Let C be a small category and k a field. There are two interesting mathematical subjects: the category algebra kC and the classifying space |C| = BC. We study the ring homomorphism HH * (kC) → H * (|C|, k) and prove it is split surjective, using the factorization category of Quillen [18] and certain techniques from functor cohomology theory. This generalizes the well-known theorems for groups and posets. Based on this result, we construct a seven-dimensional category algebra whose Hochschild cohomology ring modulo nilpotents is not finitely generated, disproving a conjecture of Snashall and Solberg [20].

We define for each small category C a category algebra RC over a base ring R and study its representations. When C is an EI-category, we develop a theory of vertices and sources for RC-mod, which parameterizes the indecomposable RC-modules. As a main application, we use our theory to find formulas for computing higher (inverse) limits over C.

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Abstract. Let C be a finite EI category and k be a field. We consider the category algebra kC. Suppose K(C) = D b (kC-mod) is the bounded derived category of finitely generated left modules. This is a tensor triangulated category and we compute its spectrum in the sense of Balmer. When C = G ∝ P is a finite transporter category, the category algebra becomes Gorenstein so we can define the stable module category CMk(G ∝ P), of maximal Cohen-Macaulay modules, as a quotient category of K(G ∝ P). Since CMk(G ∝ P) is also tensor triangulated, we compute its spectrum as well.These spectra are used to classify tensor ideal thick subcategories of the corresponding tensor triangulated categories, despite the fact that the previously mentioned tensor categories are not rigid.

Let $\mathcal{C}$ be a finite category and let k be a field. We consider the category algebra $k\mathcal{C}$ and show that $k\mathcal{C}$-mod is closed symmetric monoidal. Through comparing $k\mathcal{C}$ with a co-commutative bialgebra, we exhibit the similarities and differences between them in terms of homological properties. In particular, we give a module-theoretic approach to the multiplicative structure of the cohomology rings of small categories. As an application, we prove that the Hochschild cohomology rings of a certain type of finite category algebras are finitely generated.