Sheaves of modules on atomic sites and discrete representations of topological groups

TL;DR

This paper links sheaves on atomic sites, category representations, and discrete topological group representations, providing new classification methods and stability insights for certain infinite groups.

Contribution

It characterizes sheaves as saturated representations, relates sheaf cohomology to derived torsion functors, and classifies simple and indecomposable injective discrete representations of key infinite groups.

Findings

Sheaves of modules are equivalent to saturated representations.

Sheaf cohomology corresponds to derived torsion functors.

Explicit classification of simple and indecomposable injective representations for specific topological groups.

Abstract

The main goal of this paper is to establish close relations among sheaves of modules on atomic sites, representations of categories, and discrete representations of topological groups. We characterize sheaves of modules on atomic sites as saturated representations, which are precisely representations right perpendicular to torsion representations in the sense of Geigle and Lenzing. Consequently, the category of sheaves is equivalent to the Serre quotient of the category of presheaves by the category of torsion presheaves. We also interpret the sheaf cohomology functors as derived functors of the torsion functor and for some special cases as the local cohomology functors. These results as well as a classical theorem of Artin provides us a new approach to study discrete representations of topological groups. In particular, by importing established facts in representation stability theory, we explicitly classify simple or indecomposable injective discrete representations of some topological groups such as the infinite symmetric group, the infinite general or special linear group over a finite field, and the automorphism group of the linearly ordered set $\mathbb{Q}$. We also show that discrete representations $V$ of these topological groups satisfy a certain stability property.