nLab
localization of an (infinity,1)-category

Idea

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

$$c\to L(c)$$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

The left adjoint $(\mathrm{\infty }\mathrm{,1})$-functor

$$f:C\to C_0$$f : C \to C_0

to a reflective (∞,1)-subcategory $C_0\hookrightarrow C$ is called a localization of $C$.

Examples

• Simplicial model categories model (∞,1)-categories. localization of a simplicial model category accordingly models localization of the corresponding $(\mathrm{\infty }\mathrm{,1})$-category.

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

References

definition 5.2.7.2 of

• Jacob Lurie, Higher Topos Theory