps. Prove that the class of nearly open maps (see Definition III.4.3.1)forms an E-pretopology in (T,V)-Cat with E the class of all epimorphisms.

5. CONNECTEDNESS 449

5 Connectedness

An object in a category is connected if it has no non-trivial decomposition into a coproduct.This property becomes particularly powerful when the ambient category is extensive. Aftera brief review of extensive categories we explore the notion of connectedness in (T,V)-Catand exhibit the pivotal role of the category Top in this context. Stability under products isdiscussed at the end.

5.1 Extensive categories. In Corollary III.4.3.10 we gave an ad-hoc definition of exten-sive category and provided sufficient conditions for (T,V)-Cat to be extensive. Here weinvestigate this notion more systematically.For every small family (Yi)i∈I of objects in a category X with small-indexed coproducts andpullbacks one has the adjunction∏

i∈I X/Yi//

⊥ X/∐i∈I Yioo (5.1.i)

with the left adjoint given by coproduct (mapping (fi : Xi −→ Yi)i∈I to ∐i∈I fi : ∐i∈I Xi −→∐i∈I Yi) and the right adjoint by pullback along the coproduct injection tj of ∐i∈I Yi:

Xj

sj//

fj

X

f

Yjtj//∐i∈I Yi .

(5.1.ii)

5.1.1 Definition. A category X with small-indexed coproducts and pullbacks is extensiveif the adjunction (5.1.i) is an equivalence of categories; equivalently, if both the counits andthe units are isomorphisms, that is: if

(1) small coproducts are universal in X, so that X is a coproduct of (Xi)i∈I with injectionssj if all diagrams (5.1.ii) are pullbacks, and

(2) small coproducts are totally disjoint in X, so that all commutative diagrams (5.1.ii)are pullbacks if X ∼=

∐i∈I Xi (and therefore f = ∐

i∈i fi).

X is finitely extensive if instead of small coproducts we consider only finite coproductseverywhere.

Note that universality of finite coproducts entails in particular that the initial object 0must be strict, that is, any morphism f : X −→ 0 must be an isomorphism (just considerI = ∅). In fact, strictness of 0 follows already from the universality of binary coproducts,as the following result shows.

5.1.2 Proposition. Finite coproducts are universal in X if binary coproducts are.


Proof. It suffices to show that the initial object 0 in X is strict. For any morphism f : X −→ 0,

X1 //

f

X

f

X

f

1oo

0 1 // 0 01oo

are pullback diagrams, with the bottom arrows representing the injections of a binarycoproduct. The same is therefore true for the top arrows. To say that X ∼= X +X withisomorphic injections means equivalently that X is pre-initial, that is: |X(X, Y )| ≤ 1 for allobjects Y . In particular, the split epimorphism f must be inverse to 0 −→ X.

5.1.3 Proposition. If binary coproducts are universal in X, then the two injections of abinary coproduct are monomorphic and their pullback is pre-initial.

Proof. LetP

p//

X

s

Q

qoo

X s // X + Y Ytoo

be pullback diagrams and s, t coproduct injections. By hypothesis, p, q are coproductinjections as well, and since the left pullback property makes p a split epimorphism, thecoproduct property makes it actually an isomorphism. Consequently, having an isomorphicprojection in its kernel pair, s is a monomorphism.This shows that coproduct injections are monic; in particular, q is monic. With the trivialpullback diagrams

Q

q

1Q// Q

q

Q1Qoo

1Q

X1X // X Q

qoo

we see that, since the bottom arrows are the injections of X +Q ∼= X, universality givesQ+Q ∼= Q with isomorphic injections, that is, Q is pre-initial.

5.1.4 Lemma. Let X be a category with finite coproducts and pullbacks. If binary coproductsin X are universal, and pre-initial objects are initial, then the functor

(−) + C : X −→ X

reflects isomorphisms, for all objects C in X.

Proof. For f : A −→ B in X, assume that f + 1C : A+C −→ B+C is an isomorphism. Sincecoproduct injections are monic, the functor (−) + C is faithful and therefore reflects mono-


and epimorphisms, so that f is both monic and epic. We form the pullbacks D, E as in

Du //

a

Bw

E

c

voo

B + C

(f+1C)−1

A s//

f

BB

A+ C Ct

oo

where s, t, w are coproduct injections and a, u, c, v pullback projections. One obtainsg : A −→ D with a · g = 1A and u · g = f , making a an isomorphism as a pullback ofthe monomorphism (f + 1C)−1 · w. Consequently, with u, v being coproduct injections byuniversality, the diagram

Af// B E

voo

is also a coproduct, with f epic. With coproduct injections i, j,m, n, we have the commu-tative diagram

A m // A+ (E + E) E + Enoo

A

1A

OO

f// A+ E

1A+iOO

1A+jOO

Evoo

i

OO

j

OO

where 1A + i = 1A + j since f is epic. But then i = j because n is monic, which makes Epre-initial and thus E ∼= 0, by hypothesis. Consequently, the coproduct injection f becomesan isomorphism.

5.1.5 Theorem. The following conditions on a category X with finite coproducts andpullbacks are equivalent:

(i) X is finitely extensive, that is, finite coproducts in X are universal and totally disjoint;

(ii) binary coproducts in X are universal and pre-initial objects are initial;

(iii) binary coproducts in X are universal and disjoint (so that the pullback of the twocoproduct injections is the initial object).

Proof. (i) =⇒ (ii): For Q pre-initial, both rows of

0 //

Q

1

Q1oo

1

Q 1 // Q Q1oo

(5.1.iii)

represent binary coproducts. Total disjointness implies in particular that the left-hand sideis a pullback diagram, so that we must have Q ∼= 0.


(ii) =⇒ (iii) follows from Proposition 5.1.3.(iii) =⇒ (ii): If Q is pre-initial, the left-hand side of (5.1.iii) is a pullback diagram underthe disjointness hypothesis, and Q ∼= 0 follows.(ii) =⇒ (i): By Proposition 5.1.2, it suffices to show total disjointness of binary coproducts.Given the outer commutative diagram of

A //

h

A+B

Boo

l

Pp

77

Qq

gg

X // X + Y Yoo

(5.1.iv)

(with the horizontal arrows coproduct injections), one forms the pullbacks P and Q andobtains the comparison morphisms h, l making (5.1.iv) commutative. By hypothesis,P + Q ∼= A + B (with coproduct injections p, q), so that h + l : A + B −→ P + Q is anisomorphism. Since

h+ l = (h+ 1Q) · (1A + l) = (1P + l) · (h+ 1B) ,

h+ 1Q is split epic and h+ 1B is split monic. Consequently,

(h+ 1Q) + 1B ∼= h+ 1B+Q ∼= (h+ 1B) + 1Q

is both split epic and split monic, that is an isomorphism. With Lemma 5.1.4 we deducethat h (as well as l, by symmetry) is an isomorphism, as desired.

