One definition is in terms of reflective (∞,1)-subcategories:
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.
In other words: is a localization if it is the reflector of a reflective (∞,1)-subcategory .
This is HTT, def. 188.8.131.52.
Localizations of -categories are modeled by the notion of left Bousfield localization of model categories.
One precise statement is: localizations of (∞,1)-category of (∞,1)-presheaves are presented by the left Bousfield localizations of the global projective model structure on simplicial presheaves on the simplicial category incarnation of .
This is the topic of section 5.2.7 and 5.5.4 of