my beamer presentation - Departament de matemÃ tiques

Outline 

Definitions and examples 

Homological properties of kC-mod 

An example 

A survey on Hochschild cohomology of 

finite category algebras 

Fei Xu 

Departament de Matemàtiques, Universitat Autònoma de Barcelona 

CIRM Luminy, 10 June 2010 

Fei Xu 

Finite category algebras

Outline 



An example 

1 Definitions and examples 

Finite categories and their algebras 

Representations and modules 

Motivation 

2 Homological properties of kC-mod 

A closed symmetric monoidal category 

Adjoint functors and a spectral sequence 

Two categorical constructions 

Hochschild cohomology 

3 An example 

Fei Xu 


Outline 



An example 



Motivation 

Finite categories 

A category is finite if it has finitely many morphism. 

Typical examples are finite groups and finite posets 

(partially ordered sets). 

A non-group and non-poset example 

1 x 

x 

α 

1 y 

 

y 

g 

with g 2 = 1 y and α = gα. 

Fei Xu 


Outline 



An example 

Category algebras 



Motivation 

The k-category algebra of C, kC, is defined as a k-vector 

space k Mor C, in which the multiplication is given on base 

elements by 

{ α ◦ β, if α and β are composable in C; 

α ∗ β = 

0 , otherwise. 

The previous category 

1 x 

x 

α 

1 y 

 

y 

has the following k-category algebra with identity 

1 kC = 1 x + 1 y 

Fei Xu Finite category algebras 

g

Outline 



An example 

B. Mitchell’s theorem 



Motivation 

Given a finite category algebra kC, we are interested in 

the homological properties of kC-mod of finitely 

generated left kC-modules. 

Let Vect k be the category of finite-dimensional k-vector 

spaces, and Vectk 

C the category of covariant functors. 

Each functor is called a k-representation of C. 

Mitchell’s Theorem: kC-mod ∼ = Vect C k . 

It simply says that each kC-module corresponds uniquely 

to a k-representation of C, and vice versa. 

F ↦→ ⊕ x∈Ob C F (x) and M ↦→ F M such that 

F M (x) = 1 x · M. 

Fei Xu 


Outline 



An example 



Motivation 

Ordinary cohomology, an algebraic definition 

For any M, N ∈ kC-mod, we can consider Ext ∗ kC(M, N). 

There is a constant functor k ∈ Vectk 

C such that 

k(x) = k and k(α) = Id k for any α ∈ Mor C. It is also 

called the trivial kC-module. 

There is an augmentation map ɛ : kC → k. 

The groups Ext ∗ kC(k, N) have different interpretations. 

∗ 

They are isomorphic to both lim N, the higher limits 

←− C 

of N, and H ∗ (C; N), the cohomology of C with 

coefficients in N. 

Ext ∗ kC(k, k) possesses a cup product and is isomorphic to 

H ∗ (BC, k), the cohomology ring of the classifying 

space BC of C. It is called the ordinary cohomology ring 

of kC. 

Fei Xu 


Outline 



An example 

Transporter categories 



Motivation 

Let G be a finite group and P a finite G-poset. One can 

construct the transporter category G ⋉ P as follows: 

The objects Ob(G ⋉ P) = Ob P; 

For x, y ∈ Ob(G ⋉ P), a morphism is a pair (g, gx ≤ y) 

for some g ∈ G. 

If G acts trivially, then the category is G × P. 

P ↩→ G ⋉ P via (x ≤ y) ↦→ (e, x ≤ y) (e ∈ G is the 

identity). 

It admits a natural functor G ⋉ P → G, x ↦→ ∗ and 

(g, gx ≤ y) ↦→ g. 

It is a special situation of the Grothendieck 

construction on a functor F : C → CAT . 

Fei Xu 


A Σ 4 -poset 

Outline 



An example 



Motivation 

The poset of all non-trivial 2-subgroups of Σ 4 , S 2 (Σ 4 ): 

· · · 

· · · 

D 8 

D 8 D 8 

 

 

C 2 × C 2 C 4 V 

 

 

 

 

C 2 

C 2 

C 2 

 

C 2 

C 2 

Fei Xu 


Outline 



An example 



Motivation 

The transporter category Σ 4 ⋉ S 2 (Σ 4 ) : 

C 2 

 

8 

8 

C 2 × C 2 

8 

4 

C 

4 2 

8 

· · · 

· · · 

4 4 

D 8 D 

4 8 D 

4 8 

 

 

8 8 24 

 

24 24 

 

C 4 8 V 24 

 

 

8 24 

8 24 24 

8 8 

C 2 C 

8 2 C 

8 2 

4 

4 

8 

8 

8 

Fei Xu 


Outline 



An example 



Motivation 

Subgroups as transporter categories 

Let G be a finite group and H a subgroup. We consider 

the set of left cosets Q := G/H which can be regarded as 

a G-poset: G acts via left multiplication. 

