A Note on Homology over Functor Categories

TL;DR

This paper explores how homology in functor categories over an abelian category behaves, establishing that it aligns with the expected properties and filling a gap in the literature.

Contribution

It provides a comprehensive analysis of homology in functor categories over abelian categories, clarifying its properties and relationship to homology in the base category.

Findings

Homology in $C^I$ behaves as expected based on properties of $C$.

Functor categories over abelian categories are themselves abelian.

The paper consolidates existing ideas into a coherent framework.

Abstract

It is known that, for $C$ an abelian category and $I$ small, the functor category $C^I$ is again abelian; thus we can do homology in such categories, and examine how it relates to homology in $C$ itself. However, there does not seem to be any good reference collecting these ideas together. This article seeks to fill the gap by showing that homology in $C^I$ behaves as one would expect.