Localization of a category

In mathematics, localization of a category consists of adding to a 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. A 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 -modules. Here is the localization of R with respect to the (multiplicatively closed) subset S consisting of all powers of r, 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 . 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 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.