The transporter category G ⋉ Q is a connected groupoid 

whose skeleton is isomorphic to H. 

In this way one can recover all subgroups of G, up to 

category equivalences. 

A category equivalence D → C induces a Morita 

equivalence kD ≃ kC (and a homotopy equivalence 

BD ≃ BC as well). 

The functor G ⋉ Q → G gives rise to the usual 

restriction and transfer between H ∗ (G; k) and H ∗ (H; k). 

Fei Xu 


Outline 



An example 



Motivation 

Relevance in group representations 

The transporter categories bridge up representations of groups 

and of their local categories, through the following diagram 

 

G 

G ⋉ P 

which induces functors among module categories 

kG-mod 

k(G ⋉ P)-mod 

C 

kC-mod 

It certainly provides a framework to investigate and compare 

various representations, and cohomology as well. 

Fei Xu Finite category algebras

Outline 



An example 



Motivation 

A proper setting for ordinary cohomology 

The natural functor π : G ⋉ P → G induces a ring 

homomorphism, the restriction, 

H ∗ (G; k) → H ∗ (G ⋉ P; k). 

Becker and Gottlieb constructed a transfer map 

H ∗ (G ⋉ P; k) → H ∗ (G; k) such that tr ◦ res = χ(P), the 

Euler characteristic of P. 

Applying a spectral sequence to P ↩→ G ⋉ P → G one 

can show H ∗ (G ⋉ P; k) is a finitely generated 

H ∗ (G; k)-module, and hence is finitely generated as a 

ring. 

Fei Xu 


Outline 



An example 



Motivation 

Hochschild cohomology rings of transporter 

category algebras 

Let G be a finite group, P a finite G-poset and G ⋉ P 

the transporter category. 

The previously mentioned spectral sequence for 

P ↩→ G ⋉ P → G can be used to improve our 

observation on the finite generation of 

H ∗ (G ⋉ P; k) ∼ = Ext ∗ k(G⋉P)(k, k). 

Given a k(G ⋉ P)-module N, Ext ∗ k(G⋉P)(k, N) is finitely 

generated as a Ext ∗ k(G⋉P)(k, k)- and Ext ∗ kG(k, k)-module. 

With a bit extra work, one shows that the Hochschild 

cohomology ring HH ∗ (k(G ⋉ P)) is finitely generated 

as a Ext ∗ k(G⋉P)(k, k)-module, and hence is finitely 

generated as a ring. 


Outline 



An example 





A crash introduction to closed symmetric monoidal 

categories 

Mac Lane: Much of the force of category theory will be seen to 

reside in using categories with specified additional structures. 

One basic example will be the closed categories. The simplest 

closed symmetric monoidal category is perhaps Vect k . 

There is a tensor product − ⊗ k −, which is symmetric. 

There exists a tensor identity k. 

It is closed in the sense that for a pair of (ordered) spaces 

V , W , there is a function object, a.k.a. the internal hom, 

Hom k (V , W ) in the category Vect k . 

Hom k (U ⊗ k V , W ) ∼ = Hom k (U, Hom k (V , W )). 

Fei Xu 


Outline 



An example 





kC-mod is closed symmetric monoidal 

If A is a symmetric monoidal category, so is A C . 

The tensor product in kC-mod ≃ Vectk 

C is written as 

− ˆ⊗−. It is symmetric and has k as tensor identity. 

The tensor product induces a cup product 

∪ : Ext i kC(M, N)⊗Ext j kC (M′ , N ′ ) → Ext i+j 

kC (M ˆ⊗M ′ , N ˆ⊗N ′ ). 

For any ordered pair of kC-modules M, N, Hom k (M, N) is 

not a kC-module. However there exists an internal hom 

such that Hom kC (L ˆ⊗M, N) ∼ = Hom kC (L, hom(M, N)). 

The algebra kC behaves in many ways like a 

cocommutative Hopf algebra, but is essentially different. 

Fei Xu 


Outline 



An example 





Standard tools 

Let θ : D → C be a functor between two finite categories. It 

induces a natural functor, the restriction along θ, 

Res θ : kC-mod → kD-mod. 

