Representable functorsFrom Wikipedia, the free encyclopediaContents1 Browns representability theorem 11.1 Brown representability theorem for CW complexes . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Category of elements 32.1 The category of elements of a presheaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Classifying space 53.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Formalism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.4 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Moduli scheme 84.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Representable functor 95.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.2 Universal elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.4.1 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.4.2 Preservation of limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.4.3 Left adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.5 Relation to universal morphisms and adjoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11iii CONTENTS6 Volodin space 126.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Yoneda lemma 137.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137.2 Formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137.2.1 General version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137.2.2 Naming conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.2.3 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.2.4 The Yoneda embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.3 Preadditive categories, rings and modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 177.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Chapter 1Browns representability theoremIn mathematics, Browns representability theorem in homotopy theory[1] gives necessary and sucient conditionsfor a contravariant functor F on the homotopy category Hotc of pointed connected CW complexes, to the categoryof sets Set, to be a representable functor.More specically, we are givenF: Hotcop Set,and there are certain obviously necessary conditions for F to be of type Hom(, C), with C a pointed connectedCW-complex that can be deduced from category theory alone. The statement of the substantive part of the theoremis that these necessary conditions are then sucient. For technical reasons, the theorem is often stated for functorsto the category of pointed sets; in other words the sets are also given a base point.1.1 Brown representability theorem for CW complexesThe representability theorem for CW complexes, due to E. H. Brown,[2] is the following. Suppose that:1. The functor F maps coproducts (i.e. wedge sums) in Hotc to products in Set:F(X) =F(X),2. The functor F maps homotopy pushouts in Hotc to weak pullbacks.This is often stated as a Mayer-Vietorisaxiom: for any CW complex W covered by two subcomplexes U and V, and any elements u F(U), v F(V)such that u and v restrict to the same element of F(U V), there is an element w F(W) restricting to u andv, respectively.Then F is representable by some CW complex C, that is to say there is an isomorphismF(Z) HomHotc(Z, C)for any CW complex Z, which is natural in Z in that for any morphism from Z to another CW complex Y the inducedmaps F(Y) F(Z) and HomHot(Y, C) HomHot(Z, C) are compatible with these isomorphisms.The converse statement also holds: any functor represented by a CW complex satises the above two properties.This direction is an immediate consequence of basic category theory, so the deeper and more interesting part of theequivalence is the other implication.The representing object C above can be shown to depend functorially on F: any natural transformation from F toanother functor satisfying the conditions of the theorem necessarily induces a map of the representing objects. Thisis a consequence of Yonedas lemma.Taking F(X) to be the singular cohomology group Hi(X,A) with coecients in a given abelian group A, for xed i> 0; then the representing space for F is the Eilenberg-MacLane space K(A, i). This gives a means of showing theexistence of Eilenberg-MacLane spaces.12 CHAPTER 1. BROWNS REPRESENTABILITY THEOREM1.2 VariantsSince the homotopy category of CW-complexes is equivalent to the localization of the category of all topologicalspaces at the weak homotopy equivalences, the theorem can equivalently be stated for functors on a category denedin this way.However, the theorem is false without the restriction to connected pointed spaces, and an analogous statement forunpointed spaces is also false.