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
such that is an ∞-category, the image consists of invertible maps, the induced map on ∞-categories
is invertible.
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. For example, the inclusion ∞-Grpd ∞-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 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.
Similarly, if C is a hereditary ∞-category with weak fibrations and cofibrations, then
for each small category I.
See also ∞-topos
References
Daniel Carranza, Chris Kapulkin, Zachery Lindsey, Calculus of Fractions for Quasicategories [arXiv:2306.02218]
Further reading https://mathoverflow.net/questions/310731/localization-of-infty-categories https://ncatlab.org/nlab/show/localization+of+an+%28infinity%2C1%29-category