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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 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 <strong>de</strong>fined to be hom(M, k). When C = G is a group, hom(M, N) ∼ = Hom k (M, 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 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 recor<strong>de</strong>d as a pair (u, v) : α → β. The category is topologically the same as C in that Fei Xu Finite category algebras
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