5.1.6 Corollary. The following conditions on a category X with small-indexed coproductsand pullbacks are equivalent:

(i) X is extensive;

(ii) non-empty coproducts in X are universal and pre-initial objects are initial;

(iii) non-empty coproducts in X are universal and disjoint (so that the pullback of twocoproduct injections with distinct labels is the initial object).

Proof. For any coproduct (si : Xi −→ X)i∈I and j ∈ I, note that

Xj

sj// X

∐i 6=j Xi

soo

is a binary coproduct. Therefore (ii) =⇒ (i) follows from 5.1.5(ii) =⇒ (i), and the same istrue for (ii) =⇒ (iii); indeed, when the pullback of sj and s is 0, so is the pullback of sjand si for any i 6= j since 0 is strict by Proposition 5.1.2.


We proved in Theorem III.4.3.9 that for a cartesian closed quantale V, a monad T withT taut and an associative and right-whiskering lax extension T to V-Rel, the category(T,V)-Cat is extensive. In particular, Ord, Met, Top, App and (H, 2)-Cat (for a monoid H)are extensive.Frequently studied extensive categories which do not admit a topological functor to Setinclude Cat and Rngop (the opposite of the category of unital rings).

5.2 Connected objects. An object X in a category X is connected if X(X,−) : X −→ Setpreserves all small-indexed coproducts. This definition becomes especially efficient when Xis extensive.5.2.1 Theorem. The following assertions are equivalent for an object X in an extensivecategory X with a terminal object 1:

(i) X is connected;

(ii) every morphism f : X −→ ∐i∈i Yi factors uniquely as f = tj · g with a uniquely

determined j ∈ I (and tj the corresponding coproduct injection);

(iii) every morphism f : X −→ ∐i∈I Yi factors as f = tj · g for some g and j;

(iv) X 6∼= 0, and every morphism f : X −→ 1 + 1 factors through one of the coproductinjections of 1 + 1;

(v) X 6∼= 0, and every extremal epimorphism f : X −→ Y + Z makes one of the coproductinjections an isomorphism;

(vi) X 6∼= 0, and X ∼= Y + Z implies Y ∼= 0 or Z ∼= 0.

Proof. The implications (i)⇐⇒ (ii) =⇒ (iii) are trivial, and for (iii) =⇒ (iv) observe thatX 6∼= 0 follows since there is no coproduct injection through which the empty coproduct 0may factor.(iv) =⇒ (v): For any extremal epimorphism f : X −→ Y + Z, the composite

Xf

// Y + Z!Y +!Z // 1 + 1

factors through one of the injections of 1+1, and then f factors through one of the injectionsof Y + Z since the latter is a pullback of the former, as f = t · g with t : Y −→ Y + Z,say. But then the monomorphism t must be an isomorphism since the epimorphism f isextremal.(v) =⇒ (vi): By hypothesis, one of the coproduct injections of X ∼= Y +Z is an isomorphism,and since

0 //

Z

Y // Y + Z


is a pullback diagram, one of 0 −→ Y , 0 −→ Z is an isomorphism as well.(vi) =⇒ (ii): For f : X −→ ∐

i∈I Yi one considers the pullback diagrams (5.1.ii) and hasX ∼=

∐i∈I Xi by universality of coproducts. Since X 6∼= 0, not all Xi may be initial. In fact,

there is precisely one j ∈ I with Xj 6∼= 0 since, by hypothesis, from

X ∼= Xj +∐

i 6=j Xi

one obtains ∐i 6=j Xi∼= 0 and therefore Xi

∼= 0 for all i 6= j. Consequently, the coproductinjection Xj −→ X is an isomorphism and thus allows for the factorization f = tj · (fj · s−1

j ),which is unique since tj is a monomorphism.

Let the extensive category X now come with a factorization system (E ,M) and an E-topology P . Then one easily obtains stability of connectedness under E-images and P-denseextensions, as follows.

5.2.2 Proposition. For a morphism h : Z −→ X in X with Z connected, under each of thefollowing conditions X is also connected.

(a) h ∈ E, and coproduct injections in X lie in M;

(b) h ∈ P d, and coproduct injections in X lie in M∩P.

Proof. Since Z is connected, every morphism f : X −→ ∐i∈I Yi yields a commutative

diagramZ

g//

h

Yj

tj

Xf//∐i∈I Yi

with some g and coproduct injection tj. Each of the two conditions (a), (b) produces a“diagonal” X −→ Yj making f factor through tj.

Let now X = (T,V)-Cat, with V cartesian closed, T taut, and T associative and right-whiskering, so that X is extensive (Theorem III.4.3.9), and let P = Prop(T,V) and E be theclass of all epimorphisms. It is easy to see that, when T preserves bottom V-relations (sothat T⊥X,Y = ⊥TX,TY , for the least V-relation ⊥X,Y : X −→7 Y ), then coproduct injectionsare closed embeddings, that is, lie in M∩P : see Exercise 5.B). With Corollary 4.4.4 andProposition 5.2.2 one obtains:

5.2.3 Corollary. Let V be cartesian closed, T taut, and T associative and right-whiskering.Suppose moreover that T preserves bottom V-relations. For a (T,V)-continuous f : X −→ Y

with f(X)∞ = Y , if X is connected, then Y is connected.


It follows trivially from Theorem 5.2.1 that (T,V)-spaces X with |X| = 1 are connected.For T = I the identity monad one also obtains easily the following characterization of allconnected spaces:

5.2.4 Corollary. Let V be cartesian closed. Then (X, a) is connected in V-Cat if and onlyif X 6= ∅ and for all x, y ∈ X there are x = x0, x1, . . . , xn = y in X with

a(xi, xi+1) ∨ a(xi+1, xi) > ⊥ (i = 0, . . . , n− 1) .

Proof. The criterion is sufficient for connectedness of (X, a) since continuity of any mapf : X −→ 1 + 1 means equivalently

a(x, y) > ⊥ =⇒ f(x) = f(y)

for all x, y ∈ X. Conversely, considering the least equivalence relation on X that identifiesall x, y with a(x, y) > ⊥, for Z the subspace formed by the equivalence class of somez ∈ X 6= ∅ and Y = X \ Z, one has X ∼= Y + Z in V-Cat. Connectedness of X givesY = ∅, so that the criterion holds.

5.2.5 Examples.

(1) In Ord the categorical notion of connectedness retains the usual notion of a non-emptyconnected ordered set (X,≤): for all x, y ∈ X one finds a “zigzag”

x = x0 ≤ x1 ≥ x2 ≤ x3 . . . xn−1 ≤ xn = y .

Connectedness in Met is less interesting: every non-empty metric space (X, d) with dfinite is connected.

(2) In Top connected objects X are also characterized as expected: X = Y ∪ Z withY, Z ∈ OX and Y ∩ Z = ∅ only if Y = ∅ or Z = ∅; but note again that thecategorical notion entails X 6= ∅.

(3) For a monoid H, a connected object (X,−→) in (H, 2)-Cat is characterized by X 6= ∅and the property that for all x, y ∈ X one has

x = x0α0 // x1 x2

α1oo α2 // x3 · · · xn−1αn−1

// xn = y

for some xi ∈ X and αi ∈ H (i = 1, 2, . . . , n− 1).

5.3 Topological connectedness governs. Throughout this subsection we assume that

• V is cartesian closed;

• T is taut;


• T is associative and right-whiskering.

For the extensive category (T,V)-Cat we then have the functor

Ω : (T,V)-Cat // Top

that provides a (T,V)-space with the topology of its (T,V)-open subsets; it preserves openmaps and coproducts and also reflects coproducts: see Corollary 3.6.2. Its principal purposearises from the following consequence of Theorem 5.2.1:

5.3.1 Theorem. A (T,V)-space X is connected if and only if the topological space ΩX isconnected.

Proof. Let X be connected in (T,V)-Cat and assume that X is the disjoint union of (T,V)-open sets A,B. Then X ∼= A+B in (T,V)-Cat (with A,B considered as subspaces of X):see Exercise III.4.B and Theorem III.4.3.3. Consequently, A = ∅ or B = ∅.Conversely, assuming X ∼= A+B in (T,V)-Cat and ΩX connected, coproduct preservationby Ω gives immediately A = ∅ or B = ∅, so that X must be connected in (T,V)-Cat.

A connected component of a (T,V)-space X is a maximal (with respect to ⊆) connectedsubspace X. By Corollary 5.2.3, a connected component is always (T,V)-closed. FromTheorem 5.3.1 one obtains immediately:

5.3.2 Corollary. The following statements on a (T,V)-space X are equivalent:

(i) X is a coproduct of connected (T,V)-spaces;

(ii) X is the coproduct of its connected components;

(iii) the connected components of X are (T,V)-open;

(iv) ΩX is the coproduct of its connected components.

5.3.3 Examples.

(1) Even in Top ∼= (, 2)-Cat, a space X may fail to satisfy the equivalent conditions ofCorollary 5.3.2 (consider the subspace 0 ∪ 1

n| n = 1, 2, . . . of R, for example).

Every open subspace of X satisfies them precisely when X is locally connected, thatis: when every neighborhood of a point x contains a connected neighborhood of x.

(2) Let M be a multiplicative monoid, considered as a compact Hausdorff (L, 2)-space:see Corollary 1.4.4. A subset A ⊆ M is closed when it is closed under the binaryoperation of M , and A is open in M if

∀x, y ∈M (x · y ∈ A =⇒ x ∈ A & y ∈ A) .


Hence, non-empty open sets in M must contain the neutral element and, consequently,M is connected in (L, 2)-Cat.When M is commutative, for A ⊆M to be open means precisely that A is down-closedwith respect to the divisibility order:

x|z ⇐⇒ ∃y ∈M (x · y = z) .

Hence, the topology of the Alexandroff space ΩM is induced by this order.

Recall that a topological space is Alexandroff if intersections of open sets are open: seeExample II.5.10.5. The connected component Cx of a point x in an Alexandroff space X isnot only closed but also open, since

Cx =⋂

y∈X\CxX \ Cy .

We can now show that when T preserves intersections (that is, multiple pullbacks of sinksof monomorphisms), the values of the functor Ω are always Alexandroff.5.3.4 Proposition. Let T : Set −→ Set preserve intersections. Then, for every (T,V)-spaceX = (X, a), the topological space ΩX is Alexandroff and X is the coproduct of its connectedcomponents.

Proof. The second assertion follows from the first. To prove the first, consider open sets Ui(i ∈ I) in ΩX and x ∈ U , y ∈ TY with a(y , x) > ⊥. Then y ∈ TUi by hypothesis on Uifor all i ∈ I, hence y ∈ TU by hypothesis on T . Consequently U → X is (T,V)-open.

5.4 Products of connected spaces. Throughout this subsection, we assume

• V is cartesian closed and integral;

• T is taut and T1 ∼= 1;

• T is associative and right-whiskering.5.4.1 Proposition. Finite products of connected (T,V)-spaces are connected.

Proof. The terminal (T,V)-space 1 = (1,>) is certainly connected. For X, Y connected, itsuffices to show that any (x0, y0), (x1, y1) ∈ X × Y lie in the same connected componentof X × Y . The additional assumptions on T and V make x0 : 1 −→ X a morphism, andthe split monomorphism x0 × 1Y : Y ∼= 1 × Y −→ X × Y is preserved by Ω, makingx0 × Y a connected subspace of Ω(X × Y ), by Theorem 5.3.1; likewise for X × y1.Since (x0 × Y ) ∩ (X × y1) 6= ∅, both (x0, y0), (x1, y1) lie in the connected subspace(x0×Y )∪ (X ×y1) of Ω(X ×Y ), showing that Ω(X ×Y ) is connected. By 5.3.1 again,X × Y is connected in (T,V)-Cat.

The hypothesis T1 ∼= 1 is essential for this Proposition to hold, and so is the restriction tofinite products, as the following examples show.


5.4.2 Examples.

(1) For the list monad L, consider the (L, 2)-space (X,`) with X = x, y and ` theleast (L, 2)-structure on X with (x, y) ` x, and (Y,`) with Y = ∗ discrete. SinceX 6∼= x + y and |Y | = 1, the spaces X and Y are connected, but X × Y ∼=(x × Y ) + (y × Y ) is not.

(2) In Ord ∼= (I, 2)-Cat, the product of a family of objects in which any two elementshave a lower bound is connected (by Corollary 5.2.4). However, if we let Xn = x1 ≤x2 ≥ x3 ≤ . . . xn be a “zigzag” of length n, then Xn is connected, but X = ∏

nXn

is not since the sequences (x1)n and (xn)n lie in distinct components of X. Notethat Ω : Ord −→ Top is the coreflective embedding that provides an ordered set withits Alexandroff topology, and that the topological space ∏n(ΩXn) is connected (seeCorollary 5.4.4 below) while ΩX is not. In particular, Ω does not preserve infiniteproducts, but it does preserve finite products.

Here is a criterion for infinite products of connected (T,V)-spaces to be connected. Weassume T to preserve bottom V-relations.

5.4.3 Theorem. For a family (Xi)i∈I of connected (T,V)-spaces, X = ∏i∈I Xi is connected

if and only if there is a connected subspace A of X such that

A = (xi) ∈ X | ∃(zi) ∈ A (xi = zi for all but finitely many i ∈ I)

is Prop(T,V)-dense in X.

Proof. The condition is trivially necessary since we may choose A = X. Conversely, forz = (zi)i∈I ∈ A and F ⊆ I finite, there is a morphism 1 −→ ∏

i∈I\F Xi with constant value(zi)i∈I\F and then a split monomorphism∏

i∈F Xi −→∏i∈F Xi ×

∏i∈I\F Xi

∼= X .

