nLab localization of an (infinity,1)-category

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; another is in terms of reflective (∞,1)-subcategories:

A localization , in this sense, of an (∞,1)-category $C$ is a functor $L : C \to C_0$ to an $(\infty,1)$-subcategory $C_0 \hookrightarrow C$ such that with $c$ any object there is a morphism connecting it to its localization

$$c \to L(c)$$

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

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

Definition

Definition

An (∞,1)-functor $L : C \to C_0$ is called a localization of the (∞,1)-category $C$ if it has a right adjoint (∞,1)-functor $i : C_0 \hookrightarrow C$ that is full and faithful.

$$(L \dashv i) : C_0 \stackrel{\overset{L}{\leftarrow}}{\underset{i}{\hookrightarrow}} C \,.$$

In other words: $L$ is a localization if it is the reflector of a reflective (∞,1)-subcategory $C_0 \hookrightarrow C$.

This is HTT, def. 5.2.7.2.

Examples

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

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

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

• cohomology localization

• homotopy localization

References

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:

• 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]