The Gabriel-Popescu theorem tells us that every Grothendieck category with a generator is a left exact localization of a module category. I'm interested in a slightly different way of "representing" such categories:

**Question:** Let $\mathcal C$ be a Grothendieck category with a generator.

Does there exist a Grothendieck topos $\mathcal X$ such that $\mathcal C \simeq Ab(\mathcal X)$ is equivalent to the category of abelian group objects in $\mathcal X$?

Slightly less naively, does there exist a ringed Grothendieck topos $(\mathcal X, \mathcal O_{\mathcal X})$ such that $\mathcal C \simeq \mathcal O_{\mathcal X}\textrm{-}Mod$ is equivalent to the category of $\mathcal O_\mathcal X$-modules?

Same as (2), but with $(\mathcal X, \mathcal O_{\mathcal X})$ being locally ringed?