Since the domain is connected, as in Proposition 5.4.1, its image

Fz = (xi)i∈F | ∀i ∈ I \ F (xi = zi)

is connected as well, and so is

A =⋃

z∈A,F⊆I finite Fz .

Indeed, for x ∈ Fz, y ∈ Gw with z, w ∈ A and F,G ⊆ I finite, the connected componentsof x and z coincide since Fz is connected, and so do the connected components of y andw, but also of z and w since A is connected. With Corollary 5.2.3, connectedness of Xfollows.


Choosing for A any singleton subset of X, Theorem 5.4.3 shows in particular:

5.4.4 Corollary. c©The product of connected spaces in Top ∼= (, 2)-Cat is connected.

5.4.5 Remark. In general, the connected subspace A in Theorem 5.4.3 may not be chosento be a singleton set, not even finite. Indeed, in Ord ∼= (I, 2)-Cat, the product of countablymany copies of the set Z of integers is connected (see Example 5.4.2(2)). For any finite setA ⊆ ZN consider a point x = (xn)n∈N with xm < zm for all z = (zn)n∈N ∈ A and m ∈ N;since the idempotent grand closure is given by the up-closure, such x cannot lie in thegrand closure of A.

Exercises

5.A Connected categories. Confirm that connected objects in Cat are characterized as inExercise II.2.Q.

5.B Coproduct injections are closed embeddings. Let T and V be such that the lax extensionT : V-Rel −→ V-Rel preserves bottom V-relations (see Corollary 5.2.3). Then coproductinjections in (T,V)-RGph are O-initial (for O : (T,V)-RGph −→ Set). Moreover, when T isassociative the corresponding statement holds for (T,V)-Cat, and every coproduct injectionis proper.

5.C Infinite products of connected spaces. Consider the subspaces Xn = 0, n (n ∈ N) ofR with their Euclidean metric. Show that the product ∏n∈NXn fails to be connected inMet although each Xn is connected.

5.D Total disconnectedness. A (T,V)-space Y is called totally disconnected if each of itsconnected components has precisely one element. Show that the following statements holdunder the hypotheses of Corollary 5.2.3.

(1) Y is totally disconnected if and only if every (T,V)-continuous map f : X −→ Y withX connected is constant.

(2) A (T,V)-space X is connected if and only if every (T,V)-continuous map f : X −→ Ywith Y totally disconnected is constant.

(3) The full subcategory of totally disconnected (T,V)-spaces is strongly epireflective in(T,V)-Cat.

(4) In Top ∼= (, 2)-Cat, an extremally disconnected T0-space must be totally disconnected,but not conversely. Indiscrete spaces are both connected and extremally disconnected.

5.E Shortcomings of Ω. While the functor Ω : (T,V)-Cat −→ Top of 5.3 preserves openmaps and open embeddings, it generally fails to preserve embeddings or finite products.


5.F A right adjoint to Ω. If V = 2, T is taut, and T is associative and right-whiskering,then the functor

Ω : (T, 2)-Cat −→ Tophas a right adjoint which assigns to a topological space X the (T, 2)-space (X,−→) with

x −→ x ⇐⇒ ∀U ∈ OX (x ∈ U =⇒ x ∈ TU)

for all x ∈ X, x ∈ TX.


Notes on Chapter V

For concrete categories endowed with a notion of closed subobject, in [Manes, 1974] Manes essentiallyconsiders stably-closed maps and defines an object X to be compact and Hausdorff if, respectively, X −→ 1and X −→ X × X are stably closed, just as in the axiomatic setting of Section 4. Furthermore, for amonad T on Set endowed with its Barr extension to Rel, he in essence considers the category (T, 2)-Cat andthe notion of closed subobject as in Remark 3.6.4 and gives a relational characterization of compactnessand Hausdorff separation as in our Definition 1.1.1. He also observes that the stably-closed maps areequationally defined, in the same way as proper maps are defined in Section 3. Briefly, his paper is tobe considered an eminent precursor to large parts of Chapter V. There are two remarkable (but not wellknown or accessible) PhD theses which greatly extended Manes’ ideas in an abstract relational setting,by Kamnitzer [1974a] (written under the direction of G.C.L. Brummer; see also [Kamnitzer, 1974b]) andMobus [1981] (written under the direction of H. Schubert; see also [Mobus, 1978]). Specifically, Kamnitzerconsiders T0, T1, and Hausdorff separation and compactness as used in this chapter, and our definition ofR0, R1, regularity, normality and extremal disconnectedness follows Mobus who considers these notions inthe general relational context of [Klein, 1970] and [Meisen, 1974]. Our treatment of order-separation can betraced back to Marny’s definition of T0-separation for topological categories [Marny, 1979].In the particular case of approach or (,P+)-spaces, compactness coincides with 0-compactness as developedin [Lowen, 1988, 1997]. A study of low separation properties in that setting goes back to [Lowen and Sioen,2003], where order-separation is called T0. The R1 property was considered by Robeys [1992] where anapproach space satisfying this condition is called complemented. Regularity for approach or (,P+)-spacescoincides with the notion considered in [Robeys, 1992] and is further characterized in terms of the tower ofthe approach space in [Brock and Kent, 1998]. A notion of normality for approach spaces weaker than theone used in Section 2 was introduced in Van Olmen’s thesis [Van Olmen, 2005]; here it appears as item (iii)in Theorem 2.5.2.Versions of the Tychonoff Theorem and the Cech–Stone compactification appear in various contexts,including the ones already mentioned (see in particular [Clementino, Giuli, and Tholen, 1996; Clementinoand Tholen, 1996; Lowen, 1997; Mobus, 1981]), but its treatment in the general (T,V)-context as givenin Proposition 1.2.1 and Theorem 1.2.3 draws heavily on Proposition III.1.2.1 which first appeared in[Clementino and Hofmann, 2009].The equational definition of proper morphism as presented in (T,V)-Cat in Section 3 may be traced back to[Manes, 1974], called perfect by him and strongly closed in [Kamnitzer, 1974a], while we maintain Bourbaki’sterminology [Bourbaki, 1989]. The corresponding definition of open morphism as used here appears first in[Mobus, 1981]. Generalizations of the Kuratowski–Mrowka Theorem, named after [Kuratowski, 1931] (whoproved that product projections along a compact space are closed) and [Mrowka, 1959] (who showed thatKuratowski’s property characterizes compactness) have been considered by various authors, notably in thecontext of closure operators by Dikranjan and Giuli [1989] and for approach spaces in [Colebunders, Lowen,and Wuyts, 2005]. There is also a fairly general version of the Kuratowski–Mrowka Theorem in a differentsetting in Hofmann’s fundamental article [Hofmann, 2007] while the construction used in Theorem 3.4.1relies on Proposition III.4.9.1 that draws on [Clementino and Hofmann, 2012]. The other crucial ingredientto the characterization of proper maps, Theorem 3.2.5, appeared only recently in [Clementino and Tholen,2013], following which Solovyov [2013] proposed the notion of closed map as given in Definition 3.3.6.The characterization of proper maps of approach spaces as stably closed maps or as closed maps with0-compact fibres appears in [Colebunders, Lowen, and Wuyts, 2005]. Open maps of convergence spacesare described in [Kent and Richardson, 1973]; for approach spaces openness is introduced in [Lowen andVerbeeck, 1998, 2003] in terms of the associated distance, a notion coinciding with that of inversely closedmaps for (,P+)-spaces.Theorem 3.5.6 on the product stability of proper maps (which entails the Tychonoff Theorem) originates


