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Context

Category theory

category theory

Concepts

• category

• functor

• natural transformation

• Cat

Universal constructions

• universal construction

  • representable functor

  • adjoint functor

  • limit/colimit

  • weighted limit

  • end/coend

  • Kan extension

Theorems

• Yoneda lemma

• Isbell duality

• Grothendieck construction

• adjoint functor theorem

• monadicity theorem

• adjoint lifting theorem

• Tannaka duality

• Gabriel-Ulmer duality

• small object argument

• Freyd-Mitchell embedding theorem

• relation between type theory and category theory

Extensions

• sheaf and topos theory

• enriched category theory

• higher category theory

Applications

• applications of (higher) category theory

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Contents

• Idea
• For categories and (∞,1)-categories
  • General
  • Reflective
  • Coreflective
• For commutative rings
• For noncommutative rings
• For modules
• For monoids
• For categories of spaces
• For categories of spectra

Idea

In mathematics, localization refers to the idea of formally adding inverses to some elements in an algebraic structure.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 adds formal inverses to morphisms. This construction is known as acategory, 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 books texts (e.g., (e.g. inHigher Topos TheoryHigher 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