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

A localization in the first sense is a functor $L:C \to D$ of $\infty$-categories that is initial among the functors inverting a prescribed set of morphisms of $C$.

A localization , in the second 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.

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.

Definition

Let $C$ be an $\infty$-category and $W$ a set of morphisms of $C$. A functor $L:C\to D$ is said to exhibit $D$ as a (Dwyer–Kan) localization of $C$ with respect to $W$ if for each $\infty$-category $E$, the functor

$$\operatorname{Fun}(D,E)\to \operatorname{Fun}(C,E)$$

is fully faithful and its essential image consists of those functors $C\to E$ that carry each morphism of $W$ into equivalences of $E$.

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

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 $C_0 \hookrightarrow C$.

Remark

Reflective localizations are a special case of Dwyer–Kan localizations. This is kerodon, tag04JL.

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

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]