[3]Asimilar statement does, however, hold for spectra instead of CWcomplexes. Brown also proved a general categoricalversion of the representability theorem,[4] which includes both the version for pointed connected CW complexes andthe version for spectra.A version of the representability theorem in the case of triangulated categories is due to Amnon Neeman.[5] Togetherwith the preceding remark, it gives a criterion for a (covariant) functor F: C D between triangulated categoriessatisfying certain technical conditions to have a right adjoint functor. Namely, if C and D are triangulated categorieswith C compactly generated and F a triangulated functor commuting with arbitrary direct sums, then F is a leftadjoint. Neeman has applied this to proving the Grothendieck duality theorem in algebraic geometry.Jacob Lurie has proved a version of the Brown representability theorem[6] for the homotopy category of a pointedquasicategory with a compact set of generators which are cogroup objects in the homotopy category. For instance, thisapplies to the homotopy category of pointed connected CW complexes, as well as to the unbounded derived categoryof a Grothendieck abelian category (in view of Luries higher-categorical renement of the derived category).1.3 References[1] Switzer, Robert M. (2002), Algebraic topology---homotopy and homology, Classics in Mathematics, Berlin, New York:Springer-Verlag, ISBN 978-3-540-42750-6, MR 1886843, see pages 152157[2] Brown, Edgar H. (1962), Cohomology theories, Annals of Mathematics. Second Series 75: 467484, ISSN 0003-486X,JSTOR 1970209, MR 0138104[3] Freyd, Peter; Heller, Alex (1993), Splitting homotopy idempotents. II., Journal of Pure and Applied Algebra 89 (12):93106, doi:10.1016/0022-4049(93)90088-b[4] Brown, Edgar H. (1965), Abstract homotopy theory, Transactions of the AMS 119 (1): 7985, doi:10.2307/1994231[5] Neeman, Amnon (1996), The Grothendieck duality theorem via Bouselds techniques and Brown representability,Journal of the American Mathematical Society 9 (1): 205236, doi:10.1090/S0894-0347-96-00174-9, ISSN 0894-0347,MR 1308405[6] Lurie, Jacob (2011), Higher Algebra (PDF)Chapter 2Category of elementsIn category theory, if C is a category and F: C Set is a set-valued functor, the category of elements of F el(F)(also denoted by CF) is the category dened as follows:Objects are pairs (A, a) where A Ob(C) and a FA .An arrow (A, a) (B, b) is an arrow f: A B in C such that (Ff)a = b .A more concise way to state this is that the category of elements of F is the comma category F , where is aone-point set. The category of elements of F comes with a natural projection el(F) C that sends an object (A,a)to A, and an arrow (A, a) (B, b) to its underlying arrow in C.2.1 The category of elements of a presheafSomewhat confusingly in some texts (e.g. Mac Lane, Moerdijk), the category of elements for a presheaf is deneddierently. If P C:= SetCopis a presheaf, the category of elements of P (again denoted by el(P) , or, to makethe distinction to the above denition clear, C P) is the category dened as follows:Objects are pairs (A, a) where A Ob(C) and a P(A) .An arrow (A, a) (B, b) is an arrow f: A B in C such that (Pf)b = a .As one sees, the direction of the arrows is reversed. One can, once again, state this denition in a more concisemanner: the category just dened is nothing but ( P)op . Consequentially, in the spirit of adding a co in frontof the name for a construction to denote its opposite, one should rather call this category the category of coelementsof P.For C small, this construction can be extended into a functor C fromC to Cat , the category of small categories. Infact, using the Yoneda lemma one can show that CP= y P , where y : C C is the Yoneda embedding. Thisisomorphism is natural in P and thus the functor C is naturally isomorphic to y :C Cat .2.2 See alsoGrothendieck construction2.3 ReferencesMac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5(2nd ed.). Springer-Verlag. ISBN 0-387-98403-8.Mac Lane, Saunders; Moerdijk, Ieke (1992). Sheaves in Geometry and Logic. Universitext (corrected ed.).Springer-Verlag. ISBN 0-387-97710-4.34 CHAPTER 2. CATEGORY OF ELEMENTS2.4 External linksCategory of elements in nLabChapter 3Classifying spaceIn mathematics, specically in homotopy theory, a classifying space BG of a topological group G is the quotient ofa weakly contractible space EG (i.e. a topological space for which all its homotopy groups are trivial) by a free actionof G. It has the property that any G principal bundle over a paracompact manifold is isomorphic to a pullback of theprincipal bundle EG BG.