Background
Basic concepts
equivalences in/of -categories
Universal constructions
Local presentation
Theorems
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; another is in terms of reflective (∞,1)-subcategories:
A localization , in this 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.
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 .
This is HTT, def. 5.2.7.2.
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 .
∞-stackification (or (∞,1)-sheafification) is the localization of an (∞,1)-category of (∞,1)-presheaves to the -subcategory of (∞,1)-sheaves.
This is the topic of of
Jacob Lurie, §5.2.7 & §5.5.4 of: Higher Topos Theory (2009)
Jacob Lurie, pp. 485 of: Higher Algebra (2017)
With an eye towards modal homotopy type theory:
Via a calculus of fractions for quasi-categories:
Last revised on March 29, 2024 at 09:31:55. See the history of this page for a list of all contributions to it.