This functor comes with two adjoints, the left and right Kan 

extensions 

LK θ , RK θ : kD-mod → kC-mod. 

When C = G is a group, D = H is a subgroup and θ is the 

inclusion, we have Res θ 

∼ =↓ 

G 

H and LK θ 

∼ =↑ 

G 

H 

∼ =⇑ 

G 

H 

∼ = RKθ . 

For any N ∈ kD-mod, there is a (multiplicative) cohomology 

Grothendieck spectral sequence 

H i (C; H j (\θ; N)) ⇒ H i+j (D; N). 

Fei Xu 


Outline 



An example 





Diagonal subcategory and functor 

We can consider the algebra of the product category 

C × C, k(C × C). 

It is easy to see that k(C × C) ∼ = kC ⊗ k kC. In particular 

there are algebra homomorphisms k1 kC ⊗ kC ↩→ kC ⊗ kC 

and kC ⊗ k1 kC ↩→ kC ⊗ kC. 

The category C × C has a diagonal subcategory ∆C, 

whose objects are of the form (x, x) for any x ∈ Ob C and 

whose morphisms are of the form (α, α) for any 

α ∈ Mor C. There is an isomorphism ∆C ∼ = C. 

We shall consider ∆ : C → C × C and RK ∆ . 

Fei Xu 


Outline 



An example 





Right Kan extension and internal hom 

Let M, N ∈ kC-mod. Then M ⊗ N is a kC ⊗ kC-module. 

M ˆ⊗N = Res ∆ (M ⊗ N) where Res ∆ : kC ⊗ kC-mod 

→ kC-mod. 

Thus 

Hom kC (L ˆ⊗M, N) = Hom kC (Res ∆ (L ⊗ M), N) 

∼ = HomkC⊗kC (L ⊗ M, RK ∆ N) 

∼ = HomkC (L, Hom kC (M, RK ∆ N)). 

The internal hom is hom(M, N) = Hom kC (M, RK ∆ N). 

The dual module of M is defined to be hom(M, k). 

When C = G is a group, hom(M, N) ∼ = Hom k (M, N). 

Fei Xu 


Outline 



An example 





Category of factorizations and skew diagonal 

functor 

We also need the opposite category C op because 

kC op ∼ = (kC) op , and the category algebra 

kC e := k(C × C op ) is isomorphic to the enveloping algebra 

(kC) e = kC ⊗ k kC op . 

Note that kC as a functor C × C op → Vect k is given by 

kC(x, y) = k Hom C (y, x) (zero if Hom C (y, x) = ∅). 

There is a category of factorizations in C, named 

F (C). Its objects are the morphisms in C and there is a 

morphism from α → β if and only if α is a factor of β. If 

β = uαv, then the morphism is recorded as a pair 

(u, v) : α → β. 

The category is topologically the same as C in that 


Outline 



An example 

Using the left Kan extension 





We examine the following diagram 

F (C) 

 

τ=(t,s) 

C × C op 

t 

pr 

 

C . 

It immediately gives rise to another diagram 

Res τ 

kF (C)-mod 

k(C × C op )-mod 

Res t 

Res 

 

pr 

kC-mod 

. 

Fei Xu 


Outline 



An example 





Using the left Kan extension, continued 

It leads to a diagram that we need 

LK τ 

kF (C)-mod 

k(C × C op )-mod 

LK t 

LK 

 

pr 

kC-mod . 

k 

LK τ 

kF (C)-mod 

 

kC e -mod kC 

LK t 

LK pr ∼ =−⊗kC k 

 

kC-mod . 

Fei Xu 

k 


Outline 



An example 





Example: Hochschild cohomology of a poset 

Let P be a poset. We have HH ∗ (kP) ∼ = H ∗ (P; k) 

(Gerstenhaber-Shack). When P = x α →y, the triple 

τ : F (P) → P e becomes 

(α,1 x ) 

 

[1 x ] 

[α] (y, x) 

 

 

(1 y ,α) 

 

[1 y ] 

(α,1 op 

x ) 

 

(x, x) 

 

(1 x ,α op ) 

(1 y ,α op ) 

(α,1 op 

y ) 

 

(x, y) 

Note that 

kP(x, x) = k1 x , kP(y, y) = k1 y , kP(y, x) = kα and 

kP(x, y) = 0. Moreover Res τ kP = kP ∼ = k. 

