Editing Localization of a category

In [[mathematics]], '''localization of a category''' consists of adding to a [[Category (mathematics)|category]] inverse [[morphism]]s for some collection of morphisms, constraining them to become [[isomorphism]]s. 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{{what|date=March 2019}}. '''Calculus of fractions''' is another name for working in a localized category.

==Introduction and motivation==

A [[category (mathematics)|category]] ''C'' consists of objects and [[morphism]]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''-[[module (mathematics)|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 (ring theory)|unit]]) not an isomorphism: 
:<math>M \to M \quad m \mapsto r \cdot m.</math>
The category that is most closely related to ''R''-modules, but where this map ''is'' an isomorphism turns out to be the category of <math>R[S^{-1}]</math>-modules. Here <math>R[S^{-1}]</math> is the [[localization of a ring|localization]] of ''R'' with respect to the (multiplicatively closed) subset ''S'' consisting of all powers of ''r'',
<math>S = \{ 1, r, r^2, r^3, \dots\}</math>
The expression "most closely related" is formalized by two conditions: first, there is a [[functor]]
:<math>\varphi: \text{Mod}_R \to \text{Mod}_{R[S^{-1}]} \quad M \mapsto M[S^{-1}]</math>
sending any ''R''-module to its [[localization of a module|localization]] with respect to ''S''. Moreover, given any category ''C'' and any functor 
:<math>F: \text{Mod}_R \to C</math>
sending the multiplication map by ''r'' on any ''R''-module (see above) to an isomorphism of ''C'', there is a unique functor
:<math>G: \text{Mod}_{R[S^{-1}]} \to C</math>
such that <math>F = G \circ \varphi</math>.

==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.

==Examples==

===Serre's ''C''-theory===

[[Jean-Pierre Serre|Serre]] introduced the idea of working in [[homotopy theory]] ''[[Ideal (ring theory)|modulo]]'' some class ''C'' of [[abelian group]]s. This meant that groups ''A'' and ''B'' were treated as isomorphic, if for example ''A/B'' lay in ''C''. 

===Module theory===

In the theory of [[module (mathematics)|module]]s over a [[commutative ring]] ''R'', when ''R'' has [[Krull dimension]] ≥ 2, it can be useful to treat modules ''M'' and ''N'' as ''pseudo-isomorphic'' if ''M/N'' has [[support of a module|support]] of codimension at least two. This idea is much used in [[Iwasawa theory]].

===Derived categories===

The [[derived category]] of an [[abelian category]] is much used in [[homological algebra]]. It is the localization of the category of chain complexes (up to homotopy) with respect to the [[quasi-isomorphism]]s.

=== Quotients of abelian categories ===
Given an [[abelian category]] ''A'' and a [[Serre subcategory]] ''B,'' one can define the [[Quotient of an abelian category|quotient category]] ''A/B,'' which is an abelian category equipped with an [[exact functor]] from ''A'' to ''A/B'' that is [[Essentially surjective functor|essentially surjective]] and has kernel ''B.'' This quotient category can be constructed as a localization of ''A'' by the class of morphisms whose kernel and cokernel are both in ''B.''

===Abelian varieties up to isogeny===<!-- This section is linked from [[Elliptic curve]] -->

An [[isogeny]] from an [[abelian variety]] ''A'' to another one ''B'' is a surjective morphism with finite [[Kernel (category theory)|kernel]]. Some theorems on abelian varieties require the idea of ''abelian variety up to isogeny'' for their convenient statement. For example, given an abelian subvariety ''A<sub>1</sub>'' of ''A'', there is another subvariety ''A<sub>2</sub>'' of ''A'' such that

:''A<sub>1</sub>'' &times; ''A<sub>2</sub>''

is ''isogenous'' to ''A'' (Poincaré's reducibility  theorem: see for example ''Abelian Varieties'' by [[David Mumford]]). To call this a [[direct sum]] decomposition, we should work in the category of abelian varieties up to isogeny.

== Related concepts ==

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 equivalence (homotopy theory)|weak equivalence]]s, which is in general weaker than forcing them to become isomorphisms.<ref>Philip S. Hirschhorn: ''Model Categories and Their Localizations'', 2003, {{isbn|0-8218-3279-4}}., Definition 3.3.1</ref>

==See also==
*[[Simplicial localization]]

==References==
<references />

[[Category:Category theory]]
[[Category:Localization (mathematics)]]