nLab
localization of an (infinity,1)-category

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 reflective (∞,1)-subcategories:

A localization , in this 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.

Definition

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.

This is HTT, def. 5.2.7.2.

Examples

References

This is the topic of section 5.2.7 and 5.5.4 of

Revised on April 28, 2014 20:46:33 by Adeel Khan (132.252.136.40)