nLab colocalization

Idea

A localization is a universal functor from a given category $C$ with respect to the inversion of some family $\Sigma$ of morphisms in $C$; sometimes one says also quotient category. In topos theory and some other subjects one often restricts to the situation when $\Sigma$ is a calculus of left fractions, and the corresponding localization functor has a right adjoint (which is then necessarily fully faithful); even more, one often requires that the localization functor is also left exact, hence exact.

One may then speak of colocalization as the dual version of localization, a universal functor into a quotient category where now $\Sigma$ admits a calculus of right fractions and the corresponding quotient functor has a left adjoint. See also at right Bousfield localization.

The theory of colocalization in co-Grothendieck categories? has some features of its own as compared to the localization in Grothendieck categories. Namely, while by Gabriel-Popescu’s theorem, every Grothendieck category is a localization of a category of modules over a fixed unital ring, their dual categories may be presented in terms of the theory of linear topological rings with some compactness properties, which is the content of Gabriel-Oberst duality theory.

Related concepts

• right Bousfield delocalization

References

• C. Năstăsescu, B. Torrecillas, Colocalization on Grothendieck categories with applications to coalgebras, J. Algebra 185 (1996), 108–124 (pdf)

A textbook exposition is in the chapter 6, Duality of

• Nicolae Popescu, Abelian categories with applications to rings and modules, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. MR0340375

In derived and triangulated categories:

• Shoham Shamir, Colocalization functors in derived categories and torsion theories, arxiv/0910.4724
• Hvedri Inassaridze, Tamaz Kandelaki, Ralf Meyer, Localisation and colocalisation of triangulated categories at thick subcategories, arxiv/0912.2088

category: algebra