Higher-Dimensional Category Theory

Higher-dimensional category theory

Eugenia Cheng

University of Sheﬃeld 17th December 2010

1. Plan

1. Introduction

2. Plan

1. Introduction 2. Introduction to categories

2. Plan

1. Introduction 2. Introduction to categories 3. Enrichment

2. Plan

1. Introduction 2. Introduction to categories 3. Enrichment 4. Internalisation

2. Plan

1. Introduction 2. Introduction to categories 3. Enrichment 4. Internalisation 5. 2-vector spaces

2. Plan

1. Introduction 2. Introduction to categories 3. Enrichment 4. Internalisation 5. 2-vector spaces 6. Open questions

2. Plan

1. Introduction 2. Introduction to categories 3. Enrichment 4. Internalisation 5. 2-vector spaces 6. Open questions 7. Categories and n-categories in the UK

2. Plan

1. Introduction 2. Introduction to categories 3. Enrichment 4. Internalisation 5. 2-vector spaces 6. Open questions 7. Categories and n-categories in the UK 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 of widgets

4. 1. Introduction

Theory of studied via Some algebra widgets or other

4. 1. Introduction

Theory of studied via Some algebra widgets or other

we dream of

Higher-dimensional widgets

4. 1. Introduction

Theory of studied via Some algebra widgets or other

we we dream dream of of

Higher-dimensional Higher-dimensional widgets algebra

4. 1. Introduction

Theory of studied via Some algebra widgets or other

we we dream dream of of studied via Higher-dimensional Higher-dimensional widgets algebra

4. 1. Introduction

Theory of studied via Some algebra widgets or other

we we dream dream of of studied via Higher-dimensional Higher-dimensional widgets algebra studied via

4. 1. Introduction

Theory of loops studied via Groups or paths in a space or groupoids

5. 1. Introduction

Theory of loops studied via Groups or paths in a space or groupoids

we dream of

Theory of paths in a space and all higher homotopies

5. 1. Introduction

Theory of loops studied via Groups or paths in a space or groupoids

we we dream dream of of

Theory of paths in a space Higher-dimensional and all higher homotopies groupoids

5. 1. Introduction

Theory of loops studied via Groups or paths in a space or groupoids

we we dream dream of of studied via Theory of paths in a space Higher-dimensional and all higher homotopies groupoids studied via

5. 1. Introduction

Cohomology

6. 1. Introduction

Torsors studied via Cohomology ≡ special kinds of sheaves ≡ special functors into Gp

we we dream dream of of studied via taking all higher cohomology n-gerbes groups into account ≡ special n-stacks at the same time studied via ≡ special functors into n-Gpd

6. 1. Introduction

group G

7. 1. Introduction

studied via group G functors G −→ Vect

we we dream dream of of studied via

n-group G n-functors G −→ n-Vect studied via

7. 1. Introduction

Also: • cobordisms • topological quantum ﬁeld theory • concurrency via fundamental n-category of directed space

8. 1. Introduction

How do we add dimensions?

9. 1. Introduction

How do we add dimensions?

form a widgets are sets with extra structure ⊲ category

9. 1. Introduction

How do we add dimensions?

form a widgets are sets with extra structure ⊲ category 2-widgets categories ′′ ⊲ 2-category

9. 1. Introduction

How do we add dimensions?

form a widgets are sets with extra structure ⊲ category 2-widgets categories ′′ ⊲ 2-category 3-widgets 2-categories ′′ ⊲ 3-category

9. 1. Introduction

How do we add dimensions?

form a widgets are sets with extra structure ⊲ category 2-widgets categories ′′ ⊲ 2-category 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 strict: axioms hold on the nose (A × B) × C = A × (B × C) weak: axioms become coherent isomorphisms (A × B) × C =∼ A × (B × C)

((A × B) × C) × D A × (B × (C × D))

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)

(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: axioms hold on the nose (A × B) × C = A × (B × C) weak: axioms become coherent isomorphisms (A × B) × C =∼ A × (B × C)

• strict 2-categories model homotopy 2-types

• every weak 2-category is equivalent to a strict one

11. 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)

• strict 2-categories • strict 3-categories model homotopy 2-types do not model homotopy 3-types

• every weak 2-category • not every weak 3-category is equivalent to is equivalent to a strict one a strict one

11. 2. Introduction to categories

12. 2. Introduction to categories

A category is given by

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

12. 2. Introduction to categories

A category is given by

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

1x • identities: for every object x a morphism x −→ x f g • composition: for every pair of morphisms x −→ y −→ z gf composition: a morphism x −→ z

12. 2. Introduction to categories

A category is given by

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

1x • identities: for every object x a morphism x −→ x f g • composition: for every pair of morphisms x −→ y −→ z gf composition: a morphism x −→ z satisfying unit and associativity axioms.

