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.

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.