Localization of an ∞-categoryIn mathematics, specifically in higher category theory, a localization of an ∞-category is an ∞-category obtained by inverting some maps. An ∞-category is a presentable ∞-category if it is a localization of an ∞-presheaf category in the sense of Bousfield, by definition[1] or as a result of Simpson.[2] DefinitionLet S be a simplicial set and W a simplicial subset of it. Then the localization in the sense of Dwyer–Kan is a map such that
When W is clear form the context, the localized category is often also denoted as . A Dwyer–Kan localization that admits a right adjoint is called a localization in the sense of Bousfield.[4] For example, the inclusion ∞-Grpd ∞-Cat has a left adjoint given by the localization that inverts all maps (functors).[5] The right adjoint to it, on the other hand, is the core functor (thus the localization is Bousfield). PropertiesLet C be an ∞-category with small colimits and a subcategory of weak equivalences so that C is a category of cofibrant objects. Then the localization induces an equivalence for each simplicial set X.[6] Similarly, if C is a hereditary ∞-category with weak fibrations and cofibrations, then for each small category I.[7] See alsoReferences
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