Localization of an ∞-category

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
:

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. The right adjoint to it, on the other hand, is the core functor.

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.