nLab
localization

Contents

Idea

In general, localization is a process of adding formal inverses to an algebraic structure. The localization of a category CC at a collection WW of its morphisms is – if it exists – the result of universally making all morphisms in WW into isomorphisms.

Motivation

A classic example is the localization of a commutative ring: we can ‘localize the ring \mathbb{Z} at the prime 22’ and obtain the ring [12]\mathbb{Z}[\frac{1}{2}], or localize it at all primes and obtain its field of fractions: the field \mathbb{Q} of rational numbers.

The reason for the term ‘localization’ becomes more apparent when we consider examples of a more vividly geometric flavor.

For example, the ring [x]\mathbb{R}[x] consists of polynomial functions on the real line. If we pick a point aa \in \mathbb{R} and localize [x]\mathbb{R}[x] by putting in an inverse to the element (xa)(x-a), the resulting ring consists of rational functions defined everywhere on the real line except possibly at the point aa. This is called localization away from aa, or localization away from the ideal II generated by (xa)(x-a).

If on the other hand we put in an inverse to every element of [x]\mathbb{R}[x] that is not in the ideal II, we obtain the ring of rational functions defined somewhere on the real line at least at the point aa: namely, those without a factor of (xa)(x-a) in the denominator. This is called localizing at aa, or localizing at the ideal II.

Notice that what is literally ‘localized’ when localizing the ring is not the ring itself, but its spectrum: the spectrum becomes smaller. The spectrum of [x]\mathbb{R}[x] is the whole real line. When we localize away from aa, the resulting ring has spectrum {a}\mathbb{R} - \{a\}. When we localize at aa, the resulting ring has spectrum {a}\{a\}.

A ring is a very special case of a category, namely a one-object Ab-enriched category. This article mainly treats the more general case of localizing an arbitrary category. The localization of a category CC at a class of morphisms WW is the universal solution to making the morphisms in WW into isomorphisms; it is written C[W 1]C[W^{-1}] or W 1CW^{-1}C. It also could be called the homotopy category of CC with respect to WW.

Definition

Let CC be a category and WMor(C)W \subset Mor(C) a collection of morphisms.

General

A localization of CC by WW (or “at WW”) is

  • a (generally large, see below) category C[W 1]C[W^{-1}];

  • and a functor Q:CC[W 1]Q : C \to C[W^{-1}];

  • such that

()Q:Funct(C[W 1],A)Funct(C,A)(-)\circ Q : Funct(C[W^{-1}], A) \to Funct(C,A)

is full and faithful for every category AA.

Note:

  • if C[W 1]C[W^{-1}] exists, it is unique up to equivalence.

  • In 2-categorical language, C[W 1]C[W^{-1}] is the coinverter of the canonical natural transformation sts\to t, where s,t:WCs,t:W\to C are the “source” and “target” functors and WW is considered as a full subcategory of the arrow category C 2C ^{\mathbf{2}}.

  • size issues: If CC is large, then the existence of C[W 1]C[W^{-1}] may depend on foundations, and it will not necessarily be locally small even if CC is. The tools of homotopy theory, and in particular model categories, can be used to address this question.

Reflective localization

A special class of localizations are reflective localizations, those where the functor CL WCC \to L_W C has a full and faithful right adjoint L WCCL_W C \hookrightarrow C.

In such a case

L WCQC L_W C \stackrel{\overset{Q}{\leftarrow}}{\hookrightarrow} C

this adjoint exhibits L WCL_W C as a reflective subcategory of CC.

One shows that L WCL_W C is – up to equivalence of categories – the full subcategory on the WW-local objects, and this property precisely characterizes such reflective localizations.

More on this is at reflective subcategory and reflective sub-(∞,1)-category, and also reflective factorization system.

