higher-dimensional category theory -...

Higher-dimensional category

theory

Eugenia Cheng

University of Sheffield17th December 2010

1.

Plan

1. Introduction

2.

Plan

1. Introduction

2. Introduction to categories

2.

Plan

1. Introduction


3. Enrichment

2.

Plan

1. Introduction


3. Enrichment

4. Internalisation

2.

Plan

1. Introduction


3. Enrichment

4. Internalisation

5. 2-vector spaces

2.

Plan

1. Introduction


3. Enrichment

4. Internalisation

5. 2-vector spaces

6. Open questions

2.

Plan

1. Introduction


3. Enrichment

4. Internalisation

5. 2-vector spaces

6. Open questions

7. Categories and n-categories in the UK

2.

Plan

1. Introduction


3. Enrichment

4. Internalisation

5. 2-vector spaces

6. Open questions


8. Research areas at the University of Sheffield

2.

1. Introduction

Slogan

Categorification is the general process of

taking a theory of something, and making a

higher-dimensional version.

3.

1. Introduction

Theory ofwidgets

4.

1. Introduction

Theory ofwidgets

studied via Some algebraor other

4.

1. Introduction

Theory ofwidgets


we

dream

of

Higher-dimensionalwidgets

4.

1. Introduction

Theory ofwidgets


we

dream

of


we

dream

of

Higher-dimensionalalgebra

4.

1. Introduction

Theory ofwidgets


we

dream

of


we

dream

of


studied via

4.

1. Introduction

Theory ofwidgets


we

dream

of


we

dream

of


studied via

studied via

4.

1. Introduction

Theory of loopsor paths in a space

studied via Groupsor groupoids

5.

1. Introduction



we

dream

of

Theory of paths in a spaceand all higher homotopies

5.

1. Introduction



we

dream

of


we

dream

of

Higher-dimensionalgroupoids

5.

1. Introduction



we

dream

of


we

dream

of

Higher-dimensionalgroupoids

studied via

studied via

5.

1. Introduction

Cohomology

6.

1. Introduction

Cohomologystudied via

Torsors≡ special kinds of sheaves≡ special functors into Gp

we

dream

of

taking all higher cohomologygroups into accountat the same time

we

dream

of

n-gerbes≡ special n-stacks≡ special functors into n-Gpd

studied via

studied via

6.

1. Introduction

group G

7.

1. Introduction

group Gstudied via functors

G −→ Vect

we

dream

of

n-group G

we

dream

of

n-functorsG −→ n-Vect

studied via

studied via

7.

1. Introduction

Also:

• cobordisms

• topological quantum field theory

• concurrency via fundamental n-category of directedspace

8.

1. Introduction

How do we add dimensions?

9.

1. Introduction


form awidgets are sets with extra structure ⊲ category

9.

1. Introduction



2-widgets categories ′′ ⊲ 2-category

9.

1. Introduction




3-widgets 2-categories ′′ ⊲ 3-category

9.

1. Introduction




3-widgets 2-categories ′′ ⊲ 3-category...

n-widgets (n − 1)-categories ′′ ⊲ n-category

9.

1. Introduction

Strict vs weak

10.

1. Introduction

Strict vs weak

strict: axioms hold on the nose (A × B) × C = A × (B × C)

weak: axioms become coherent isomorphisms (A × B) × C ∼= A × (B × C)

10.

1. Introduction

Strict vs weak



((A × B) × C) × D A × (B × (C × D))

10.

1. Introduction

Strict vs weak



((A × B) × C) × D A × (B × (C × D))

(A × (B × C)) × D A × ((B × C) × D)

(A × B) × (C × D)

10.

1. Introduction

Strict vs weak



• strict 2-categoriesmodel homotopy 2-types

• every weak 2-categoryis equivalent toa strict one

11.

1. Introduction

Strict vs weak



• strict 2-categoriesmodel homotopy 2-types

• strict 3-categoriesdo not model homotopy 3-types

• every weak 2-categoryis equivalent toa strict one

• not every weak 3-categoryis equivalent toa strict one

11.


12.


A category is given by

• a collection of “objects”

• for every pair of objects x, y a collection of “morphisms”x −→ y

12.





equipped with

• identities: for every object x a morphism x1x

−→ x

• composition: for every pair of morphisms xf

−→ yg

−→ z

composition: a morphism xgf−→ z

12.





equipped with

• identities: for every object x a morphism x1x

−→ x

• composition: for every pair of morphisms xf

−→ yg

−→ z

composition: a morphism xgf−→ z

satisfying unit and associativity axioms.

12.


