Localization of a category

In mathematics, localization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become isomorphisms. This is formally similar to the process of localization of a ring; it in general makes objects isomorphic that were not so before. In homotopy theory, for example, there are many examples of mappings that are invertible up to homotopy; and so large classes of homotopy equivalent spaces. Calculus of fractions is another name for working in a localized category.

Some significant examples follow.

erre's "C"-theory

Serre introduced the idea of working in homotopy theory "modulo" some class "C" of abelian groups. This meant that groups "A" and "B" were treated as isomorphic, if for example "A/B" lay in "C". Later Dennis Sullivan had the bold idea instead of using the localization of a topological space, which took effect on the underlying topological spaces.

Module theory

In the theory of modules over a commutative ring "R", when "R" has Krull dimension ≥ 2, it can be useful to treat modules "M" and "N" as "pseudo-isomorphic" if "M/N" has support of codimension at least two. This idea is much used in Iwasawa theory.

Derived categories

The construction of a derived category in homological algebra proceeds by a step of adding inverses of quasi-isomorphisms.

Abelian varieties up to isogeny

An isogeny from an abelian variety "A" to another one "B" is a surjective morphism with finite kernel. Some theorems on abelian varieties require the idea of "abelian variety up to isogeny" for their convenient statement. For example, given an abelian subvariety "A₁" of "A", there is another subvariety "A₂" of "A" such that

:"A₁" × "A₂"

is "isogenous" to "A" (Poincaré's theorem: see for example "Abelian Varieties" by David Mumford). To call this a direct sum decomposition, we should work in the category of abelian varieties up to isogeny.

et-theoretic issues

In general, given a category "C" and some class "w" of morphisms in the category, there is some question as to whether it is possible to form a localization "w^-1 C" by inverting all the morphisms in "w". The typical procedure for constructing the localization might result in a pair of objects with a proper class of morphisms between them. Avoiding such set-theoretic issues is one of the initial reasons for the development of the theory of model categories.

References

P. Gabriel and M. Zisman. "Calculus of fractions and homotopy theory". Springer-Verlag New York, Inc., New York, 1967. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35.