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.

Dual to that, othus, one often speaks of colocalization as the dual case (of the universal functor into the quotient category) where $\Sigma$ 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.