Higher-dimensional category theory - Edinburgh University - School
Transcript of Higher-dimensional category theory - Edinburgh University - School
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 ofwidgets
studied via Some algebraor other
we
dream
of
Higher-dimensionalwidgets
4.
1. Introduction
Theory ofwidgets
studied via Some algebraor other
we
dream
of
Higher-dimensionalwidgets
we
dream
of
Higher-dimensionalalgebra
4.
1. Introduction
Theory ofwidgets
studied via Some algebraor other
we
dream
of
Higher-dimensionalwidgets
we
dream
of
Higher-dimensionalalgebra
studied via
4.
1. Introduction
Theory ofwidgets
studied via Some algebraor other
we
dream
of
Higher-dimensionalwidgets
we
dream
of
Higher-dimensionalalgebra
studied via
studied via
4.
1. Introduction
Theory of loopsor paths in a space
studied via Groupsor groupoids
we
dream
of
Theory of paths in a spaceand all higher homotopies
5.
1. Introduction
Theory of loopsor paths in a space
studied via Groupsor groupoids
we
dream
of
Theory of paths in a spaceand all higher homotopies
we
dream
of
Higher-dimensionalgroupoids
5.
1. Introduction
Theory of loopsor paths in a space
studied via Groupsor groupoids
we
dream
of
Theory of paths in a spaceand all higher homotopies
we
dream
of
Higher-dimensionalgroupoids
studied via
studied via
5.
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 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?
form awidgets are sets with extra structure ⊲ category
2-widgets categories ′′ ⊲ 2-category
9.
1. Introduction
How do we add dimensions?
form awidgets 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 awidgets 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
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-categoriesmodel homotopy 2-types
• every weak 2-categoryis equivalent toa 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-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.
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
• identities: for every object x a morphism x1x
−→ x
• composition: for every pair of morphisms xf
−→ yg
−→ z
composition: a morphism xgf−→ 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
• 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.
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 ofcontinuous maps.
13.
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 objectand 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.
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
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 ∈ obC 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
• 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)
satisfying unit and associativity axioms.
18.
3. Enrichment
Definition
A (small) 2-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 ∈ obC a function
C(y, z) × C(x, y) −→ C(x, z)
satisfying unit and associativity axioms.
category
18.
3. Enrichment
Definition
A (small) 2-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 ∈ obC a function
C(y, z) × C(x, y) −→ C(x, z)
satisfying unit and associativity axioms.
category
functor
functor
18.
3. Enrichment
Definition
A (small) 2-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 ∈ obC a function
C(y, z) × C(x, y) −→ C(x, z)
satisfying unit and associativity axioms.
category
functor
functor
—a category enriched in Cat.
18.
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
• the category C(x, y) has objects x y// “1-cells”
morphisms x y""
<<�� “2-cells”
composition x y��
CC//��
��“vertical comp.”
• 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
• the category C(x, y) has objects x y// “1-cells”
morphisms x y""
<<�� “2-cells”
composition x y��
CC//��
��“vertical comp.”
• 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
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
Definition
A (small) category C is given by
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
A (small) category C is given by
C0 ∈ SetC1
s
ttogether with
• identities: a function C0e
−→ 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
A (small) category C is given by
C0 ∈ SetC1
s
ttogether with
• identities: a function C0e
−→ 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
satisfying unit and associativity axioms.
23.
4. Internalisation
Definition
A (small) double category C is given by a diagram
C0 ∈ SetC1
s
ttogether with
• identities: a function C0e
−→ 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
satisfying unit and associativity axioms.
24.
4. Internalisation
Definition
A (small) double category C is given by a diagram
C0 ∈ SetC1
s
ttogether with
• identities: a function C0e
−→ 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
satisfying unit and associativity axioms.
Cat
24.
4. Internalisation
Definition
A (small) double category C is given by a diagram
C0 ∈ SetC1
s
ttogether with
• identities: a function C0e
−→ 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
satisfying unit and associativity axioms.