Fei Xu Finite category algebras 

(y, y)

Outline 



An example 





Thus we have 

HH ∗ (kP) ∼ = Ext ∗ kPe(kP, kP) 

∼ = Ext ∗ kF (P)(k, k) 

∼ = Ext ∗ kP(k, k) 

∼ = 

H ∗ (P; k). 

Fei Xu 


Outline 



An example 





Theorem 

For any M ∈ kC e -mod, 

Ext ∗ kC e(kC, M) ∼ = Ext ∗ kF (C)(k, Res τ M); 

For any N ∈ kC-mod, Ext ∗ kC(k, N) ∼ = Ext ∗ kF (C)(k, Res t N); 

The module kC ∈ kC e -mod restricts to 

Res τ kC ∼ = k ⊕ U ∈ kF (C)-mod; 

There exists a natural split surjective ring homomorphism 

Ext ∗ kC e(kC, kC) → Ext∗ kC(k, k) ∼ = Ext ∗ kF (C)(k, k). 

We can understand Hochschild cohomology via ordinary 

cohomology. 

Fei Xu 


Outline 



An example 

 

 

 

 

Example: a 7-dimensional algebra 

A category constructed by Aurélien Djament: E 

h 

g 

x 

1 x 

gh 

α 

β 

y {1 y } , 

where g 2 = h 2 = 1 x , gh = hg, αh = βg = α, and 

αg = βh = β. Its category algebra is 7-dimensional and we 

can compute the mod-2 ordinary and Hochschild cohomology 

rings. 

BE is homotopy equivalent to B(C 2 × C 2 )/BC 2 . 

Suppose chark = 2. Then H ∗ (BE, k) is isomorphic to a 

subring of the polynomial ring k[u, v] with base elements 

of the form u i , i ≥ 1, removed. 

Fei Xu 


Outline 



An example 

Infinitely generated HH ∗ 

The ordinary cohomology ring is infinitely 

generated and has no nilpotents. Thus HH ∗ (kE) is 

not finitely generated either, even modulo nilpotents. 

This calculation is very similar to that of the Hochschild 

cohomology ring of a Koszul algebra. In fact, kE is 

Koszul. 

Let A be a Koszul algebra and r its radical. Then the 

image of the following canonical map 

Ext ∗ −⊗ 

Ae(A, A) 

A A/r 

Ext ∗ A(A/r, A/r) 

is exactly the (graded) center. 

Snashall constructed Koszul algebras with infinitely 

generated Hochschild cohomology rings (modulo 

nilpotents) independent Fei Xuof the Finitechoice category algebras of a base field.

Compatibility 

Outline 



An example 

There exists a commutative diagram 

Ext ∗ kE e(kE, kE) 

−⊗ kE k 

−⊗ kE kE/r 

 

Ext ∗ kE(k, k) 

Ext ∗ kE(kE/r, kE/r). 

− ˆ⊗kE/r 

Fei Xu 


Outline 



An example 

J. C. Becker, D. H. Gottlieb, The transfer map and fibre 

bundles, Topology 14 (1975) 1-12. 

W. G. Dwyer, H.-W. Henn, Homotopy Theoretic Methods 

in Group Cohomology, Birkhäuser 2001. (Grothendieck 

constr.) 

P. Hilton, U. Stammbach, A Course in Homological 

Algebra (Second edition), GTM 4, Springer 1997. (Kan 

extensions, Grothendieck s.s.) 

S. Mac Lane, Categories for the Working Mathematician 

(Second edition), GTM 5, Springer 1998. (various 

categorical constructions) 

I. Moerdijk, J. A. Svensson, A Shapiro lemma for diagram 

of spaces with applications to equivariant topology, 

Compositio Math. 96 (1995) 249-282. (Grothendieck 

s.s.) 

D. Quillen, Higher algebraic K-theory I, in: Lecture Notes 


Outline 



An example 

J. Thévenaz, G-algebras and Modular Representation 

Theory, Oxford University Press 1995. (transporter 

categories, local categories) 

R. W. Thomason, Homotopy colimits in the category of 

small categories, Math. Proc. Camb. Phil. Soc. 85 

(1979) 91-109. (Grothendieck construction) 

P. J. Webb, An introduction to the representations and 

cohomology of categories, in: Group Representation 

Theory, EPFL Press 2007, pp. 149-173. 

F. Xu, Hochschild and ordinary cohomology rings of small 

categories, Adv. Math. 219 (2008) 1872-1893. 

F. Xu, Tensor structure on kC-mod and cohomology, 

preprint 2009. 

Fei Xu

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