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.
Grothendieck topologies on posets
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abstract
Lindenhovius has studied Grothendieck topologies on posets and has given a complete classification in the case that the poset is Artinian. We extend his approach to more general posets, by translating known results in locale and domain theory to the study of Grothendieck topologies. In particular, explicit descriptions are given for the family of Grothendieck topologies with enough points and the family of Grothendieck topologies of finite type. As an application, we compute the cardinalities of these families in various examples.
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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.