with Schubert’s thesis [Schubert, 2006] who also observed the crucial role of complete distributivity as inProposition 3.5.4. Conditions for the openness of coproduct injections are considered in [Mobus, 1981]. Ourpresentation of stability of openness under coproducts relies on Mahmoudi, Schubert, and Tholen [2006].The first axiomatic categorical treatment of compactness and Hausdorff separation depending on a parameterP as in Section 4 was given by Penon [1972]; in fact the axioms we have imposed on a topology P may beconsidered as a finitary version of Penon’s axioms, except that he does not require closure under composition.But as emphasized in [Tholen, 1999], closure under composition is essential to fully exhibit the beautifulinterplay of the two notions. A slightly different axiomatic approach, starting with a notion of closed mapfrom which proper is derived as stably-closed, is presented in [Clementino, Giuli, and Tholen, 2004a] wherefurther topological themes like local compactness and exponentiability are being pursued in greater depth.Of course, there is a rich supply of articles recognizing and axiomatizing the key role of the class of properor perfect maps, including [Herrlich, 1974; Herrlich, Salicrup, and Strecker, 1987; Manes, 1974] which, oncea definite notion of categorical closure operator had been introduced by Dikranjan and Giuli [1987], led tomany investigations of compactness and separation in that context; see in particular [Clementino, Giuli,and Tholen, 1996; Clementino and Tholen, 1996; Dikranjan and Giuli, 1989]. The term topology on acategory as in Section 4 appears in [Tholen, 2003] and is adopted in [Schubert, 2006] and published in[Hofmann and Tholen, 2012].Extensive categories (in the finitary sense) were studied by Carboni, Lack, and Walters [1993]. Theirelegant definition as given in Section 5 is due to Steve Schanuel, and their characterization as given inTheorem 5.1.5 draws also on [Borger, 1994], an extended preprint of which appeared as [Borger, 1987].Connected objects as defined in Section 5.2 appeared in Hoffmann’s thesis [Hoffmann, 1972], and theircharacterization as given in Theorem 5.2.1 draws on [Janelidze, 2004]. Theorem 5.3.1 (for V = 2) and itsconsequences are due to Clementino, Hofmann, and Montoli [2013].

Suggestions for further reading: Jager [2012], Hohle [2001], Manes [2002], Manes [2007], Manes [2010].

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Index

absolute coequalizer, 79G-absolute pair, 80abstract basis, 357

ordered, 358action, 92, 100

left, 100right, 101

additive category, 112adjoint

left, 24, 49, 107left-inverse, 448

right, 23, 49, 107adjunction, 24, 49, 107

descent type, 94functor, 45rules, 148vs. Galois correspondence, 24

admissible, 181Alexander’s Subbase Lemma, 268Alexandroff

space, 136topology, 136

(T,V)-algebra, see (T,V)-categoryT-algebra, 76

free, 76algebraic

closure space, see finitaryfunctor, 208

antisymmetric, see separatedapproach distance, 192approach space, 192

0-compact, 372core-compact, 275pre-, 220pseudo-, 220stable, 275

arrow, see morphismassociated monad, 74associative lax extension, 165associativity law, 43-atomic, 33auxiliary relation, 313

Barr extension, 194, 200, 309base

filter, 35for a topology, 30ideal, 35

BC conditionfor a functor, 173for a natural transformation, 173

BC-square, 172Beck Criterion, 80Beck–Chevalley condition, see BC conditionBeck–Chevalley square, see BC-squarebicontinuous map, 200bicoreflective, 118V-bimodule, see V-modulebimorphism, 98biproduct, 109bireflective, 118birelation, 147bitopology, 200bottom, 29bound

greatest lower, 27least upper, 26lower, 27upper, 26

cartesianlifting, 130monad, 185morphism, 128product, 19square, 129

cartesian closedcategory, 102locally, 227quantale, 226

category, 43additive, 112cartesian closed, 102cocomplete, 61comma, 55

478

INDEX 479

comma, lax, 303complete, 61concrete, 58connected, 72discrete, 56dual, 44Eilenberg–Moore, 76enriched, 105finite, 61finitely cocomplete, 61finitely complete, 61functor, 47has coequalizers, 61has coproducts, 61has equalizers, 61has finite coproducts, 61has finite products, 61has partial products, 228has products, 61has pullbacks, 61has pushouts, 61Kleisli, 84locally small, 46meta-, 45monadic, 77monadic, strictly, 77monoidal, 97monoidal closed, 102monoidal, strict, 98opposite, 44ordered, 105self-dual, 45small, 43small-cocomplete, 61small-complete, 61symmetric monoidal, 97thin, 105

(T,V)-category, 158discrete, 163separated, 262

V-category, 150tensored, 215underlying, 208

V-category, 1112-category, 105, 112Cayley representation, 46Cech–Stone compactification, 265, 374cell, 178chain, 22

three-, 31two-, 31

change-of-base functor, 210chaotic structure, see indiscretecharacteristic morphism, 238class, 17, 39, 43

large, 17proper, 17small, 17

classifierpartial-map, 241M-partial-map, 239partial-map, regular, 241subobject, 239M-subobject, 237subobject, regular, 239

clique, 296monad, 296ultra-, 314

closed, 25cartesian category, 102(T,V)-functor, 413inversely, 414monoidal category, 102stably, 415subobject, 124under M-morphisms, 135under M-sources, 135under limits, 64, 117under products, 117

closure, 124finitary, 199grand, 410Kuratowski, 40

v-closure, 410closure operation, 25

down-, 26finitely additive, 40metric, 200up-, 27

closure operator, 124continuous, 124extensive, 124hereditary, 125hereditary, weakly, 125idempotent, 125monotone, 124

closure space, 25algebraic, see finitary

480 INDEX

finitary, 199metric, 200

cloven fibration, 130co-dcpo, 351

cocontinuous, 351coarser, 282cocartesian

lifting, 130morphism, 128

cocomplete, 61finitely, 61

T-cocomplete, 263cocone, 60cocontinuous

co-dcpo, 351functor, 63lattice, 340ordered set, 351

codomain, 43, 114functor, 129

coequalizer, 60absolute, 79

cofibration, 130cogenerating class, 127cogenerator, 127coherence conditions

for cartesian product, 20for tensor product, 98

cokernel pair, 60colax idempotent monad, 111colimit, 60

creation, 78directed, 63

comma category, 55lax, 303

commutativemonoid, 20quantale, 30

comonoid, 100compact, 191

core-, 267locally, 269locally, Hausdorff, 446(T,V)-space, 370stably, 269super-, 371