[1]For a discrete group G, BG is, roughly speaking, a path-connected topological space X such that the fundamentalgroup of X is isomorphic to G and the higher homotopy groups of X are trivial, that is, BG is an Eilenberg-Maclanespace, or a K(G,1).3.1 MotivationAn example for G innite cyclic is the circle as X. When G is a discrete group, another way to specify the conditionon X is that the universal cover Y of X is contractible. In that case the projection map: Y Xbecomes a ber bundle with structure group G, in fact a principal bundle for G. The interest in the classifying spaceconcept really arises from the fact that in this case Y has a universal property with respect to principal G-bundles, inthe homotopy category. This is actually more basic than the condition that the higher homotopy groups vanish: thefundamental idea is, given G, to nd such a contractible space Y on which G acts freely. (The weak equivalence ideaof homotopy theory relates the two versions.) In the case of the circle example, what is being said is that we remarkthat an innite cyclic group C acts freely on the real line R, which is contractible. Taking X as the quotient spacecircle, we can regard the projection from R = Y to X as a helix in geometrical terms, undergoing projection fromthree dimensions to the plane. What is being claimed is that has a universal property amongst principal C-bundles;that any principal C-bundle in a denite way 'comes from' .3.2 FormalismA more formal statement takes into account that G may be a topological group (not simply a discrete group), and thatgroup actions of G are taken to be continuous; in the absence of continuous actions the classifying space concept canbe dealt with, in homotopy terms, via the EilenbergMacLane space construction. In homotopy theory the denitionof a topological space BG, the classifying space for principal G-bundles, is given, together with the space EG whichis the total space of the universal bundle over BG. That is, what is provided is in fact a continuous mapping: EG BG.Assume that the homotopy category of CW complexes is the underlying category, from now on. The classifyingproperty required of BG in fact relates to . We must be able to say that given any principal G-bundle56 CHAPTER 3. CLASSIFYING SPACE: Y Zover a space Z, there is a classifying map from Z to BG, such that is the pullback of along . In less abstractterms, the construction of by 'twisting' should be reducible via to the twisting already expressed by the constructionof .For this to be a useful concept, there evidently must be some reason to believe such spaces BG exist. In abstract terms(which are not those originally used around 1950 when the idea was rst introduced) this is a question of whether thecontravariant functor from the homotopy category to the category of sets, dened byh(Z) = set of isomorphism classes of principal G-bundles on Zis a representable functor. The abstract conditions being known for this (Browns representability theorem) ensurethat the result, as an existence theorem, is armative and not too dicult.3.3 Examples1. The circle S1is a classifying space for the innite cyclic group Z.2. The n-torus Tnis a classifying space for Zn, the free abelian group of rank n.3. The wedge of n circles is a classifying space for the free group of rank n.4. A closed (that is compact and without boundary) connected surface S of genus at least 1 is a classifying spacefor its fundamental group 1(S) .5. The innite-dimensional projective space RP is a classifying space for Z/2Z.6. A closed (that is compact and without boundary) connected hyperbolic manifold M is a classifying space forits fundamental group 1(M) .7. A nite connected locally CAT(0) cubical complex is a classifying space of its fundamental group.8. CP is the classifying space BS1for the circle S1thought of as a compact topological group.3.4 ApplicationsThis still leaves the question of doing eective calculations with BG; for example, the theory of characteristic classesis essentially the same as computing the cohomology groups of BG, at least within the restrictive terms of homotopytheory, for interesting groups G such as Lie groups (H Cartans theorem). As was shown by the Bott periodicity the-orem, the homotopy groups of BG are also of fundamental interest. The early work on classifying spaces