nLab localization of an enriched category

Idea

Given a symmetric closed monoidal category $V$, a $V$-enriched category $A$ with underlying ordinary category $A_0$ and a subcategory $\Sigma$ of $A_0$ containing the identities of $A_0$, one may consider the enriched generalization of the notion of localization of a category.

References

• Harvey Wolff: $V$-localizations and $V$-triples, Dissertation, University of Illinois-Urbana (1970)

• Harvey Wolff: $V$-localizations and $V$-monads, J. Alg. 24, 405-438, 1973, MR310041, doi;

• Harvey Wolff: V-localizations and $V$-monads. II, Pacific J. Math. 63 (1976), no. 2, 579–589, MR412253, euclid;

• Harvey Wolff: $V$-localizations and $V$-Kleisli algebras, Manuscripta Math. 16 (1975), no. 3, 203–228, MR382383, doi

While Wolff in principle defines localizations more generally, most of the theory is developed for reflective localizations, i.e. when the counit of the 2-adjunction is iso of $V$-categories. For such a $V$-enriched category $C$,

• F. Borceux, C. Quinteiro, A theory of enriched sheaves, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 37 no. 2 (1996), p. 145-162, MR1394507, numdam

consider reflective $V$-localizations which preserve finite limits of the enriched category of presheaves $[C^{op},V]$, and relate them to an enriched version of Grothendieck topology on $C$, and to a “universal closure operation” on $[C^{op},V]$. See also under enriched sheaf.