0-compact, 372P-compact, 433compactification, Cech–Stone, 265, 374

companionleft, 118right, 118

comparison functor, 77compatible, 210

strictly, 328complement, 25complemented, 148complete

category, 61finitely, 61large-, 133lattice, 30ordered set, 26small-, 61

completely distributive, 33constructively, 34

completely prime, 361composite monad, 87composition, 43, 111

horizontal, 47, 107Kleisli, 84of functors, 44vertical, 47, 107

concrete category, 58cone, 56confluent, 276conglomerate, 17conjugate ordered category, 106connected

category, 72component, 72, 456limit, 134locally, 456object, 453

constantdiagram, 56sequence, 42

constructively completely distributive, see completelydistributive

continuousclosure operator, 124dcpo, 351functor, 63lattice, 340map, 25, 26, 30

(T,V)-continuous, 369V-continuous, 369contravariant

INDEX 481

hom-functor, 46powerset functor, 50

convergent sequence, 42convolution

distance distribution maps, 186Kleisli, 161

coproduct, 60disjoint, 227injection, 60pullback stable, 227totally disjoint, 449universal, 449

core-compact, 267, 275coreflection, 54coreflective

bi-, 118mono-, 118

M-coreflective, 118coreflective subcategory, 54coreflector, 54coseparator, see cogeneratorcounit, 49couniversal

arrow, 52property, 53

covariant hom-functor, 46cowellpowered, 123creation

of colimits, 78of limits, 78

dcpo, continuous, 351dense

finally, 136functor, 69initially, 136map, 439morphism, 124subobject, 124

P-dense, 438densely generating, 140derived

class, 431topology, 431

descent type, 94diagonal morphism, 46diagram, 56

constant, 56shape, 56

diamond lattice, 40

Dirac morphism, 211direct limit, 63directed

colimit, 63down-, 34graph, 114limit, 63up-, 35

disconnectedextremally, 393extremally, topology, 392totally, 459

discretecategory, 56(T,V)-category, 163object, 128order, 30, 50presheaf monad, 320structure, 203topology, 50

P-discrete, 447disjoint coproduct, 227disjointness preservation, 246distance

approach, 192distribution map, 186

distributive law, 86lax, 182

V-distributor, see V-moduledomain, 43, 114

functor, 141double category, 179double-powerset monad, 75down

-closed, 26-closure, 26-directed, 34-interior, 41

down-set, 26functor, 45monad, 111

down-set-filter monad, 341dual

category, 44V-category, 154lax extension, 182relation, 21V-relation, 146statement, 53

482 INDEX

dualization, 53functor, 45

Duskin Criterion, 82

edge, 114Eilenberg–Moore

algebra, see T-algebracategory, 76

embeddingfull, 45open, 224order-, 29Yoneda, 48

enriched category, 105epi

-reflective, 118-reflective, regular, 118-sink, 123-sink, extremal, 127-sink, strict, 140

epic, 60epimorphism, 60

extremal, 118regular, 71split, 71strong, 138

equalizer, 58generalized, 66

equivalence, 23, 54equivalent categories, 54equivariant, 92, 101evaluation, 50, 228

functor, 64exponentiable, 102⊗-exponentiable, 101exponential, 102, 234extension, 109

Kleisli, 288lax, see lax extensionof a functor, 93of a monad, 90operation, 85

extensive, 449category, 227category, finitely, 449closure operator, 124

extremalepi-sink, 127epimorphism, 118mono-source, 127

monomorphism, 117

factorization classleft, 118right, 118

factorization systemfor morphisms, 116for sinks, 121for sources, 123proper, 116trivial, 118

faithful functor, 45fibration, 130

mono-, 131E0-fibration, 131M0-fibration, 131fibre, 21, 130, 408

finite, 247fibred product, 59fibrewise

pretopology, 431topology, 431

filterbase, 35coarser, 282finer, 282functor, 75image, 35in ordered set, 34inverse image, 35monad, 75neighborhood, 42neighborhood, open, 361on a set, 35principal, 35, 36proper, 34restriction, 35round, 359ultra-, 36vs. sequence, 42

filtered sum, 36finally dense, 136finer, 282finitary

-up-set monad, 296closure, 199closure space, 199metric, 151V-relation, 150up-set, 295

INDEX 483

finite-powerset monad, 199finitely additive, 40, 193fixpoints, 25flat lax extension, 156, 157forgetful functor, 45fork, 92

split, 79frame, 29

homomorphism, 30sub-, 30

free-group functor, 66-monoid functor, 50-monoid monad, 75T-algebra, 76

Frobenius lawleft, 226right, 226

fullfunctor, 45subcategory, 45

fully faithfulfunctor, 45monotone map, 22

functor, 44adjunction, 45algebraic, 208category, 47change-of-base, 210cocontinuous, 63codomain, 129comparison, 77composition, 44continuous, 63contravariant powerset, 50dense, 69domain, 141double category, 179down-set, 45dualization, 45evaluation, 64extension, 93faithful, 45filter, 75forgetful, 45free-group, 66free-monoid, 50full, 45full embedding, 45

fully faithful, 45hom-, contravariant, 46hom-, covariant, 46identity, 44, 75inclusion, 45inverse image-, 119Kuratowski-closure, 45lax, 106lifting, 93module, 46monadic, 77open-set, 45oplax, 106V-powerset, 180powerset, contravariant, 45, 50powerset, covariant, 45preserves colimits, 63preserves disjointness, 246preserves isomorphisms, 70preserves limits, 63preserves monomorphisms, 63pseudo-, 106reflects isomorphisms, 54reflects limits, 63reflects monomorphisms, 63representable, 71self-adjoint, 50set-of-objects, 141topological, 131M0-topological, 132transportable, 133ultrafilter, 75underlying-order, 45up-set, 75whiskering, 47

(T,V)-functor, 159, 217closed, 413open, 401proper, 401

V-functor, 150V-functor, 1122-functor, 106functorial, 124

Galois correspondence, see adjunctiongenerating, 127

densely, 140strongly, 127

generatoras a class, 127

484 INDEX

as an object, 127strong, 127

generic point, 263graph

directed, 114edge, 114of a map, 21vertices, 114

(T,V)-graph, 217greatest lower bound, 27group, 46

topological, 138

Hausdorffmetric, 253(T,V)-space, 369topology, 131

P-Hausdorff, 433morphism, 433

hereditaryclosure operator, 125closure operator, weakly, 125morphisms class, 430sink, 245

M-hereditary, 119hom

-functor, contravariant, 46-functor, covariant, 46-functor, internal, 102-object, 111-set, 43

homeomorphism, 361local, 443

homomorphismlax, 159of T-algebras, 76of frames, 30of join-semilattices, 29of meet-semilattices, 29of monoidal categories, 98of monoids, 20, 99of quantales, 30of quantales, lax, 30of quantaloids, 108of semilattices, 29