introducedconstructions (for example, the bar construction), that gave concrete descriptions as a simplicial complex.An example of a classifying space is that when G is cyclic of order two; then BG is real projective space of innitedimension, corresponding to the observation that EG can be taken as the contractible space resulting from removingthe origin in an innite-dimensional Hilbert space, with G acting via v going to v, and allowing for homotopyequivalence in choosing BG. This example shows that classifying spaces may be complicated.In relation with dierential geometry (ChernWeil theory) and the theory of Grassmannians, a much more hands-onapproach to the theory is possible for cases such as the unitary groups that are of greatest interest. The constructionof the Thom complex MG showed that the spaces BG were also implicated in cobordism theory, so that they assumeda central place in geometric considerations coming out of algebraic topology. Since group cohomology can (in manycases) be dened by the use of classifying spaces, they can also be seen as foundational in much homological algebra.Generalizations include those for classifying foliations, and the classifying toposes for logical theories of the predicatecalculus in intuitionistic logic that take the place of a 'space of models.3.5. REFERENCES 73.5 References[1] Stashe, James D. (1971), "H-spaces and classifying spaces:foundations and recent developments, Algebraic topology(Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970), Providence, R.I.: American MathematicalSociety, pp. 247272, Theorem 2J.P. May, A concise course in algebraic topology3.6 External linksHazewinkel, Michiel, ed.(2001), Classifying space, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-43.7 See alsoClassifying space for O(n), BO(n)Classifying space for U(n), BU(n)Classifying stackBorels theoremEquivariant cohomologyChapter 4Moduli schemeIn mathematics, a moduli scheme is a moduli space that exists in the category of schemes developed by AlexanderGrothendieck. Some important moduli problems of algebraic geometry can be satisfactorily solved by means ofscheme theory alone, while others require some extension of the 'geometric object' concept (algebraic spaces, algebraicstacks of Michael Artin).Work of Grothendieck and David Mumford (see geometric invariant theory) opened up this area in the early 1960s.The more algebraic and abstract approach to moduli problems is to set them up as a representable functor question,then apply a criterion that singles out the representable functors for schemes. When this programmatic approachworks, the result is a ne moduli scheme. Under the inuence of more geometric ideas, it suces to nd a schemethat gives the correct geometric points. This is more like the classical idea that the moduli problem is to express thealgebraic structure naturally coming with a set (say of isomorphism classes of elliptic curves). The result is then acoarse moduli scheme. Its lack of renement is, roughly speaking, that it doesn't guarantee for families of objectswhat is inherent in the ne moduli scheme.As Mumford pointed out in his book Geometric Invariant Theory, onemight want to have the ne version, but there is a technical issue (level structure and other 'markings) that must beaddressed to get a question with a chance of having such an answer.4.1 ReferencesHazewinkel, Michiel, ed. (2001), Moduli theory, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-48Chapter 5Representable functorIn mathematics, particularly category theory, a representable functor is a functor of a special form from an arbitrarycategory into the category of sets. Such functors give representations of an abstract category in terms of knownstructures (i.e. sets and functions) allowing one to utilize, as much as possible, knowledge about the category of setsin other settings.From another point of view, representable functors for a category C are the functors given with C. Their theory is avast generalisation of upper sets in posets, and of Cayleys theorem in group theory.5.1 DenitionLet C be a locally small category and let Set be the category of sets. For each object A of C let Hom(A,) be thehom functor that maps objects X to the set Hom(A,X).A functor F : C Set is said to be representable if it is naturally isomorphic to Hom(A,) for some object A of C.A representation of F is a pair (A, ) where : Hom(A,) Fis a natural isomorphism.A contravariant functor G from C to Set is the same thing