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Outline Definitions and examples Homological properties of kC-mod An example A closed symmetric monoidal category Adjoint functors and a spectral sequence Two categorical constructions Hochschild cohomology 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 Finite category algebras
Outline Definitions and examples Homological properties of kC-mod An example A closed symmetric monoidal category Adjoint functors and a spectral sequence Two categorical constructions Hochschild cohomology Diagonal subcategory and functor We can consi<strong>de</strong>r 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 consi<strong>de</strong>r ∆ : C → C × C and RK ∆ . Fei Xu Finite category algebras
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