T-homomorphism, 76horizontal composition, 47, 107hull, idempotent, 439

ideal, 35

base, 35principal, 35proper, 35

idempotent, 29closure operator, 125hull, 439monad, 95

identity, 43, 111functor, 44, 75law, 43lax, 161monad, 75

M-image, 119image filter, 35inclusion functor, 45indiscrete

object, 128order, 50structure, 204topology, 50

inducedmonad, 74order, 30, 204

induction, 38inf-map, 27infimum, 27

preservation, 27initial

extension, 298lifting, 131object, 55object, strict, 449pre-, 450source, 128

initially dense, 136injection of coproduct, 60M-injective, 140injective, locally, 435P-injective, locally, 447integral, 148interior

down-, 41operation, 25

internal hom-functor, 102interpolating, 301, 302M-intersection, 122νT-invariant, 333inverse

-image functor, 119

INDEX 485

image, 119image filter, 35limit, 63

M-inverse image, 119irreducible, 267, 275isomorphism, 23, 43

natural, 48preservation, 70reflection, 54

join, 26join-semilattice, 29

homomorphism, 29joint kernel pair, 82jointly surjective sink, 123

Kan extensionleft, 68pointwise, 68, 69right, 67

kernel pair, 59, 82joint, 82

Kleislicategory, 84composition, 84convolution, 161extension, 288, 309monoid, see T-monoid(T,V)-tower, 311triple, 85

morphism, 85KM-test space, 415Kock–Zoberlein monad, 110

dual, 111Kowalsky sum, 36Kuratowski

-closure functor, 45closure, 40

large class, 17large-complete, 133lattice, 29

cocontinuous, 340complete, 30completely distributive, 33continuous, 340diamond, 40

lawassociativity, 43distributive, 86

distributive, lax, 182identity, 43middle unit, 87middle-interchange, 19, 48, 178multiplication, 74unit, 74

laxalgebra, see (T,V)-categorycomma category, 303distributive law, 182functor, 106homomorphism, 159homomorphism of quantales, 30homomorphism, compatible, 210idempotent, 110identity, 161transformation, 107

lax extensionassociative, 165dual, 182flat, 157initial, 298morphism, 207of functor, 154of functor, flat, 156of monad, 156symmetric, 391

lean, 148least upper bound, 26left

-cancellable, weakly, 117adjoint, 24, 49, 107companion, 118factorization class, 118unitary, 163

left-cancellable, 117lifting, 109

cartesian, 130cocartesian, 130initial, 131of a functor, 93of a monad, 86

limitclosed under, 64, 117connected, 134creation, 78direct, 63directed, 63inverse, 63

486 INDEX

of a diagram, 57preservation, 63reflection, 63

list monad, 75locally

cartesian closed, 227compact, 269P-compact Hausdorff, 444injective, 435P-injective, 447P-perfect, 443small, 46

lower bound, 27lower-Vietoris space, 273

map, 107(T,V)-continuous, 369V-continuous, 369pointed, 243

mate, 51maximal, 36, 41maximum, 29meet, 27

topological, 352meet-semilattice, 29

homomorphism, 29metacategory, 45metric, 150

closure operation, 200closure space, 200finitary, 151Hausdorff, 253probabilistic, 186relation, 146separated, 150symmetric, 150ultra-, 199

metric space, 150probabilistic, 186ultra-, 199

middle-interchange law, 19, 48, 178unit law, 87

minimum, 29modular, widely, 420module, 22

functor, 46V-module, 153monad, 74

associated, 74

cartesian, 185clique, 296composite, 87discrete presheaf, 320double-powerset, 75down-set, 111down-set-filter, 341extension, 90filter, 75finitary-up-set, 296finite-powerset, 199free-monoid, 75idempotent, 95identity, 75induced, 74lifting, 86list, 75morphism, 74multiplication, 74open-filter, 336ordered-filter, 340power-enriched, 285powerset, 75V-powerset, 149, 180terminal, 92trivial, 92ultraclique, 315ultrafilter, 75unit, 74up-set, 75, 111

monadiccategory, 77category, strictly, 77functor, 77pre-, 94strictly, 77

monic, 59mono

-coreflective, 118-coreflective, regular, 118-fibration, 131-source, 123

mono-source, extremal, 127

monoidcommutative, 20homomorphism, 20, 99in Set, 20in a category, 99

INDEX 487

ordered, 103, 254separated, 338

T-monoid, 286monoidal category, 97

closed, 102homomorphism, 98strict, 98symmetric, 97

monomorphism, 59extremal, 117regular, 71split, 71strong, 138

monotone, 22closure operator, 124

morphism, 43bi-, 98cartesian, 128couniversal, see couniversal arrowdiagonal, 46epi-, 60epic, 60iso-, 43monic, 59mono-, 59of Kleisli triples, 85of lax extensions, 207of monads, 74quotient, 118, 141universal, see universal arrow

multi-ordered set, 348, 380multiple pullback, 59

stable, 117multiplication

law, 74of a monad, 74

naturalisomorphism, 48number object, 38transformation, 46transformation, lax, 107transformation, oplax, 107transformation, pseudo-, 107

V-natural transformation, 112naturality diagram, 46nearly open, 223neighborhood, 42

filter, 42non-expansive map, 151, 192, 199, 200

probabilistically, 186norm, H-valued, 379normal

(T,V)-space, 392topology, 266

object, 43discrete, 128hom-, 111indiscrete, 128initial, 55terminal, 55trivial, 92zero-, 55

observable specialization system, 357open, 25, 223

-closed interchange, 444-filter monad, 336-set functor, 45embedding, 224(T,V)-functor, 401nearly, 223neighborhood filter, 361neighborhood filter map, 361Scott-, 351

P-open, 4392-open, 440(T,V)-open map, 401opfibration, 130oplax

functor, 106transformation, 107

oppositecategory, 44relation, 21V-relation, 146

order, 21-compatible topology, 351-embedding, 29-preserving, see monotoneantisymmetric, see separateddiscrete, 30, 50indiscrete, 50induced, 30, 204partial, 22pointwise, 23pre-, see orderrefinement, 282, 286separated, 21specialization, 30

488 INDEX

total, 21underlying, 30, 204

ordered-filter monad, 340category, 105category, conjugate, 106hull, 22monoid, 103, 254

ordered set, 22cocontinuous, 351complete, 26discrete, 30multi-, 348, 380

ordinal recursion, 39orthogonal

factorization system, see factorization systemmorphism, 116sink, 121

partial-map classifier, 241-map classifier, regular, 241order, see separated orderproduct, 228

M-partial-map classifier, 239map, 239

P-perfectlocally, 443

P-perfect, 433point

-separating source, 123of a frame, 361

pointed, 38map, 243set, 243

pointwise order, 23power-enriched

monad, 285morphism, 285

powersetfunctor, contravariant , 45functor, covariant, 45monad, 75monad, finite-, 199

