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.

See also MR4328537.

Sufficient conditions when the base change of enriching subcategory commutes with sheafification are studied in

• Ariel E. Rosenfield, Enriched Grothendieck topologies under change of base, arXiv:2405.19529