12. 2. Introduction to categories

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 of continuous maps.

13. 2. Introduction to categories

Examples 2

Algebraic objects as categories:

14. 2. Introduction to categories

Examples 2

Algebraic objects as categories:

monoid category with one object groupoid category in which every morphism is invertible group category with one object and every morphism invertible poset category with a −→ b given by a ≤ b.

14. 2. Introduction to categories

There is a large category Cat of small categories.

15. 2. Introduction to categories

There is a large category Cat of small categories. The morphisms are functors.

15. 2. Introduction to categories

There is a large category Cat of small categories. 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. 2. Introduction to categories

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 (small) category C is given by

• 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 ∈ ob C a function

C(y, z) × C(x, y) −→ C(x, z) satisfying unit and associativity axioms.

17. 3. Enrichment

Deﬁnition A (small) 2-category C is given by

C(y, z) × C(x, y) −→ C(x, z) satisfying unit and associativity axioms.

18. 3. Enrichment

Deﬁnition A (small) 2-category C is given by

• a set obC of objects category • 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 ∈ ob C a function

C(y, z) × C(x, y) −→ C(x, z) satisfying unit and associativity axioms.

18. 3. Enrichment

Deﬁnition A (small) 2-category C is given by

• a set obC of objects category • for every pair of objects x, y a set C(x, y) of morphisms equipped with functor • identities: for every object x a function 1 −→ C(x, x) • composition: for all x, y, z ∈ ob C a function functor C(y, z) × C(x, y) −→ C(x, z) satisfying unit and associativity axioms.

18. 3. Enrichment

Deﬁnition A (small) 2-category C is given by

18. 3. Enrichment Unravel

19. 3. Enrichment Unravel

• the category C(x, y) has objects x / y “1-cells” " x " y morphisms < “2-cells” x / y composition C “vertical comp.” C

19. 3. Enrichment Unravel

• the category C(x, y) has objects x / y “1-cells” " x " y morphisms < “2-cells” x / y composition C “vertical comp.” C

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

f g gf on objects x / y / z 7→ x/ z f g gf " " " on morphisms x α y β z 7→ xβ∗α z “horizontal comp.” < < < f ′ g′ g′f ′

19. 3. Enrichment Unravel

• the category C(x, y) has objects x / y “1-cells” " x " y morphisms < “2-cells” x / y composition C “vertical comp.” C

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

f g gf on objects x / y / z 7→ x/ z f g gf " " " on morphisms x α y β z 7→ xβ∗α z “horizontal comp.” < < < f ′ g′ g′f ′ functoriality says ./ ./ . . / . . / . ./ ./ . = / F / F F F 19. 3. Enrichment

Iteration

20. 3. Enrichment

Iteration

A 2-category is a category enriched in Cat.

20. 3. Enrichment

Iteration

A 2-category is a category enriched in Cat.

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

20. 3. Enrichment

Iteration

A 2-category is a category enriched in Cat.

A 3-category is a category enriched in 2-Cat: C(x, y) ∈ 2-Cat

▽ 0-cells ⊲ 1-cells of C 1-cells ⊲ 2-cells of C 2-cells ⊲ 3-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 A (small) category C is given by

23. 4. Internalisation Deﬁnition A (small) category C is given by s C1 C0 ∈ Set t together with

23. 4. Internalisation Deﬁnition A (small) category C is given by s C1 C0 ∈ Set C0 t together with e 1 1 e C1 • identities: a function C0 −→ C1 such that s t

C0 C0

23. 4. Internalisation Deﬁnition A (small) category C is given by s C1 C0 ∈ Set C0 t together with e 1 1 e C1 • identities: a function C0 −→ C1 such that s t

C0 C0

C1 ×C0 C1 • c composition: a function C1 ×C0 C1 −→ C1 c

sπ1 tπ2 e C1 identities: a function C0 −→ C1 such that s t

C0 C0

23. 4. Internalisation Deﬁnition A (small) category C is given by s C1 C0 ∈ Set C0 t together with e 1 1 e C1 • identities: a function C0 −→ C1 such that s t