as a functor G : Cop Set and is commonly called apresheaf. A presheaf is representable when it is naturally isomorphic to the contravariant hom-functor Hom(,A) forsome object A of C.5.2 Universal elementsAccording to Yonedas lemma, natural transformations from Hom(A,) to F are in one-to-one correspondence withthe elements of F(A). Given a natural transformation : Hom(A,) F the corresponding element u F(A) is givenbyu = A(idA).Conversely, given any element u F(A) we may dene a natural transformation : Hom(A,) F viaX(f) = (Ff)(u)where f is an element of Hom(A,X). In order to get a representation of F we want to know when the natural trans-formation induced by u is an isomorphism. This leads to the following denition:910 CHAPTER 5. REPRESENTABLE FUNCTORA universal element of a functor F : C Set is a pair (A,u) consisting of an object A of C and anelement u F(A) such that for every pair (X,v) with v F(X) there exists a unique morphism f : A X such that (Ff)u = v.A universal element may be viewed as a universal morphism from the one-point set {} to the functor F or as aninitial object in the category of elements of F.The natural transformation induced by an element u F(A) is an isomorphismif and only if (A,u) is a universal elementof F. We therefore conclude that representations of F are in one-to-one correspondence with universal elements ofF. For this reason, it is common to refer to universal elements (A,u) as representations.5.3 ExamplesConsider the contravariant functor P : Set Set which maps each set to its power set and each function to itsinverse image map. To represent this functor we need a pair (A,u) where A is a set and u is a subset of A, i.e.an element of P(A), such that for all sets X, the hom-set Hom(X,A) is isomorphic to P(X) via X(f) = (Pf)u= f1(u). Take A = {0,1} and u = {1}. Given a subset S X the corresponding function from X to A is thecharacteristic function of S.Forgetful functors to Set are very often representable. In particular, a forgetful functor is represented by (A,u) whenever A is a free object over a singleton set with generator u.The forgetful functor Grp Set on the category of groups is represented by (Z, 1).The forgetful functor Ring Set on the category of rings is represented by (Z[x], x), the polynomialring in one variable with integer coecients.The forgetful functor Vect Set on the category of real vector spaces is represented by (R, 1).The forgetful functor Top Set on the category of topological spaces is represented by any singletontopological space with its unique element.A group G can be considered a category (even a groupoid) with one object which we denote by . A functorfrom G to Set then corresponds to a G-set. The unique hom-functor Hom(,) from G to Set corresponds tothe canonical G-set G with the action of left multiplication. Standard arguments from group theory show that afunctor from G to Set is representable if and only if the corresponding G-set is simply transitive (i.e. a G-torsoror heap). Choosing a representation amounts to choosing an identity for the heap.Let C be the category of CW-complexes with morphisms given by homotopy classes of continuous functions.For each natural number n there is a contravariant functor Hn: C Ab which assigns each CW-complexits nth cohomology group (with integer coecients). Composing this with the forgetful functor we have acontravariant functor from C toSet. Browns representability theorem in algebraic topology says that thisfunctor is represented by a CW-complex K(Z,n) called an EilenbergMac Lane space.5.4 Properties5.4.1 UniquenessRepresentations of functors are unique up to a unique isomorphism. That is, if (A1,1) and (A2,2) represent thesame functor, then there exists a unique isomorphism : A1 A2 such that11 2= Hom(, )as natural isomorphisms from Hom(A2,) to Hom(A1,). This fact follows easily from Yonedas lemma.Stated in terms of universal elements: if (A1,u1) and (A2,u2) represent the same functor, then there exists a uniqueisomorphism : A1 A2 such that(F)u1= u2.5.5. RELATION TO UNIVERSAL MORPHISMS AND ADJOINTS 115.4.2 Preservation of limitsRepresentable functors are naturally isomorphic to Hom functors and therefore share their properties. In particular,(covariant) representable functors preserve all limits. It follows that any functor which fails to preserve some limit isnot representable.Contravariant representable functors take colimits to limits.5.4.3 Left adjointAny functor K : C Set with a left adjoint F : Set C