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Outline Definitions and examples Homological properties of kC-mod An example Finite categories and their algebras Representations and modules 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 Finite category algebras
Outline Definitions and examples Homological properties of kC-mod An example Transporter categories Finite categories and their algebras Representations and modules 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 Finite category algebras
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