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 $[C^{op},V]$.

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