nLab enriched sheaf

Enriched sheaf theory has been introduced in

  • F. Borceux, C. Quinteiro, A theory of enriched sheaves, Cahiers Topologie Gรฉom. Diffรฉrentielle Catรฉg. 37 (1996), no. 2, 145โ€“162 MR1394507

They consider a locally finitely presentable symmetrical monoidal closed category ๐’ฑ\mathcal{V} and a small ๐’ฑ\mathcal{V}-enriched category ๐’ž\mathcal{C}. The category [๐’ž op,๐’ฑ][\mathcal{C}^\mathrm{op},\mathcal{V}] of ๐’ฑ\mathcal{V}-valued ๐’ฑ\mathcal{V}-enriched functors on the dual of ๐’ž\mathcal{C} is considered as a category of enriched presheaves. Axioms for ๐’ฑ\mathcal{V}-enriched Grothendieck topologies are introduced in terms of ๐’ฑ\mathcal{V}-subfunctors of representable functors. The main result of the article is a bijection between reflective ๐’ฑ\mathcal{V}-enriched localizations of [๐’ž op,๐’ฑ][\mathcal{C}^\mathrm{op},\mathcal{V}] preserving finite limits and ๐’ฑ\mathcal{V}-enriched Grothendieck topologies on ๐’ž\mathcal{C} and also a bijection with universal ๐’ฑ\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. See also MR4328537.

Created on September 14, 2022 at 14:33:13. See the history of this page for a list of all contributions to it.