The core groupoid can suffice

TL;DR

This paper investigates the categorical conditions under which functor categories over a category and its core groupoid are equivalent, extending the understanding of representation theory in a categorical context.

Contribution

It abstracts the necessary categorical structure to establish an equivalence between functor categories over a category and its core groupoid, generalizing previous results.

Findings

Identifies conditions for equivalence of functor categories

Extends representation theory to new categorical settings

Provides a framework for further categorical analysis

Abstract

This work results from a study of Nicholas Kuhn's paper entitled "Generic representation theory of finite fields in nondescribing characteristic". Our goal is to abstract the categorical structure required to obtain an equivalence between functor categories $[\mathscr{F},\mathscr{V}]$ and $[\mathscr{G},\mathscr{V}]$ where $\mathscr{G}$ is the core groupoid of the category $\mathscr{F}$ and $\mathscr{V}$ is a category of modules over a commutative ring.