nLab localization

Idea

In mathematics, localization refers to the idea of adjoining formal inverses for some elements in an algebraic structure.

In particular, the structure itself can be a category, in which case localization adjoins formal inverses to a prescribed class of morphisms. This construction is known as a category of fractions and also ambiguously as a localization of a category. An important special case of this idea is the notion of a reflective localization / left Bousfield localization, in which case the localization functor has a fully faithful right adjoint. In some texts (e.g. in Higher Topos Theory) the term “localization” (when applied to categories or (∞,1)-categories) refers specifically to reflective localizations.

For categories and (∞,1)-categories

General

• category of fractions, homotopy category

• quasicategory of fractions, simplicial localization, hammock localization, simplicial homotopy category

Reflective

• reflective localization, left Bousfield localization

Coreflective

• coreflective localization, right Bousfield localization

For commutative rings

• localization of a commutative ring

• field of fractions, ring of fractions

For noncommutative rings

• localization of a ring, noncommutative localization

• Cohn localization, Ore localization

For modules

• localization of an abelian group

• localization of a module

For monoids

• localization of a monoid

• group completion

For categories of spaces

• localization of a space

For categories of spectra

• Bousfield localization of spectra