Representations of categories and their applications

Abstract

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.

A generalized Koszul theory and its application

“…The following fact, which is well known in the context of finite EI categories (see [24]), is essential in the proof.…”

On simple modules over twisted finite category algebras

“…This raises the question how to generalise vertices and sources to arbitrary indecomposable modules over the category algebra kC. For finite EI-categories, Fei Xu has developed in [26] a notion of vertices and sources, where the vertices are certain full subcategories.…”

On the cohomology rings of small categories

“…We want to compare the cohomology rings of C and of its various subcategories. In [5,16,12,32] the authors studied the case where D ⊂ C is a full subcategory which has fewer objects, and showed under certain assumptions one can have H * (C) ∼ = H * (D). Here we investigate subcategories D ⊂ C with the same set of objects but with fewer morphisms.…”

“…We denote by C-mod the abelian category of all covariant functors from C to Ab. The nth cohomology group of C with coefficients in a functor F ∈ C-mod, H n (C; F), can be defined as the nth higher inverse limit lim ← − n C F [2,14,26,32]. If A is an abelian group and A is the corresponding constant functor which sends every object to A and every morphism to the identity, then H n (C; A) ∼ = H n (|C|, A), where |C| is the topological realization of NC-the nerve of C. We are particularly interested in the case where A is a commutative ring with identity, because then H * (C; A) ∼ = H * (|C|, A) will become a graded commutative ring.…”

“…For each small category C and a fixed ring R, one can define an associative R-algebra, called the category algebra RC of C (see Xu [32]), generalizing the notion of the group algebra of a group and the notion of the incidence algebra of a poset.…”