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
*References
* * * {{cite book , doi=10.1007/978-3-030-61524-6_2 , title=Introduction to Infinity-Categories , series=Compact Textbooks in Mathematics , date=2021 , last1=Land , first1=Markus , isbn=978-3-030-61523-9, zbl= 1471.18001 * Daniel Carranza, Chris Kapulkin, Zachery Lindsey, Calculus of Fractions for Quasicategories rXiv:2306.02218Further reading
* https://mathoverflow.net/questions/310731/localization-of-infty-categories * https://ncatlab.org/nlab/show/localization+of+an+%28infinity%2C1%29-category Higher category theory