nLab enriched sheaf

Enriched sheaf theory has been introduced in

  • F. Borceux, C. Quinteiro, A theory of enriched sheaves, Cahiers Topologie GΓ©om. DiffΓ©rentielle CatΓ©g. 37 (1996), no. 2, 145–162 MR1394507

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

See also MR4328537.

Last revised on October 12, 2023 at 19:02:03. See the history of this page for a list of all contributions to it.