is represented by (FX, X()) where X = {} is a singletonset and is the unit of the adjunction.Conversely, if K is represented by a pair (A, u) and all small copowers of A exist in C then K has a left adjoint Fwhich sends each set I to the Ith copower of A.Therefore, if C is a category with all small copowers, a functor K : C Set is representable if and only if it has a leftadjoint.5.5 Relation to universal morphisms and adjointsThe categorical notions of universal morphisms and adjoint functors can both be expressed using representable func-tors.Let G : D C be a functor and let X be an object of C. Then (A,) is a universal morphism from X to G if and onlyif (A,) is a representation of the functor HomC(X,G) from D to Set. It follows that G has a left-adjoint F if andonly if HomC(X,G) is representable for all X in C. The natural isomorphism X : HomD(FX,) HomC(X,G)yields the adjointness; that isX,Y: HomD(FX, Y ) HomC(X, GY )is a bijection for all X and Y.The dual statements are also true. Let F : C D be a functor and let Y be an object of D. Then (A,) is a universalmorphism from F to Y if and only if (A,) is a representation of the functor HomD(F,Y) from C to Set. It followsthat F has a right-adjoint G if and only if HomD(F,Y) is representable for all Y in D.5.6 See alsoSubobject classier5.7 ReferencesMac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5(2nd ed.). Springer. ISBN 0-387-98403-8.Chapter 6Volodin spaceIn mathematics, more specically in topology, the Volodin space X of a ring R is a subspace of the classifying spaceBGL(R) given byX=n,B(Un(R))where Un(R) GLn(R) is the subgroup of upper triangular matrices with 1s on the diagonal (i.e., the unipotentradical of the standard Borel) and a permutation matrix thought of as an element in GLn(R) and acting (superscript)by conjugation.[1] The space is acyclic and the fundamental group 1X is the Steinberg group St(R) of R. In fact,Suslins 1981 paper [2] explains that X yields a model for the Quillens plus-construction BGL(R)/X BGL+(R)in algebraic K-theory.6.1 Notes[1] Weilbel 2013, Ch. IV. Example 1.3.2.[2] A. A. Suslin, On the equivalence of K-theories, Comm. Algebra 9 (1981), no. 15, 15591566.6.2 ReferencesC. Weibel, The K-book: an introduction to algebraic K-theoryI. Volodin, Algebraic K-theory as extraordinary homology theory on the category of associative rings withunity, Izv. Akad. Nauk. SSSR, 35, (Translation: Math. USSR Izvestija Vol. 5 (1971) No. 4, 859887)12Chapter 7Yoneda lemmaIn mathematics, specically in category theory, theYonedalemma is an abstract result on functors of the typemorphisms into a xed object. It is a vast generalisation of Cayleys theorem from group theory (viewing a groupas a particular kind of category with just one object). It allows the embedding of any category into a category offunctors (contravariant set-valued functors) dened on that category. It also claries how the embedded category, ofrepresentable functors and their natural transformations, relates to the other objects in the larger functor category. Itis an important tool that underlies several modern developments in algebraic geometry and representation theory. Itis named after Nobuo Yoneda.7.1 GeneralitiesThe Yoneda lemma suggests that instead of studying the (locally small) category C, one should study the category ofall functors of C into Set (the category of sets with functions as morphisms). Set is a category we think we understandwell, and a functor of C into Set can be seen as a representation of C in terms of known structures. The originalcategory C is contained in this functor category, but new objects appear in the functor category, which were absentand hidden in C. Treating these new objects just like the old ones often unies and simplies the theory.This approach is akin to (and in fact generalizes) the common method of studying a ring by investigating the modulesover that ring. The ring takes the place of the category C, and the category of modules over the ring is a category offunctors dened on C.7.2 Formal statement7.2.1 General versionYonedas lemma concerns functors froma xed category C to the category of sets, Set. If C is a locally small category(i.e. the hom-sets are actual sets and not proper classes), then each object A of C gives rise to a natural functor to Setcalled a hom-functor. This functor is denoted:hA= Hom(A, ).The (covariant) hom-functor hAsends X to the set of morphisms Hom(A,X) and sends a morphism f from X to Y tothe morphismf (composition with f on the left) that sends a morphism g in Hom(A,X) to the morphism f o g inHom(A,Y). That is,f Hom(A, f) = [[Hom(A, X) g f g Hom(A, Y )]]Let F be an arbitrary functor from C to Set. Then Yonedas lemma says that:1314 CHAPTER 7. YONEDA LEMMAFor each object A of C, the natural transformations from hAto F are in one-to-one correspondence with the elementsof F(A). That is,Nat(hA, F) = F(A).Moreover this isomorphism is natural in A and F when both sides are regarded as functors from SetCx C to Set.