nLab localization of model categories

Idea

In the way that model categories serve as presentations for their homotopy categories or more generally of their associated $\infty$-categories so the general notion of localization of a model category [Hirschhorn (2002), Def. 3.1.1] serves to invert morphisms in the homotopy category such as to present localization of that $\infty$-category.

To quote Hirschhorn (2002), p. xi:

‘Localizing a model category with respect to a class of maps does not mean making the maps into isomorphisms; instead, it means making the images of those maps in the homotopy category into isomorphisms. Since the image of a map in the homotopy category is an isomorphism if and only if the map is a weak equivalence, localizing a model category with respect to a class of maps means making maps into weak equivalences.’

In practice, the most prominent notion of localization of model categories is Bousfield localization of model categories (see there for more), which is a special case, by Hirschhorn (2002), Thm. 3.3.19.

Examples

• localization of a simplicial model category

• Bousfield localization of model categories

References

The main account on the general notion of localization of model categories is:

• Philip Hirschhorn, Section 3 of: Model Categories and Their Localizations, AMS Math. Survey and Monographs 99 (2002) [ISBN:978-0-8218-4917-0, pdf toc, pdf, pdf]

[p. 47:] In Section 3.1 we define left and right localizations of model categories, and explain the connection between left localizations, local objects, and local equivalences […] in Section 3.3 we discuss (left and right) Bousfield localizations of categories, a special case of left and right localizations (see Theorem 3.3.19).

For more references dedicated to the special case of Bousfield localization of model categories see there.