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.
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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.