Localizations of enriched categories

Given a symmetric closed monoidal category VV, a VV-enriched category AA with underlying ordinary category A 0A_0 and a subcategory Σ\Sigma of A 0A_0 containing the identities of A 0A_0, H. Wolff defines the corresponding theory of localizations. See localization of an enriched category.

Construction

There is a general construction of C[W 1]C[W^{-1}], if it exists, which is however hard to use. When the system WW has special properties, most notably when WW admits a calculus of fractions or a factorization system, then there are more direct formulas for the hom-sets of C[W 1]C[W^{-1}].

General construction

If CC is a category and WW is a set of arrows, we construct the localization of CC. Let W opW^{op} be the set in C opC^{op} corresponding to WW (it isn’t necessarily a category).

Let GG be the following directed graph:

  • the vertices of GG are the objects of CC,
  • the arrows of GG between two vertices x,yx,y are given by the disjoint union C(x,y)W op(x,y)C(x,y)\coprod W^{op}(x,y).

The arrows in W op(x,y)W^{op}(x,y) are written as f¯\overline{f} for fW(y,x)f\in W(y,x).

Let 𝒫G\mathcal{P}G be the free category on GG. The identity arrows are given by the empty path beginning and ending at a given object. We introduce a relation on the arrows of 𝒫G\mathcal{P}G and quotient by the equivalence relation \sim generated by it to get C[W 1]C[W^{-1}].

The equivalence relation \sim is generated by

  • for all objects xx of CC,
    (x;id x;x)(x;;x)(x;id_x;x) \sim (x;\emptyset;x)
  • for all f:xyf:x\to y and g:yzg:y\to z in CC,
    (x;f,g;z)(x;gf;z)(x;f,g;z)\sim (x;g\circ f;z)
  • for all f:xyf:x\to y in WW,
    (x;f,f¯;x)(x;id x;x)(x;f,\overline{f};x)\sim (x;id_x;x)

    and

    (y;f¯,f;y)(y;id y;y)(y;\overline{f},f;y)\sim (y;id_y;y)

(Continue to show the quotient by \sim gives a category, that it is locally small, and that if CC is small, the quotient is small.)

David Roberts: This could probably be described as the fundamental category of 2-dimensional simplicial complex with the directed space structure coming from the 1-skeleton, which will be the path category above. In that case, we could/should probably leave out the paths of zero length.

Construction when there is a calculus of fractions

If the class WW admits a calculus of fractions, then there is a simpler description of C[W 1]C[W^{-1}] in terms of spans instead of zig-zags. The idea is that any morphism f:xzf: x \to z in C[W 1]C[W^{-1}] is built from a morphism f 2:yzf_2 : y \to z in CC and a morphism f 1:yxf_1 : y \to x in WW:

xf 1yf 2z x \stackrel{f_1}{\longleftarrow} y \stackrel{f_2}{\longrightarrow} z

For more on this, see the entry calculus of fractions.

Dorette Pronk has extended this idea to construct a bicategories of fractions where a class of 1-arrows is sent to equivalences.

In abelian categories

Localization is especially well developed in abelian setup where several competing formalisms and input data are used. See localization of an abelian category.

In higher category theory

The notion of localization of a category has analogs in higher category theory.

For (∞,1)-categories and the special case of reflective embeddings this is discussed in

Every locally presentable (∞,1)-category is presented by a combinatorial model category. Accordingly, there is a model for the localization of (,1)(\infty,1)-categories in terms of these models. This is called

See also localization of a simplicial model category.

References

The classical reference to localization for categories is the book by Gabriel and Zisman:

  • P. Gabriel, M. Zisman, Calculus of fractions and homotopy theory, Springer, New York, 1967. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35.

A more modern account of localization with a calculus of fractions is section 7 of

The pioneering work on abelian categories, having large part about the localization in abelian categories is

A terminological discussion prompted by question in which sense “localization” is a descriptive term or not is archived ion nnForum here.

Revised on November 12, 2013 08:38:36 by Urs Schreiber (145.116.131.45)