Editing Localization of a category (section)

==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 (mathematics)|category]] ''C'' and some class ''W'' of [[morphisms]] in ''C'', the localization ''C''[''W''<sup>&minus;1</sup>] 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'' &rarr; ''C''[''W''<sup>&minus;1</sup>] and given another category ''D'', a functor ''F'': ''C'' &rarr; ''D'' factors uniquely over ''C''[''W''<sup>&minus;1</sup>] 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'',<ref>{{cite book
  | last1=Gabriel | first1=Pierre | author1-link=Pierre Gabriel
  | last2=Zisman | first2=Michel
  | title=Calculus of fractions and homotopy theory
  | url=https://www.uio.no/studier/emner/matnat/math/MAT9580/v18/documents/gabriel-zisman-calculus-of-fractions-and-homotopy-theory.pdf| publisher=Springer-Verlag| location= New York
  | year=1967
  | page=12
  | series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]], Band 35}}
</ref> the morphisms from object ''X'' to object ''Y'' are given by [[span (category theory)|''roofs'']]
:<math>X \stackrel f \leftarrow X' \rightarrow Y</math>
(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.

===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 category|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 equivalence (homotopy theory)|weak equivalence]]s. 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(l<sub>X</sub>),l<sub>L(X)</sub>:L(X) → LL(X)'' are isomorphisms. It can be proven that in this case, both maps are equal.<ref>[http://www.math.uchicago.edu/~mitya/idempotents.pdf Idempotents in Monoidal Categories]</ref>

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 <math>C \to C[W^{-1}]</math>, but also a functor in the opposite direction,
:<math>C[W^{-1}] \to C.</math>
For example, modules over the localization <math>R[S^{-1}]</math> of a ring are also modules over ''R'' itself, giving a functor
:<math>\text{Mod}_{R[S^{-1}]} \to \text{Mod}_R </math>
In this case, the composition
:<math>L : C \to C[W^{-1}] \to C</math>
is a localization of ''C'' in the sense of an idempotent and coaugmented functor.