Localization Of A Category
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, localization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become
isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
s. This is formally similar to the process of
localization of a ring Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is aff ...
; 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 In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A ...
spaces. Calculus of fractions is another name for working in a localized category.


Introduction and motivation

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: :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 In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
:\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 mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
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 In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
: 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 f ...
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: 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 ''
modulo In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the '' modulus'' of the operation. Given two positive numbers and , mo ...
'' 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 (mathematics), 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 prope ...
''R'', when ''R'' has
Krull dimension In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally ...
≥ 2, it can be useful to treat modules ''M'' and ''N'' as ''pseudo-isomorphic'' if ''M/N'' has support of codimension at least two. This idea is much used in Iwasawa theory.


Derived categories

The derived category 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 o ...
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 o ...
''A'' and a Serre subcategory ''B,'' one can define the
quotient category In mathematics, a quotient category is a category (mathematics), category obtained from another category by identifying sets of morphisms. Formally, it is a quotient object in the category of small categories, category of (locally small) categories ...
''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 ''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. As an example, the direct sum of two abelian groups A and B is anothe ...
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 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


References

Category theory Localization (mathematics)