In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, localization of a category consists of adding to a
category inverse
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s 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
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolog ...
, for example, there are many examples of mappings that are invertible
up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' wi ...
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
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s 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