A torsion theoretic interpretation for sheaves of modules and Grothendieck topologies on directed categories

We prove that every Grothendieck topology induces a hereditary torsion pair in the category of presheaves of modules on a ringed site, and obtain a homological characterization of sheaves of modules: a presheaf of modules is a sheaf of modules if and only if it is saturated with respect to torsion presheaves, or equivalently, it is right perpendicular to torsion presheaves in the sense of Geigle and Lenzing. We also study Grothendieck topologies on directed categories $\mathscr{C}$ satisfying certain finiteness condition, and show that every Grothendieck topology on $\mathscr{C}$ is a subcategory topology if and only if $\mathscr{C}$ is an artinian EI category. Consequently, in this case every sheaf category is equivalent to the presheaf category over a full subcategory of $\mathscr{C}$. Finally, we classify all Grothendieck topologies on a special type of noetherian EI categories, and extend the locally self-injective property of representations of $\mathrm{F}$ and $\mathrm{VI}$ to representations of their infinite full subcategories. Some potential applications in group representation theory are given at the end of this paper.