In 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 or as a result of Simpson.
Definition.
Let "S" be a simplicial set and "W" a simplicial subset of it. Then the localization in the sense of Dwyer–Kan is a map
formula_1
such that
is invertible.
When "W" is clear form the context, the localized category formula_5 is often also denoted as formula_6.
A Dwyer–Kan localization that admits a right adjoint is called a localization in the sense of Bousfield. For example, the inclusion ∞-Grpd formula_7 ∞-Cat has a left adjoint given by the localization that inverts all maps (functors). The right adjoint to it, on the other hand, is the core functor (thus the localization is Bousfield).
Properties.
Let "C" be an ∞-category with small colimits and formula_8 a subcategory of weak equivalences so that "C" is a category of cofibrant objects. Then the localization formula_9 induces an equivalence
formula_10
for each simplicial set "X".
Similarly, if "C" is a hereditary ∞-category with weak fibrations and cofibrations, then
formula_11
for each small category "I".