Examples 1

Large categories of mathematical structures:

Objects Morphisms

Set sets functions

Top topological spaces continuous maps

Gp groups group homomorphisms

Ab abelian groups group homomorphisms

ChCpx chain complexes chain maps

Htpy topological spaces homotopy classes ofcontinuous maps.

13.


Examples 2

Algebraic objects as categories:

14.


Examples 2

Algebraic objects as categories:

monoid category with one object

groupoid category in which every morphism is invertible

group category with one objectand every morphism invertible

poset category with a −→ b given by a ≤ b.

14.


There is a large category Cat of small categories.

15.



The morphisms are functors.

15.



The morphisms are functors.

Some examples of functors:

fundamental group Top∗ −→ Gp

(co)homology Top −→ Gp

a representation G −→ Vect

n-dimensional TQFT nCob −→ Vect

a sheaf on X O(X)op −→ Rng.

15.


Idea

We can use categories as a framework

for building higher-dimensional widgets.

16.

3. Enrichment

17.

3. Enrichment

Definition

A (small) category C is given by

17.

3. Enrichment

Definition


• a set obC of objects

• for every pair of objects x, y a set C(x, y) of morphisms

equipped with

• identities: for every object x a function 1 −→ C(x, x)

• composition: for all x, y, z ∈ obC a function

C(y, z) × C(x, y) −→ C(x, z)


17.

3. Enrichment

Definition

A (small) 2-category C is given by



equipped with



C(y, z) × C(x, y) −→ C(x, z)


18.

3. Enrichment

Definition




equipped with



C(y, z) × C(x, y) −→ C(x, z)


category

18.

3. Enrichment

Definition




equipped with



C(y, z) × C(x, y) −→ C(x, z)


category

functor

functor

18.

3. Enrichment

Definition




equipped with



C(y, z) × C(x, y) −→ C(x, z)


category

functor

functor

—a category enriched in Cat.

18.

3. Enrichment

Unravel

19.

3. Enrichment

Unravel

• the category C(x, y) has objects x y// “1-cells”

morphisms x y""

<<�� “2-cells”

composition x y��

CC//��

��“vertical comp.”

19.

3. Enrichment

Unravel


morphisms x y""

<<�� “2-cells”


CC//��


• the functor C(y, z) × C(x, y) −→ C(x, z) gives

on objects x y zf

// g// 7→ x z

gf//

on morphisms x y z

f""

f ′

<<α

��

g

""

g′

<<β�� 7→ x z

gf

""

g′f ′

<<β∗α�� “horizontal comp.”

19.

3. Enrichment

Unravel


morphisms x y""

<<�� “2-cells”


CC//��


• the functor C(y, z) × C(x, y) −→ C(x, z) gives

on objects x y zf

// g// 7→ x z

gf//

on morphisms x y z

f""

f ′

<<α

��

g

""

g′

<<β�� 7→ x z

gf

""

g′f ′

<<β∗α�� “horizontal comp.”

functoriality says

. .�� FF//��

��. .�� FF//

��

��=. . .��//�� //��

. . .FF//��

FF//��

19.

3. Enrichment

Iteration

20.

3. Enrichment

Iteration

A 2-category is a category enriched in Cat.

20.

3. Enrichment

Iteration


A 3-category is a category enriched in 2-Cat:

20.

3. Enrichment

Iteration


A 3-category is a category enriched in 2-Cat:

C(x, y) ∈ 2-Cat

▽

0-cells ⊲ 1-cells of C



An n-category is a category enriched in (n-1)-Cat.

20.

3. Enrichment

Other popular categories in which to enrich:

Ab Abelian groups

Vect vector spaces

Hilb Hilbert spaces

ChCpx chain complexes

Top topological spaces

sSet simplicial sets

Poset posets.

21.

3. Enrichment

Warning

To get weak higher-dimensional structures

we have to do something more subtle

—but we’re not going to do it now.

22.

4. Internalisation

23.

4. Internalisation

Definition


23.

4. Internalisation

Definition


C0 ∈ SetC1

s

ttogether with

23.

4. Internalisation

Definition


C0 ∈ SetC1

s

ttogether with

• identities: a function C0e

−→ C1 such that

C0

C1

e

C0

s

1

C0

t

1

23.

4. Internalisation

Definition


C0 ∈ SetC1

s

ttogether with


−→ C1 such that

C0

C1

e

C0

s

1

C0

t

1

• composition: a function C1 ×C0C1

c−→ C1

identities: a function C0

e−→ C1 such that

C1 ×C0C1

C1

c

C0

s

sπ1

C0

t

tπ2

23.

4. Internalisation

Definition


C0 ∈ SetC1

s

ttogether with


−→ C1 such that

C0

C1

e

C0

s

1

C0

t

1


c−→ C1



C1 ×C0C1

C1

c

C0

s

sπ1

C0

t

tπ2


23.

