Representation Cohomology of a Small Category

TL;DR

This paper introduces and studies the concept of representation cohomology for small categories, using simplicial objects and Grothendieck groups to analyze their algebraic properties.

Contribution

It defines representation cohomology for small categories via simplicial objects and explores its fundamental properties and computations in specific cases.

Findings

Established basic properties of representation cohomology.

Performed theoretical computations in favorable cases.

Linked representation cohomology to simplicial structures in categories.

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.