nLab
localization of a simplicial model category

Idea

A simplicial model category is a presentation of an (infinity,1)-category.

The localization of a simplicial model category accordingly models the localization of the corresponding (infinity,1)-category, i.e. the passage to the a reflective (infinity,1)-subcategory).

Examples

• A famous example of localizations of simplicial model categories appears in the study of model structures on simplicial presheaves, where it yields a presentation of (infinity,1)-sheafification completely analogous to how ordinary localization described ordinary sheafification:

  • the global projective/injective model structures on simplicial presheaves on a site $S$ presents the $(\mathrm{\infty }\mathrm{,1})$-category of (infinity,1)-presheaves on $S$;

  • the Bousfield localization of these global model structures at morphisms which are local (stalkwise) weak equivalences yields the local projective/injective model structure on simplicial presheaves on $S$, which then presents the (infinity,1)-category of (infinity,1)-sheaves on $S$.

Remarks

• see also Bousfield localization

References

chapter 9.4 in

• P. Goerss and J. Jardine, Simplicial homotopy theory

section A.3.7 of

• Jacob Lurie, Higher Topos Theory