On sheaves in finite group representations

TL;DR

This paper explores sheaf categories in finite group representations, providing explicit computations and new proofs of classical equivalences using Grothendieck topologies and sheafification techniques.

Contribution

It introduces a novel approach to relate group representations with fixed-point sheaves via sheafification on transporter categories.

Findings

Explicit computation of sheaf categories for transporter categories

Identification of G-representations with fixed-point sheaves

New proof of Artin's equivalence using sheaf-theoretic methods

Abstract

Given a general finite group $G$, we consider several categories built on it, their Grothendieck topologies and resulting sheaf categories. For a certain class of transporter categories and their quotients, equipped with atomic topology, we explicitly compute their sheaf categories via sheafification. This enables us to identify $G$-representations with various fixed-point sheaves. As a consequence, it provides an intrinsic new proof to the equivalence of M. Artin between the category of sheaves on the orbit category and that of group representations.