Localization of a category

Summary

Localizing a category at a class of morphisms means adding formal inverses so that these morphisms become isomorphisms, in the most economical way. The result is a new category with a functor from the old one, characterized by a universal property. Homotopy categories and derived categories are localizations: one inverts weak equivalences or quasi-isomorphisms. The construction generalizes the localization of a commutative ring, where chosen elements are made invertible.