Localization of a category

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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.

Contents 1 Serre's C-theory 2 Module theory 3 Derived categories 4 Abelian varieties up to isogeny 5 Set-theoretic issues 6 References

[edit] Serre'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.

[edit] 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.

[edit] Derived categories

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

[edit] 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.

[edit] Set-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.

[edit] 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.