nLab
localization of an enriched category

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 $A_0$, H. Wolff defines the corresponding theory of localizations.

• H. Wolff, $V$-localizations and $V$-triples, Dissertation, University of Illinois-Urbana, 1970.
• H. Wolff, $V$-localizations and $V$-monads, J. Alg. 24, 405-438, 1973, MR310041, doi; V-localizations and $V$-monads. II, Pacific J. Math. 63 (1976), no. 2, 579–589, MR412253, euclid; $V$-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 $[C^{op},V]$.