Localization of a 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:

${\displaystyle M\to M\quad m\mapsto r\cdot m.}$

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

${\displaystyle \varphi$ :{\text{Mod}}_{R}\to {\text{Mod}}_{R[S^{-1}]}\quad M\mapsto M[S^{-1}]}

sending any R-module to its localization with respect to S. Moreover, given any category C and any functor

${\displaystyle F:{\text{Mod}}_{R}\to C}$

sending the multiplication map by r on any R-module (see above) to an isomorphism of C, there is a unique functor

${\displaystyle G:{\text{Mod}}_{R[S^{-1}]}\to C}$

such that ${\displaystyle F=G\circ \varphi }$.

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,[1] the morphisms from object X to object Y are given by roofs

${\displaystyle X{\stackrel {f}{\leftarrow }}X'\rightarrow Y}$

(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 "calculus of fractions" can be seen as a generalization of the construction of rational numbers as equivalence classes of pairs of integers.

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.

The localization of a topological space, introduced by Dennis Sullivan, 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.[3]

• Simplicial localization