(Here the notation SetCdenotes the category of functors from C to Set.)Given a natural transformation from hAto F, the corresponding element of F(A) is u = A(idA) .[lower-alpha 1]There is a contravariant version of Yonedas lemma, which concerns contravariant functors from C toSet. Thisversion involves the contravariant hom-functorhA= Hom(, A),which sends X to the hom-set Hom(X,A). Given an arbitrary contravariant functor G from C to Set, Yonedas lemmaasserts thatNat(hA, G) = G(A).7.2.2 Naming conventionsThe use of "hA" for the covariant hom-functor and "hA" for the contravariant hom-functor is not completely standard.Many texts and articles either use the opposite convention or completely unrelated symbols for these two functors.However, most modern algebraic geometry texts starting with Alexander Grothendieck's foundational EGA use theconvention in this article.[lower-alpha 2]The mnemonic falling into something can be helpful in remembering that "hA" is the contravariant hom-functor.When the letter "A" is falling (i.e. a subscript), hA assigns to an object X the morphisms from X into A.7.2.3 ProofThe proof of Yonedas lemma is indicated by the following commutative diagram:Proof of Yonedas lemma7.3. PREADDITIVE CATEGORIES, RINGS AND MODULES 15This diagram shows that the natural transformation is completely determined byA(idA) =u since for eachmorphism f : A X one hasX(f) = (Ff)u.Moreover, any element uF(A) denes a natural transformation in this way.The proof in the contravariant case iscompletely analogous.In this way, Yonedas Lemma provides a complete classication of all natural transformations from the functorHom(A,-) to an arbitrary functor F:CSet.7.2.4 The Yoneda embeddingAn important special case of Yonedas lemma is when the functor F from C to Set is another hom-functor hB. In thiscase, the covariant version of Yonedas lemma states thatNat(hA, hB) = Hom(B, A).That is, natural transformations between hom-functors are in one-to-one correspondence with morphisms (in thereverse direction) between the associated objects. Given a morphism f : B A the associated natural transformationis denoted Hom(f,).Mapping each object A in C to its associated hom-functor hA= Hom(A,) and each morphism f : B A to thecorresponding natural transformation Hom(f,) determines a contravariant functor h from C to SetC, the functorcategory of all (covariant) functors from C to Set. One can interpret h as a covariant functor:h: Cop SetC.The meaning of Yonedas lemma in this setting is that the functor h is fully faithful, and therefore gives an embeddingof Cop in the category of functors toSet. The collection of all functors {hA, A in C} is a subcategory ofSetC.Therefore, Yoneda embedding implies that the category Copis isomorphic to the category {hA, A in C}.The contravariant version of Yonedas lemma states thatNat(hA, hB) = Hom(A, B).Therefore, h gives rise to a covariant functor from C to the category of contravariant functors to Set:h: C SetCop.Yonedas lemma then states that any locally small category C can be embedded in the category of contravariantfunctors from C to Set via h. This is called the Yoneda embedding.7.3 Preadditive categories, rings and modulesA preadditive category is a category where the morphism sets form abelian groups and the composition of morphismsis bilinear; examples are categories of abelian groups or modules. In a preadditive category, there is both a multipli-cation and an addition of morphisms, which is why preadditive categories are viewed as generalizations of rings.Rings are preadditive categories with one object.The Yoneda lemma remains true for preadditive categories if we choose as our extension the category of additivecontravariant functors from the original category into the category of abelian groups; these are functors which arecompatible with the addition of morphisms and should be thought of as forming a module category over the original16 CHAPTER 7. YONEDA LEMMAcategory. The Yoneda lemma then yields the natural procedure to enlarge a preadditive category so that the enlargedversion remains preadditive in fact, the enlarged version is an abelian category, a much more powerful condition.In the case of a ring R, the extended category is the category of all right modules over R, and the statement of theYoneda lemma reduces to the well-known isomorphismM HomR(R,M) for all right modules M over R.7.4 HistoryThe Yoneda lemma was introduced but not proved in a 1954 paper by Nobuo Yoneda.