Skew category algebras and modules on ringed finite sites

Abstract

Let C be a small category. We investigate ringed sites (C,R) on C and the resulting module categories Mod-R. When C is finite, based on Grothendieck and Verdier's classification of finite topoi, we prove that each Mod-R is equivalent to Mod-R|D[D], where R|D[D] is the skew category algebra, canonically defined on (C,R), for a uniquely determined full subcategory D⊂C and the restriction R|D of R to D.