Sheaves of modules are exactly the J-saturated presheaves, every Grothendieck topology on a noetherian EI directed category is rigid, and all topologies on type N/Z categories are classified.
On cohomological characterizations of endotrivial modules
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abstract
Given a general finite group $G$, there are various finite categories whose cohomology theories are of great interests. Recently Balmer and Grodal gave some new characterizations of the groups of endotrivial modules, via \v{C}ech cohomology and category cohomology, respectively, defined on certain orbit categories. These two seemingly different approaches share a common root in topos theory. We shall demonstrate the connection, which leads to a better understanding as well as new characterizations of the group of endotrivial modules.
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A torsion theoretic interpretation for sheaves of modules and Grothendieck topologies on directed categories
Sheaves of modules are exactly the J-saturated presheaves, every Grothendieck topology on a noetherian EI directed category is rigid, and all topologies on type N/Z categories are classified.