nLab localization of an (infinity,1)-category

Contents

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:CDL: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:CC 0L : C \to C_0 to an (,1)(\infty,1)-subcategory C 0CC_0 \hookrightarrow C such that with cc any object there is a morphism connecting it to its localization

cL(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.

Definition

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

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

is fully faithful and its essential image consists of those functors CEC\to E that carry each morphism of WW into equivalences of EE.

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

Definition

An (∞,1)-functor L:CC 0L : C \to C_0 is called a localization of the (∞,1)-category CC if it has a right adjoint (∞,1)-functor i:C 0Ci : C_0 \hookrightarrow C that is full and faithful.

(Li):C 0iLC. (L \dashv i) : C_0 \stackrel{\overset{L}{\leftarrow}}{\underset{i}{\hookrightarrow}} C \,.

In other words: LL is a localization if it is the reflector of a reflective (∞,1)-subcategory C 0CC_0 \hookrightarrow C.

Remark

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

Examples

References

Reflective localization is the topic of

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

With an eye towards modal homotopy type theory:

Via a calculus of fractions for quasi-categories:

Last revised on October 28, 2024 at 00:31:53. See the history of this page for a list of all contributions to it.