Skew category algebras and modules on ringed finite sites

TL;DR

This paper explores the structure of module categories over ringed sites on finite categories, showing they are equivalent to modules over a skew category algebra derived from the site and a specific subcategory.

Contribution

It establishes an equivalence between module categories on finite ringed sites and modules over a uniquely determined skew category algebra, extending the understanding of such algebraic structures.

Findings

Module categories are equivalent to skew category algebra modules.

The skew algebra is canonically defined on the site.

Uniqueness of the subcategory $\\mathcal{D}$ is proven.

Abstract

Let $\mathcal{C}$ be a small category. We investigate ringed sites $(\mathbf{C},\mathfrak{R})$ on $\mathcal{C}$ and the resulting module categories $\mathfrak{M}{\rm od}\text{-}\mathfrak{R}$. When $\mathcal{C}$ is finite, based on Grothendieck and Verdier's classification of finite topoi, we prove that each $\mathfrak{M}{\rm od}\text{-}\mathfrak{R}$ is equivalent to ${\rm Mod}\text{-}\mathfrak{R}|_{\mathcal{D}}[\mathcal{D}]$, 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 $\mathcal{D}$.