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The main goal of this paper is to establish close relations among sheaves ofmodules on atomic sites, representations of categories, and discreterepresentations of topological groups. We characterize sheaves of modules onatomic sites as saturated representations, and show that the category ofsheaves is equivalent to the Serre quotient of the category of presheaves bythe category of torsion presheaves. Consequently, the sheafification functorand sheaf cohomology functors are interpreted by localization functors, sectionfunctors, and derived functors of torsion functor in representation theory.These results as well as a classical theorem of Artin provides us a newapproach to study discrete representations of topological groups. Inparticular, by importing established facts in representation stability theory,we explicitly classify simple or indecomposable injective discreterepresentations of some discrete topological groups such as the infinitesymmetric group, the infinite general or special linear group over a finitefield, and the automorphism group of the linearly ordered set $\mathbb{Q}$. Wealso show that discrete representations of these topological groups satisfy acertain stability property.

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