Cat
functor
functor
24.
4. Internalisation
Definition
A (small) double category C is given by a diagram
C0 ∈ SetC1
s
ttogether with
• identities: a function C0e
−→ 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
satisfying unit and associativity axioms.
Cat
functor
functor
—an internal category in Cat.
24.
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
tA0 = “0-cells”A1 = “vertical 1-cells”
B0 = “horizontal 1-cells”B1 = “2-cells”
• 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
tA0 = “0-cells”A1 = “vertical 1-cells”
B0 = “horizontal 1-cells”B1 = “2-cells”
• 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.
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
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
Internal categories in Ab are just 2-term chain complexes
G −→ H ∈ Ab.
26.
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 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
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 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
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:
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
A0 ∈ SetA1
s
tA2
s
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
A0 ∈ SetA1
s
tA2
s
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)))...
Catn(Gp)
2. Take internal n-categories
e.g. data underlying a 2-category is
A0 ∈ SetA1
s
tA2
s
t—we can put this inside other categories.
these model n-types
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
A0 ∈ SetA1
s
tA2
s
t—we can put this inside other categories.
these model n-types
but these don’t
28.
5. 2-vector spaces
Recap: two methods of categorification
enrichment categories enriched in Vectk
internalisation
29.
5. 2-vector spaces
Recap: two methods of categorification
enrichment categories enriched in Vectk
internalisation categories internal to Vectk
vector spaces internal to Cat
29.
5. 2-vector spaces
Recap: two methods of categorification
enrichment categories enriched in Vectk
internalisation categories internal to Vectk
vector spaces internal to Cat KV94
29.
5. 2-vector spaces
Recap: two methods of categorification
enrichment categories enriched in Vectk B96
internalisation categories internal to Vectk
vector spaces internal to Cat KV94
29.
5. 2-vector spaces
Recap: two methods of categorification
enrichment categories enriched in Vectk B96
internalisation categories internal to Vectk BC04
vector spaces internal to Cat KV94
29.
5. 2-vector spaces
Recap: two methods of categorification
enrichment categories enriched in Vectk B96
internalisation categories internal to Vectk BC04
vector spaces internal to Cat KV94
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
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)
Baez (1996) —and further work by Bartlett
“A 2-vector space is a category enriched in Vectk.”
30.
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)
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
• What is a good definition of n-category?
• Are different definitions equivalent?
32.
6. Open questions
• What is a good definition of n-category?
• Are different definitions equivalent?
• What is the (n + 1)-category of n-categories?
32.
6. Open questions
• What is a good definition of n-category?
• Are different definitions equivalent?
• What is the (n + 1)-category of n-categories?
• Coherence and strictification theorems.
32.
6. Open questions
• What is a good definition of n-category?
• Are different definitions equivalent?
• What is the (n + 1)-category of n-categories?
• Coherence and strictification theorems.
• Modelling homotopy types (Grothendieck).
32.
6. Open questions
• What is a good definition of n-category?
• Are different definitions equivalent?
• What is the (n + 1)-category of n-categories?
• Coherence and strictification theorems.
• Modelling homotopy types (Grothendieck).
• The periodic table (Baez-Dolan).
• The stabilisation hypothesis (BD).
• The tangle hypothesis. (BD)
• The TQFT hypothesis (BD).
32.
6. Open questions
• What is a good definition of n-category?
• Are different definitions equivalent?
• What is the (n + 1)-category of n-categories?
• Coherence and strictification theorems.
• Modelling homotopy types (Grothendieck).
• 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.
7. Categories and n-categories in the UK
Mathematics:
• Cambridge: Martin Hyland, Peter Johnstone
• Glasgow: Tom Leinster, Danny Stevenson
• Sheffield: EC, Nick Gurski, Simon Willerton, NeilStrickland, John Greenlees
33.
7. Categories and n-categories in the UK
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]
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