localization of an (infinity,1)-category in nLab

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Contents

Idea

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 L:C→DL:C \to D of ∞\infty-categories that is initial among the functors inverting a prescribed set of morphisms of CC.

A localization , in the second sense, of an (∞,1)-category CC is a functor L:C→C 0L : C \to C_0 to an (∞,1)(\infty,1)-subcategory C 0↪CC_0 \hookrightarrow C such that with cc any object there is a morphism connecting it to its localization

c→L(c) c \to L(c)

in a suitable way. This “suitable way” just says that ff is left adjoint to the fully faithful inclusion functor.

Since localizations are entirely determined by which morphisms in CC are sent to equivalences in C 0C_0, they can be thought of as sending CC to the result of “inverting” all these morphisms, a process familiar from forming the homotopy category of a homotopical category.

Definitions

As explained in Idea, there are two common definitions that are referred to as localizations of ∞\infty-categories.

The following definition appears in kerodon, tag01MP.

The following second definition appears in HTT, def. 5.2.7.2:

Examples

• Localizations of (∞,1)(\infty,1)-categories are modeled by the notion of left Bousfield localization of model categories.

  One precise statement is: localizations of (∞,1)-category of (∞,1)-presheaves C=PSh (∞,1)(K)C = PSh_{(\infty,1)}(K) are presented by the left Bousfield localizations of the global projective model structure on simplicial presheaves on the simplicial category incarnation of KK.

• ∞-stackification (or (∞,1)-sheafification) is the localization of an (∞,1)-category of (∞,1)-presheaves to the (∞,1)(\infty,1)-subcategory of (∞,1)-sheaves.

• cohomology localization

• homotopy localization

References

Reflective localization is the topic of

• Jacob Lurie, §5.2.7 & §5.5.4 of: Higher Topos Theory (2009)
• Jacob Lurie, Kerodon,[Reflective Localizations, tag:02FY]

Dwyer–Kan localization (also called simplicial localizations or quasicategory of fractions) are treated in

• Jacob Lurie, Kerodon,[Localization, tag:01M4]
• Markus Land, Section 2.4 of Introduction to Infinity-Categories, Compact Textbooks in Mathematics,Birkh"auser/Springer, Cham, (2021) [doi:10.1007/978-3-030-61524-6]
• Jacob Lurie, pp. 485 of: Higher Algebra (2017)

With an eye towards modal homotopy type theory:

• Marco Vergura, Localization Theory in an Infinity Topos, 2019 (pdf, ir.lib.uwo.ca:etd/6257)

Via a calculus of fractions for quasi-categories:

• Daniel Carranza, Chris Kapulkin, Zachery Lindsey, Calculus of Fractions for Quasicategories [arXiv:2306.02218]