[1] Yoshiki Kinoshita stated in1996 that the termYoneda lemma was coined by Saunders Mac Lane following an interviewhe had with Yoneda.[2]7.5 See alsoRepresentation theorem7.6 Notes[1] Recall that A: Hom(A, A) F(A) so the last expression is well-dened and sends a morphism from A to A, to anelement in F(A).[2] A notable exception to modern algebraic geometry texts following the conventions of this article is Commutative algebrawith a view toward algebraic geometry / David Eisenbud (1995), which uses "hA" to mean the covariant hom-functor.However, the later book The geometry of schemes / David Eisenbud, Joe Harris (1998) reverses this and uses "hA" to meanthe contravariant hom-functor.7.7 References[1] Nobuo, Yoneda (1954). On the homology theory of modules.J. Fac.Sci.Univ.Tokyo.Sect.I 7: 193227. Retrieved21 December 2013. (subscription required)[2] Kinoshita, Yoshiki (23 April 1996). Prof. Nobuo Yoneda passed away. Retrieved 21 December 2013.Freyd, Peter (1964), Abelian categories, Harpers Series in Modern Mathematics (2003 reprint ed.), Harperand Row, Zbl 0121.02103.Mac Lane, Saunders (1998), Categories for the Working Mathematician, Graduate Texts in Mathematics 5 (2nded.), New York, NY: Springer-Verlag, ISBN 0-387-98403-8, Zbl 0906.18001Yoneda lemma in nLab7.8. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 177.8 Text and image sources, contributors, and licenses7.8.1 Text Browns representability theorem Source: https://en.wikipedia.org/wiki/Brown{}s_representability_theorem?oldid=645943484 Con-tributors: Michael Shulman, Charles Matthews, Giftlite, Gauge, Rjwilmsi, Bluebot, Headbomb, Jakob.scholbach, LokiClock, Rswarbrick,Yobot, Erik9bot, NSH002, Compact3dmanifold, K9re11 and Anonymous: 4 Category of elements Source: https://en.wikipedia.org/wiki/Category_of_elements?oldid=607160144 Contributors: Charles Matthews,Smimram, Algebraist, CBM, RogierBrussee, Owlcatowl, Yobot, Erik9bot, TheLaeg, Cesme'es, Mark viking and Anonymous: 2 Classifying space Source: https://en.wikipedia.org/wiki/Classifying_space?oldid=634291419 Contributors: AxelBoldt, Michael Hardy,TakuyaMurata, Charles Matthews, Tobias Bergemann, Giftlite, Lethe, Fropu, Sam nead, Gauge, Linas, Frankie1969, BD2412, Gilliam,Nbarth, Jon Awbrey, Agol, Jakob.scholbach, David Eppstein, STBot, Katzmik, Nsk92, Addbot, AnomieBOT, Materialscientist, Erik9bot,FrescoBot, SuperJew, Boriaj, Yoavlen and Anonymous: 18 Moduli scheme Source: https://en.wikipedia.org/wiki/Moduli_scheme?oldid=319458419 Contributors: Charles Matthews, Zvika andAnonymous: 1 Representable functor Source: https://en.wikipedia.org/wiki/Representable_functor?oldid=616800526 Contributors: Michael Hardy,Charles Matthews, Lethe, Fropu, Edcolins, Smimram, Mat cross, Salix alba, Mathbot, Chobot, Crasshopper, Mets501, Vaughan Pratt,AndrewHowse, Avakar, Anonymous Dissident, Ocsenave, Thehotelambush, Addbot, Yobot, Delilahblue, Citation bot, and Anonymous:12 Volodinspace Source: https://en.wikipedia.org/wiki/Volodin_space?oldid=648645652 Contributors: Michael Hardy, TakuyaMurata,David Eppstein, Santryl, Cyphoidbomb and K9re11 Yoneda lemma Source: https://en.wikipedia.org/wiki/Yoneda_lemma?oldid=675969739 Contributors: AxelBoldt, Bryan Derksen, TobyBartels, Michael Hardy, Chinju, AugPi, Charles Matthews, Greenrd, Fredrik, Giftlite, Lethe, MathKnight, Fropu, Waltpohl, Semor-rison, DemonThing, DefLog~enwiki, Smimram, Paul August, Gauge, Viriditas, Julien Tuerlinckx, Graham87, Chobot, Hairy Dude,Michael Slone, Gaius Cornelius, FF2010, SmackBot, Acepectif, Quentin72, Almeo, Michael Kinyon, Tyrrell McAllister, WAREL, Zerosharp, Konstantin.Solomatov, HStel, Blaisorblade, Kilva, Aretakis, Download, Legobot, FrescoBot, BenzolBot, Magmalex, EmausBot,Leslie.Hetherington, IkamusumeFan, APerson, Deltahedron and Anonymous: 297.8.2 Images File:E-to-the-i-pi.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/35/E-to-the-i-pi.svg License: CC BY 2.5 Contribu-tors: ? 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