4. Internalisation

Definition

A (small) double category C is given by a diagram

C0 ∈ SetC1

s

ttogether with


−→ C1 such that

C0

C1

e

C0

s

1

C0

t

1


c−→ C1



C1 ×C0C1

C1

c

C0

s

sπ1

C0

t

tπ2


24.

4. Internalisation

Definition


C0 ∈ SetC1

s

ttogether with


−→ C1 such that

C0

C1

e

C0

s

1

C0

t

1


c−→ C1



C1 ×C0C1

C1

c

C0

s

sπ1

C0

t

tπ2


Cat

24.

4. Internalisation

Definition


C0 ∈ SetC1

s

ttogether with


−→ C1 such that

C0

C1

e

C0

s

1

C0

t

1


c−→ C1



C1 ×C0C1

C1

c

C0

s

sπ1

C0

t

tπ2


Cat

functor

functor

24.

4. Internalisation

Definition


C0 ∈ SetC1

s

ttogether with


−→ C1 such that

C0

C1

e

C0

s

1

C0

t

1


c−→ C1



C1 ×C0C1

C1

c

C0

s

sπ1

C0

t

tπ2


Cat

functor

functor

—an internal category in Cat.

24.

4. Internalisation

Unravel

25.

4. Internalisation

Unravel

We get sets

A0

A1

ts

B0

B1

ts

s

s

t

tA0 = “0-cells”A1 = “vertical 1-cells”

B0 = “horizontal 1-cells”B1 = “2-cells”

25.

4. Internalisation

Unravel

We get sets

A0

A1

ts

B0

B1

ts

s

s

t



• 2-cells have the following shape

z

x

y

w

25.

4. Internalisation

Unravel

We get sets

A0

A1

ts

B0

B1

ts

s

s

t



• 2-cells have the following shape

z

x

y

w

• there is horizontal and vertical composition

25.

4. Internalisation

Another example

An internal category in Gp is a 2-group.

26.

4. Internalisation

Another example


2-groups can be equivalently characterised as

• internal groups in Cat

• 2-categories with only one object, and all cells invertible

• monoidal categories with objects and morphisms invertible

• crossed modules

26.

4. Internalisation

Another example


2-groups can be equivalently characterised as

• internal groups in Cat

• 2-categories with only one object, and all cells invertible

• monoidal categories with objects and morphisms invertible

• crossed modules

Internal categories in Ab are just 2-term chain complexes

G −→ H ∈ Ab.

26.

4. Internalisation

More examples

We can also internalise monoids:

27.

4. Internalisation

More examples


• monoids internal to Mon are commutative monoids

• monoids internal to Cat are (strictly) monoidal categories

• monoids internal to n-Cat are (strictly) monoidal n-categories

27.

4. Internalisation

More examples





We can internalise groups:

• groups internal to Cat are 2-groups

• groups internal to Gp are abelian groups

27.

4. Internalisation

More examples





We can internalise groups:

• groups internal to Cat are 2-groups

• groups internal to Gp are abelian groups

Hence we can also internalise rings and modules.

27.

4. Internalisation

There are (at least) two ways to continue this process:

28.

4. Internalisation


1. Iterate: Cat(Gp) = categories internal to Gp = 2-Gp

Cat(Cat(Gp))

Cat(Cat(Cat(Gp)))...

Catn(Gp)

28.

4. Internalisation



Cat(Cat(Gp))


Catn(Gp)

2. Take internal n-categories

28.

4. Internalisation



Cat(Cat(Gp))


Catn(Gp)


e.g. data underlying a 2-category is

A0 ∈ SetA1

s

tA2

s

t

28.

4. Internalisation



Cat(Cat(Gp))


Catn(Gp)



A0 ∈ SetA1

s

tA2

s

t—we can put this inside other categories.

28.

4. Internalisation



Cat(Cat(Gp))


Catn(Gp)



A0 ∈ SetA1

s

tA2

s


these model n-types

28.

4. Internalisation



Cat(Cat(Gp))


Catn(Gp)



A0 ∈ SetA1

s

tA2

s


these model n-types

but these don’t

28.

5. 2-vector spaces

29.

5. 2-vector spaces

Recap: two methods of categorification

enrichment

internalisation

29.

5. 2-vector spaces


enrichment categories enriched in Vectk

internalisation

29.

5. 2-vector spaces



internalisation categories internal to Vectk

vector spaces internal to Cat

29.

5. 2-vector spaces




vector spaces internal to Cat KV94

29.

5. 2-vector spaces


enrichment categories enriched in Vectk B96



29.