V-powerset, 180functor, 180monad, 149, 180

pre-approach space, 220premonadic, 94

preorder, see orderpreservation

of colimits, 63of infima, 27of isomorphisms, 70of limits, 63of suprema, 27

pretopological space, 219E-pretopology, 430pretopology, fibrewise, 431prime, completely, 361principal

filter, 35, 36filter monad morphism, 283ideal, 35

probabilisticmetric, 186metric space, 186

probabilistically non-expansive map, 186product

bi-, 109cartesian, 19closed under, 117fibred, 59in a category, 57partial, 228projection, 57tensor, 98

V-profunctor, see V-moduleprojection, 228

of product, 57projective object, 141proper

class, 17factorization system, 116filter, 34(T,V)-functor, 401ideal, 35quasi-, 427super-, 418

(T,V)-proper map, 401P-proper morphism, 433pseudo

-approach space, 220-functor, 106-natural transformation, 107

pseudotopological space, 213pullback, 58–60

multiple, 59

INDEX 489

stable, 117, 227stable, multiple, 117

pushout, 60

quantale, 30cartesian closed, 226commutative, 30homomorphism, 30integral, 148lax homomorphism, 30lean, 148non-trivial, 147superior, 370

quantaloid, 108homomorphism, 108right-sided, 317

quasi-proper, 427quasitopos, 244quotient morphism, 118, 141

recursion, ordinal, 39refinement order, 282, 286reflection, 54

of colimits, 63of isomorphisms, 54of limits, 63separated, 22

reflectivebi-, 118epi-, 118subcategory, 54

E-reflective, 118reflector, 54reflexive

Kleisli morphism, 286relation, 21, 233relation, numerical, 233

regular-subobject classifier, 239completely, Hausdorff, 446epi-reflective, 118epimorphism, 71mono-coreflective, 118monomorphism, 71partial-map classifier, 241(T,V)-space, 388topology, 388

relation, 20auxiliary, 313bi-, 147

dual, 21metric, 146opposite, 21order, 21reflexive, 21, 233reflexive, numerical, 233transitive, 21

(T,V)-relation, 161reflexive, 158transitive, 158

V-relation, 145dual, 146finitary, 150opposite, 146reflexive, 150separated, 150symmetric, 150transitive, 150

replete subcategory, 54representable functor, 71restriction of a filter, 35retraction, 71right

-cancellable, weakly, 121adjoint, 23, 49, 107companion, 118factorization class, 118unitary, 163unitary (T,V)-graph, 217

round filter, 359

Scott-open, 351topology, 342, 351

section, 71self

-adjoint, 50-dual, 45

semilattice, 29homomorphism, 29join-, 29meet-, 29

separated, 22metric, 150monoid, 338order, 21reflection, 22V-relation, 150(T,V)-space, 262, 383

separator, see generator

490 INDEX

sequence, 42constant, 42eventually constant, 42sub-, 42

set, 17-of-objects functor, 141pointed, 243

Sierpinski space, 257sink, 120

epi-, 123hereditary, 245jointly surjective, 123

sliceover, 56under, 56

smallcategory, 43class, 17

sober, 267, 275, 361weakly, 267, 275

sobrification, 361solution set, 65source, 120

initial, 128mono-, 123point-separating, 123

spaceAlexandroff, 136approach, 192bitopological, 200closure, 25KM-test, 415metric, 150Sierpinski, 257T0-, 30topological, 30ultrametric, 199

(T,V)-space, 158, 369compact, 370connected, 456extremally disconnected, 393Hausdorff, 369normal, 392R0, 385R1, 385regular, 388separated, 262, 383T0, 385T1, 385

totally disconnected, 459V-space, 369span, 169spatialization, 361specialization order, 30specialize, 351split

epimorphism, 71fork, 79monomorphism, 71pair, 80

splitting morphism, 79stable, 269, 275

under multiple pullbacks, 117under pullbacks, 117

stablyclosed, 415compact, 269

v-stratum, 307strict

finitely, 247initial object, 449monoidal category, 98

strictly monadic, 77strong

epimorphism, 138generator, 127monomorphism, 138

strongly generating, 127structure

discrete, 203indiscrete, 204morphism, 76

subbase, 268subcategory, 72

coreflective, 54M-coreflective, 118full, 45reflective, 54E-reflective, 118replete, 54

subframe, 30subobject, 119

classifier, 239classifier, regular-, 239

M-subobject, 119classifier, 237

subsequence, 42sup

INDEX 491

-dense, 301-map, 27

sup-dense, 301supercompact, 371superior, 370superproper, 418supremum, 26

preservation, 27symmetric

cartesian product, 20extension, lax, 391metric, 150monoidal category, 97V-relation, 150

taut, 225tensor

in a monoidal category, 98of a quantale, 30

tensored V-category, 215terminal

monad, 92object, 55weakly, 249

thin category, 105three-chain, 31top, 29topological

functor, 131group, 138meet, 352pre-, space, 219pseudo-, space, 213space, 30space, bi-, 200

M0-topologicalfunctor, 132

topology, 30Alexandroff, 136base, 30compact, 191compact, locally, 269connected, 455derived, 431discrete, 50extremally disconnected, 392fibrewise, 431Hausdorff, 131normal, 266on a category, 429

order-compatible, 351R0, 386R1, 386regular, 388Scott, 342, 351subbase, 268T0, 30, 383T1, 386Tychonoff, 446Zariski, 313

E-topology, 429topos, 245

quasi-, 244total order, 21totally below, 33transfinite induction, see ordinal recursiontransitive

Kleisli morphism, 286relation, 21

transportable functor, 133triangular identity, 49trivial

factorization system, 118monad, 92object, 92

two-chain, 31Tychonoff

space, 446Theorem, 373, 438

P-Tychonoff, 443, 444

ultraclique, 314monad, 315

ultrafilter, 36functor, 75monad, 75

ultrametric, 199space, 199

underlying-order functor, 45order, 30, 204

unique diagonalization property, 116unit

law, 74law, middle, 87of a monad, 74of an adjunction, 49

unitary(T,V)-graph, 217left, 163

492 INDEX

(T,V)-relation, 163right, 163(T,V)-graph, 217

universal, 227arrow, 52arrow, weakly, 65property, 52

up-closed, 27-closure, 27-directed, 35

up-set, 27finitary, 295finitary-, monad, 296functor, 75monad, 75, 111

upper bound, 26

vertical composition, 47, 107vertices, 114Vietoris space, lower-, 273

way-below, 267weak

pullback, 172pullback, preservation, 184

weaklyleft-cancellable, 117right-cancellable, 121sober, 267universal arrow, 65

weakly sober, 275wellpowered, 123whiskering

functor, 47left-, 405right-, 405

Yonedaembedding, 48Lemma, 48

Zariski topology, 313zero-object, 55Zorn’s Lemma, 41

monoidal topology - sweetsweet.ua.pt/dirk/artigos/hst14_monoidal_topology_a...monoidal topology...

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