Decompositions of categories over posets and cohomology of categories

Abstract

Let C be a small category. We present some results which describe cohomology groups and homotopy colimits of functors defined over C using cohomology groups and homotopy colimits over certain categories associated to functors from C to posets.

On the cohomology rings of small categories

“…Let π : C → [C] be the natural functor from C to its underlying poset. According to Słomińska's main result in [27], one has a homotopy equivalence…”

“…In this section, we first recall the main construction and theorem in Słomińska's paper [27]. Using her decomposition formula for classifying spaces of categories, we are able to determine the homotopy types of the classifying spaces of certain categories.…”

“…Hence, to some extent a finite EI-category may be regarded as an amalgam of a finite poset and several finite groups. In fact, Słomińska [27] showed that the classifying space of C, |C|, is homotopy equivalent to the homotopy colimit of some functor from a finite poset to the category of finite groupoids. Since cohomology rings of finite groups and finite posets are finitely generated, one wishes to generalize the finite generation property of cohomology rings to finite EI-categories.…”

A family of acyclic functors

“…Here, Ab denote the category of abelian groups and C is a small category. By the results of Słomińska in [14] these higher limits over C can be understood through higher limits over partially ordered sets (poset for short). In fact, if C is an EI-category there are further relations with posets, see [13,8].…”