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

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 provide 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We also show that discrete representations <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of these topological groups satisfy a certain stability property described in Theorem 1.9.