5. 2-vector spaces



internalisation categories internal to Vectk BC04


29.

5. 2-vector spaces



internalisation categories internal to Vectk BC04


However, these methods don’t necessarily go smoothly.

29.

5. 2-vector spaces

Kapranov–Voevodsky (1994)

“A 2-vector space is a vector space internal to Cat.”

• good for K-theory

• strict representation theory (Barrett–Mackay)

• weak representation theory (Elgueta)

30.

5. 2-vector spaces






Baez (1996) —and further work by Bartlett

“A 2-vector space is a category enriched in Vectk.”

30.

5. 2-vector spaces






Baez (1996) —and further work by Bartlett

“A 2-vector space is a category enriched in Vectk.”

Baez–Crans (2004)

“A 2-vector space is a category internal to Vectk.”

• Lie 2-algebras (Baez–Crans)

• Representation theory (Forrester-Barker)

30.

5. 2-vector spaces

Moral

“Categorification” is not a straightforward process.

Different approaches can give different results

that are useful for different things.

31.

6. Open questions

32.

6. Open questions

• What is a good definition of n-category?

• Are different definitions equivalent?

32.

6. Open questions



• What is the (n + 1)-category of n-categories?

32.

6. Open questions




• Coherence and strictification theorems.

32.

6. Open questions





• Modelling homotopy types (Grothendieck).

32.

6. Open questions






• The periodic table (Baez-Dolan).

• The stabilisation hypothesis (BD).

• The tangle hypothesis. (BD)

• The TQFT hypothesis (BD).

32.

6. Open questions






• The periodic table (Baez-Dolan).

• The stabilisation hypothesis (BD).

• The tangle hypothesis. (BD)

• The TQFT hypothesis (BD).

• A “calculus” for n-categories.

• n-category theory.

32.


Mathematics:

• Cambridge: Martin Hyland, Peter Johnstone

• Glasgow: Tom Leinster, Danny Stevenson

• Sheffield: EC, Nick Gurski, Simon Willerton, NeilStrickland, John Greenlees

33.


Mathematics:

• Cambridge: Martin Hyland, Peter Johnstone

• Glasgow: Tom Leinster, Danny Stevenson

• Sheffield: EC, Nick Gurski, Simon Willerton, NeilStrickland, John Greenlees

Computer Science:

• Cambridge: Marcelo Fiore, Glynn Winskel

• Oxford: Bob Coecke, Samson Abramsky

• Bath: John Power

• Strathclyde: Neil Ghani

33.

Pure Maths at Sheffield

• Algebra — commutative: tight closure, local cohomology;noncommutative: Weyl algebras, quantum algebras

• Analysis — real, complex, functional, harmonic, numericaland stochastic analysis

• Category theory — higher-dimensional category theory,model categories, triangulated categories, operads, applications

• Differential geometry — Lie groupoids and Lie algebroids,foliation theory, etale groupoids, orbifolds;

• Number theory — elliptic curves, modular forms, Eisensteincohomology

• Topology — stable homotopy theory, equivariant versions,generalised cohomology rings, categorical foundations ofhomotopy theory

34.

Applied Maths at Sheffield

• Computational fluid dynamics — vortex dynamics,turbulence, engineering fluid dynamics, acoustic waves

• Environmental dynamics — synthetic aperture radar,turbulent diffusion, meteorology, oceanography

• Nonlinear control — adaptive backstepping control, thesecond-order sliding mode, nonlinear discrete-time systems

• Particle astrophysics and gravitation — cosmology,gravitation, classical and quantum behaviour of black holes,fundamental theory of space and time

• Solar physics and space plasma research centre —theoretical and observational issues, helioseismology,coronal-seismology, magnetohydrodynamics

35.

Probability and Statistics at Sheffield

• Bayesian Statistics — medical statistics, quantifyinguncertainty in computer models, Bayesian time series analysis,

• Mathematical modelling — genetic epidemiology, statisticalgenetics, evolutionary conflicts

• Statistical modelling and applied statistics —environmental statistics, scientific dating methods, calibrationand model uncertainty, particle size distributions and pollutionmonitoring.

• Probability — fractals, random graphs, stochastic processes

36.

Applying to the University Sheffield

See http://www.shef.ac.uk/postgraduate/research/

Sources of funding:

• University Prize Scholarships — open to all, deadline28th January

• SoMaS Graduate Teaching Assistantships — open toUK and EU applicants, deadlines 25/2, 29/4, 29/7

• SoMaS Studentships — open to UK and EUapplicants, deadlines 25/2, 29/4, 29/7

Contact:

Prof. Caitlin Buck, Director of Post-Graduate [email protected]

37.

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