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.
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Let A be a category. A localization is an idempotent and coaugmented functor. A coaugmented functor is a pair (L,l) where L:A → A is an endofunctor and l:Id → L is a natural transformation from the identity functor to L (called the coaugmentation). A coaugmented functor is idempotent if, for every X, both maps L(lX),lL(X):L(X) → LL(X) are isomorphisms. It can be proven that in this case, both maps are equal.
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.
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.
The construction of a derived category in homological algebra proceeds by a step of adding inverses of quasi-isomorphisms.
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
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.
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.
Category:Localization (mathematics)
Gabriel, Peter; Zisman, Michel (1967). Calculus of fractions and homotopy theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35. Springer-Verlag New York, Inc.. ISBN 978-0387037776.