C0 C0

C1 ×C0 C1 • c composition: a function C1 ×C0 C1 −→ C1 c

sπ1 tπ2 e C1 identities: a function C0 −→ C1 such that s t C C satisfying unit and associativity axioms. 0 0

23. 4. Internalisation Deﬁnition A (small) double category C is given by a diagram s C1 C0 ∈ Set C0 t together with e 1 1 e C1 • identities: a function C0 −→ C1 such that s t

C0 C0

C1 ×C0 C1 • c composition: a function C1 ×C0 C1 −→ C1 c

sπ1 tπ2 e C1 identities: a function C0 −→ C1 such that s t C C satisfying unit and associativity axioms. 0 0

24. 4. Internalisation Deﬁnition A (small) double category C is given by a diagram s Cat C1 C0 ∈ Set C0 t together with e 1 1 e C1 • identities: a function C0 −→ C1 such that s t

C0 C0

C1 ×C0 C1 • c composition: a function C1 ×C0 C1 −→ C1 c

sπ1 tπ2 e C1 identities: a function C0 −→ C1 such that s t C C satisfying unit and associativity axioms. 0 0

24. 4. Internalisation Deﬁnition A (small) double category C is given by a diagram s Cat C1 C0 ∈ Set C0 t together with e functor 1 1 e C1 • identities: a function C0 −→ C1 such that s t

C0 C0 functor C1 ×C0 C1 • c composition: a function C1 ×C0 C1 −→ C1 c

sπ1 tπ2 e C1 identities: a function C0 −→ C1 such that s t C C satisfying unit and associativity axioms. 0 0

24. 4. Internalisation Deﬁnition A (small) double category C is given by a diagram s Cat C1 C0 ∈ Set C0 t together with e functor 1 1 e C1 • identities: a function C0 −→ C1 such that s t

C0 C0 functor C1 ×C0 C1 • c composition: a function C1 ×C0 C1 −→ C1 c

sπ1 tπ2 e C1 identities: a function C0 −→ C1 such that s t C C satisfying unit and associativity axioms. 0 0 —an internal category in Cat.

24. 4. Internalisation

Unravel

25. 4. Internalisation

Unravel We get sets s B1 A1 A0 = “0-cells” t A1 = “vertical 1-cells” s t s t s B0 = “horizontal 1-cells” B0 A0 B = “2-cells” t 1

25. 4. Internalisation

Unravel We get sets s B1 A1 A0 = “0-cells” t A1 = “vertical 1-cells” s t s t s B0 = “horizontal 1-cells” B0 A0 B = “2-cells” t 1 w x • 2-cells have the following shape y z

25. 4. Internalisation

Unravel We get sets s B1 A1 A0 = “0-cells” t A1 = “vertical 1-cells” s t s t s B0 = “horizontal 1-cells” B0 A0 B = “2-cells” t 1 w x • 2-cells have the following shape y z • 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

An internal category in Gp is a 2-group. 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

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

We can also internalise monoids: • 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

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

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

28. 4. Internalisation

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

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

2. Take internal n-categories

28. 4. Internalisation

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

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

2. Take internal n-categories e.g. data underlying a 2-category is s s A2 A1 A0 ∈ Set t t

28. 4. Internalisation

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

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

2. Take internal n-categories e.g. data underlying a 2-category is s s A2 A1 A0 ∈ Set t t —we can put this inside other categories.

28. 4. Internalisation

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

1. Iterate: Cat(Gp) = categories internal to Gp = 2-Gp Cat(Cat(Gp)) Cat(Cat(Cat(Gp))) . . these model n-types Catn(Gp)

2. Take internal n-categories e.g. data underlying a 2-category is s s A2 A1 A0 ∈ Set t t —we can put this inside other categories.

28. 4. Internalisation

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

1. Iterate: Cat(Gp) = categories internal to Gp = 2-Gp Cat(Cat(Gp)) Cat(Cat(Cat(Gp))) . . these model n-types Catn(Gp) but these don’t 2. Take internal n-categories e.g. data underlying a 2-category is s s A2 A1 A0 ∈ Set t t —we can put this inside other categories.

28. 5. 2-vector spaces

29. 5. 2-vector spaces