localization of an enriched category

Given a symmetric closed monoidal category VV, a VV-enriched category AA with underlying ordinary category A 0A_0 and a subcategory Σ\Sigma of A 0A_0 containing the identities of A 0A_0, H. Wolff defines the corresponding theory of localizations.

  • H. Wolff, VV-localizations and VV-triples, Dissertation, University of Illinois-Urbana, 1970.
  • H. Wolff, VV-localizations and VV-monads, J. Alg. 24, 405-438, 1973, MR310041, doi; V-localizations and VV-monads. II, Pacific J. Math. 63 (1976), no. 2, 579–589, MR412253, euclid; VV-localizations and VV-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 VV-categories. For such a VV-enriched category CC,

  • 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 VV-localizations which preserve finite limits of the enriched category of presheaves [C op,V][C^{op},V], and relate them to an enriched version of Grothendieck topology on CC, and to a “universal closure operation” on [C op,V][C^{op},V].

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