Skew category algebras and modules on ringed finite sites

Abstract

Let $\mathcal{C}$ be a small category. We investigate ringed sites $(\mathbf{C},\mathfrak{R})$ on $\mathcal{C}$ and the resulting module categories $\mathfrak{M}\mathrm{od}\text{-}\mathfrak{R}$. When $\mathcal{C}$ is finite, based on Grothendieck and Verdier's classification of finite topoi, we prove that each $\mathfrak{M}\mathrm{od}\text{-}\mathfrak{R}$ is equivalent to $\mathrm{Mod}\text{-}\mathfrak{R}{|}_{\mathcal{D}}[\mathcal{D}]$, where $\mathfrak{R}{|}_{\mathcal{D}}[\mathcal{D}]$ is the skew category algebra, canonically defined on $(\mathbf{C},\mathfrak{R})$, for a uniquely determined full subcategory $\mathcal{D}\subset \mathcal{C}$ and the restriction $\mathfrak{R}{|}_{\mathcal{D}}$ of $\mathfrak{R}$ to $\mathcal{D}$.