Representation Cohomology of a Small Category

Abstract

Let $C_\bullet$ be a simplicial object in the category $Cat$ of small categories. For a field $k$, taking the Grothendieck groups of isomorphism classes of $kC_n$-modules gives rise to a cochain complex, whose cohomology, which we refer to as representation cohomology, is the object studied in this article. In particular, to any small category $C$, we associate a simplicial object in $Cat$, where for each $n\ge 0$ the objects of the level $n$ category are the simplices of the nerve of $C$. The basic properties of the resulting representation cohomology of these simplicial objects and certain subobjects are then studied in detail. We present some general theoretical computations in favourable cases.