Giraud's Theorem and Categories of Representations

Abstract

We present an alternate proof of Giraud's Theorem based on the fact that given the conditions on a category E for being a topos, its objects are sheaves by construction. Generalizing sets to R-modules for R a commutative ring, we prove that a category with small hom-sets and finite limits is equivalent to a category of sheaves of R-modules on a site if and only if it satisfies Giraud's axioms and in addition is enriched in a certain symmetric monoidal category parametrized by an R-module.