nLab enriched sheaf

Enriched sheaf theory has been introduced in

F. Borceux, C. Quinteiro, A theory of enriched sheaves, Cahiers Topologie Géom. Différentielle Catég. 37 (1996), no. 2, 145–162 MR1394507

They consider a locally finitely presentable symmetrical monoidal closed category $\mathcal{V}$ and a small $\mathcal{V}$ -enriched category $\mathcal{C}$ . The category $[\mathcal{C}^\mathrm{op},\mathcal{V}]$ of $\mathcal{V}$ -valued $\mathcal{V}$ -enriched functors on the dual of $\mathcal{C}$ is considered as a category of enriched presheaves. Axioms for $\mathcal{V}$ -enriched Grothendieck topologies are introduced in terms of $\mathcal{V}$ -subfunctors of representable functors (one could say $\mathcal{V}$ -sieves). The main result of the article is a bijection between reflective $\mathcal{V}$ -enriched localizations of $[\mathcal{C}^\mathrm{op},\mathcal{V}]$ preserving finite limits and $\mathcal{V}$ -enriched Grothendieck topologies on $\mathcal{C}$ and also a bijection with universal $\mathcal{V}$ -closure operations.

This is a generalization of a Gabriel-Popescu theorem and of a characterization of Grothendieck topoi as left exact reflective localizations of presheaf categories.

Gabriel filters (Gabriel topologies) are the case of $\mathcal{V}$ -enriched Grothendieck topologies when the enrichment is over the category of abelian groups.