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

Contents

### Context

#### $(\infty,1)$-Category theory

(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

# 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 $C$ 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

This is the topic of of

With an eye towards modal homotopy type theory: