Background Basic concepts Universal constructions Local presentation Theorems (∞,1)-Yoneda lemma (∞,1)-Grothendieck construction adjoint (∞,1)-functor theorem (∞,1)-monadicity theorem Extra stuff, structure, properties Models As for localization of ordinary categories, there are slightly different notions of what a localization of an (∞,1)-category is. One definition is in terms of simplicial localizations or quasicategory of fractions; another is in terms of reflective (∞,1)-subcategories: A localization in the first sense is a functor of -categories that is initial among the functors inverting a prescribed set of morphisms of . A localization , in the second sense, of an (∞,1)-category is a functor to an -subcategory such that with any object there is a morphism connecting it to its localization in a suitable way. This “suitable way” just says that is left adjoint to the fully faithful inclusion functor. Since localizations are entirely determined by which morphisms in are sent to equivalences in , they can be thought of as sending to the result of “inverting” all these morphisms, a process familiar from forming the homotopy category of a homotopical category. As explained in Idea, there are two common definitions that are referred to as localizations of -categories. The following definition appears in kerodon, tag01MP. Let be an -category and a set of morphisms of . A functor is said to exhibit as a (Dwyer–Kan) localization of with respect to if for each -category , the functor is fully faithful and its essential image consists of those functors that carry each morphism of into equivalences of . The following second definition appears in HTT, def. 5.2.7.2: An (∞,1)-functor is called a localization of the (∞,1)-category if it has a right adjoint (∞,1)-functor that is full and faithful. In other words: is a localization if it is the reflector of a reflective (∞,1)-subcategory . Reflective localizations are a special case of Dwyer–Kan localizations. This is...