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.
Introduction and motivation.
A category "C" consists of objects and morphisms between these objects. The morphisms reflect relations between the objects. In many situations, it is meaningful to replace "C" by another category "C"' in which certain morphisms are forced to be isomorphisms. This process is called localization.
For example, in the category of "R"-modules (for some fixed commutative ring "R") the multiplication by a fixed element "r" of "R" is typically (i.e., unless "r" is a unit) not an isomorphism: 
The category that is most closely related to "R"-modules, but where this map "is" an isomorphism turns out to be the category of formula_2-modules. Here formula_2 is the localization of "R" with respect to the (multiplicatively closed) subset "S" consisting of all powers of "r",
formula_4
The expression "most closely related" is formalized by two conditions: first, there is a functor
sending any "R"-module to its localization with respect to "S". Moreover, given any category "C" and any functor 
sending the multiplication map by "r" on any "R"-module (see above) to an isomorphism of "C", there is a unique functor
such that formula_8.
Localization of categories.
The above examples of localization of "R"-modules is abstracted in the following definition. In this shape, it applies in many more examples, some of which are sketched below.
Given a category "C" and some class "W" of morphisms in "C", the localization "C"["W"−1] is another category which is obtained by inverting all the morphisms in "W". More formally, it is characterized by a universal property: there is a natural localization functor "C" → "C"["W"−1] and given another category "D", a functor "F": "C" → "D" factors uniquely over "C"["W"−1] if and only if "F" sends all arrows in "W" to isomorphisms.
Thus, the localization of the category is unique up to unique isomorphism of categories, provided that it exists. One construction of the localization is done by declaring that its objects are the same as those in "C", but the morphisms are enhanced by adding a formal inverse for each morphism in "W". Under suitable hypotheses on "W", the morphisms between two objects "X", "Y" are given by "roofs"
(where "X"' is an arbitrary object of "C" and "f" is in the given class "W" of morphisms), modulo certain equivalence relations. These relations turn the map going in the "wrong" direction into an inverse of "f". This procedure, however, in general yields a proper class of morphisms between "X" and "Y". Typically, the morphisms in a category are only allowed to form a set. Some authors simply ignore such set-theoretic issues.
Model categories.
A rigorous construction of localization of categories, avoiding these set-theoretic issues, was one of the initial reasons for the development of the theory of model categories: a model category "M" is a category in which there are three classes of maps; one of these classes is the class of weak equivalences. The homotopy category Ho("M") is then the localization with respect to the weak equivalences. The axioms of a model category ensure that this localization can be defined without set-theoretical difficulties.
Alternative definition.
Some authors also define a "localization" of a category "C" to be an idempotent and coaugmented functor. A coaugmented functor is a pair "(L,l)" where "L:C → C" 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.
This definition is related to the one given above as follows: applying the first definition, there is, in many situations, not only a canonical functor formula_10, but also a functor in the opposite direction,
For example, modules over the localization formula_2 of a ring are also modules over "R" itself, giving a functor
In this case, the composition
is a localization of "C" in the sense of an idempotent and coaugmented functor.
Examples.
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.
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 derived category of an abelian category is much used in homological algebra. It is the localization of the category of chain complexes (up to homotopy) with respect to the 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
is "isogenous" to "A" (Poincaré's reducibility 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.
Related concepts.
The localization of a topological space produces another topological space whose homology is a localization of the homology of the original space.
A much more general concept from homotopical algebra, including as special cases both the localization of spaces and of categories, is the "Bousfield localization" of a model category. Bousfield localization forces certain maps to become weak equivalences, which is in general weaker than forcing them to become isomorphisms.