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.

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 ${\displaystyle R[S^{-1}]}$-modules. Here ${\displaystyle R[S^{-1}]}$ is the localization of R with respect to the (multiplicatively closed) subset S consisting of all powers of r, ${\displaystyle S=\{1,r,r^{2},r^{3},\dots \}}$ The expression "most closely related" is formalized by two conditions: first, there is a functor

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