Localization Of A Category
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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: :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 R ^/math>-modules. Here R ^/math> is the localization of ''R'' with respect to the (multiplicatively closed) subset ''S'' consisting of all powers of ''r'', S = \ The expression "most closely related" is formalized by two conditions: first, there is a functor :\varphi: \text_R \to \text_ \quad M \mapsto M ^/math> sending any ''R''-module to its localization with respect to ''S''. Moreover, given any category ''C'' and any functor :F: \text_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 :G: \text_ \to C such that F = G \circ \varphi.


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 ''C'' and some class ''W'' of morphisms in ''C'', the localization ''C'' 'W''−1is 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''−1and given another category ''D'', a functor ''F'': ''C'' → ''D'' factors uniquely over ''C'' 'W''−1if 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'', the morphisms from object ''X'' to object ''Y'' are given by ''roofs'' :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 Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...
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 categories In mathematics, particularly in homotopy theory, a model category is a category theory, category with distinguished classes of morphisms ('arrows') called 'weak equivalence (homotopy theory), weak equivalences', 'fibrations' and 'cofibrations' sati ...
: a model category ''M'' is a category in which there are three classes of maps; one of these classes is the class of weak equivalences. 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(lX),lL(X):L(X) → LL(X)'' are isomorphisms. It can be proven that in this case, both maps are equal. 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 C \to C ^/math>, but also a functor in the opposite direction, :C ^\to C. For example, modules over the localization R ^/math> of a ring are also modules over ''R'' itself, giving a functor :\text_ \to \text_R In this case, the composition :L : C \to C ^\to C is a localization of ''C'' in the sense of an idempotent and coaugmented functor.


Examples


Serre's ''C''-theory

Serre introduced the idea of working 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 ...
''
modulo In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is t ...
'' some class ''C'' of abelian groups. 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 modules over a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
''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 Support may refer to: Arts, entertainment, and media * Supporting character Business and finance * Support (technical analysis) * Child support * Customer support * Income Support Construction * Support (structure), or lateral support, a ...
of codimension at least two. This idea is much used in Iwasawa theory.


Derived categories

The
derived category In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction pro ...
of an
abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...
is much used in
homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
. It is the localization of the category of chain complexes (up to homotopy) with respect to the quasi-isomorphisms.


Quotients of abelian categories

Given an
abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...
''A'' and a Serre subcategory ''B,'' one can define the quotient category ''A/B,'' which is an abelian category equipped with an exact functor from ''A'' to ''A/B'' that is 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

An isogeny from an
abelian variety In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular func ...
''A'' to another one ''B'' is a surjective morphism with finite 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 ''A1'' of ''A'', there is another subvariety ''A2'' of ''A'' such that :''A1'' × ''A2'' is ''isogenous'' to ''A'' (Poincaré's reducibility theorem: see for example ''Abelian Varieties'' by David Mumford). To call this a
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
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 In mathematics, homotopical algebra is a collection of concepts comprising the ''nonabelian'' aspects of homological algebra as well as possibly the abelian aspects as special cases. The ''homotopical'' nomenclature stems from the fact that a com ...
, including as special cases both the localization of spaces and of categories, is the ''
Bousfield localization In category theory, a branch of mathematics, a (left) Bousfield localization of a model category replaces the model structure with another model structure with the same cofibrations but with more weak equivalences. Bousfield localization is named ...
'' of a model category. Bousfield localization forces certain maps to become weak equivalences, which is in general weaker than forcing them to become isomorphisms.Philip S. Hirschhorn: ''Model Categories and Their Localizations'', 2003, {{isbn, 0-8218-3279-4., Definition 3.3.1


See also

*
Simplicial localization In category theory, a branch of mathematics, the simplicial localization of a category ''C'' with respect to a class ''W'' of morphisms of ''C'' is a simplicial category ''LC'' whose \pi_0 is the localization C ^/math> of ''C'' with respect to ' ...


References

Category theory Localization (mathematics)