Enriched sheaf theory has been introduced in
They consider a locally finitely presentable symmetrical monoidal closed category and a small -enriched category . The category of -valued -enriched functors on the dual of is considered as a category of enriched presheaves. Axioms for -enriched Grothendieck topologies are introduced in terms of -subfunctors of representable functors (one could say -sieves). The main result of the article is a bijection between reflective -enriched localizations of preserving finite limits and -enriched Grothendieck topologies on and also a bijection with universal -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 -enriched Grothendieck topologies when the enrichment is over the category of abelian groups.
See also MR4328537.
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