nLab category of fractions

Idea

In general, localization is a process of freely adjoining inverses to an algebraic structure. In category theory, the localization $\mathcal{C}[W^{-1}]$ of a category $\mathcal{C}$ at a class $W$ of its morphisms is – if it exists (see below) – the result of universally making all morphisms in $W$ into isomorphisms. A localization functor is any functor $C\to C[W^{-1}]$ satisfying the universal property of the localization at some $W$.

Some authors call any localization of a category a category of fractions (Gabriel & Zisman, §1.1; Borceux 1994 I, Def. 5.2.1), while other authors reserve this name for the case that $W$ admits a “calculus of fractions” (e.g. Fritz 2011). Conversely, some authors use “localization” to refer exclusively to the latter case (e.g. Verdier 1996, Def. 2.2.5; Stacks Project).

Yet other authors take “localization” to mean specifically “reflective localizations”, by default (e.g. Lurie 2009).

Motivation

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

The terminology is odd for historical and geometric reasons: localizing at a prime means inverting things not divisible by that prime, while inverting the prime itself is called localizing away from that prime. The reason for this, as well as for the term ‘localization’, becomes more apparent when we consider examples of a more vividly geometric flavor.

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

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

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 $\mathbb{R}[x]$ is the whole real line. When we localize away from $a$, the resulting ring has spectrum $\mathbb{R} - \{a\}$. When we localize at $a$, the resulting ring has spectrum $\{a\}$.

The case of localizing $\mathbb{Z}$ can also be interpreted geometrically in a similar way, using scheme theory and arithmetic geometry.

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 $C$ at a class of morphisms $W$ is the universal solution to making the morphisms in $W$ into isomorphisms; it is variously written $C[W^{-1}]$, $W^{-1}C$ or $L_W C$. In some contexts, it also could be called the homotopy category of $C$ with respect to $W$.

Definition

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

General

A localization of $C$ by $W$ (or “at $W$”) is

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

• and a functor $Q : C \to C[W^{-1}]$ (localization functor);

• such that

  • for all $w \in W$, $Q(w)$ is an isomorphism;

  • for any category $A$ and any functor $F : C \to A$ such that $F(w)$ is an isomorphism for all $w \in W$, there exists a functor $F_W : C[W^{-1}] \to A$ and a natural isomorphism $F \simeq F_W \circ Q$;

  • the map between functor categories

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

    is full and faithful for every category $A$.

Note:

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

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

Remark

Contrary to what is sometimes asserted, the localization $C[W^{-1}]$ may generally be constructed when $C$ is large, even only in ZF where “large” means definable by class formulas. See Remark below. However, the localization might not be locally small, even if $C$ is. The tools of homotopy theory, and in particular model categories, can be used to address this issue (see also at homotopy category of a model category).

Reflective localization

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

In such a case

$$L_W C \stackrel{\overset{Q}{\leftarrow}}{\hookrightarrow} C$$

this adjoint exhibits $L_W C$ as a reflective subcategory of $C$.

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

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

Localizations of enriched categories

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

Construction

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

General construction

Given $C$ a category and $W \,\subset\, Mor(C)$ is a set of morphisms (not necessarily forming a subcategory), we construct the localization of $C$. Let $W^{op}$ denote the corresponding set in the opposite category $C^{op}$.

Let $G$ be the following directed graph:

• the vertices of $G$ are the objects of $C$,

• the morphisms of $G$ between two vertices $x,y$ are given by the disjoint union $C(x,y)\coprod W^{op}(x,y)$.

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

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

The equivalence relation $\sim$ is that generated by

• for all objects $x$ of $C$,

  $$(x;id_x;x) \sim (x;\emptyset;x)$$

• for all $f:x\to y$ and $g:y\to z$ in $C$,

  $$(x;f,g;z)\sim (x;g\circ f;z)$$

• for all $f:x\to y$ in $W$,

  $$(x;f,\overline{f};x)\sim (x;id_x;x)$$

  and

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

Remark

With a little care, this general construction can be enacted even for large categories $C$, where “large” means given by a class formula in ZF. First, to define the free category $\mathcal{P} G$ on a large graph $G$, notice that an arrow of $\mathcal{P} G$ is a certain partial function or functional relation from $\mathbb{N}$ to $Arr(G)$ satisfying some conditions given by ZF formulas; each such relation is a “set” by the replacement axiom, and the collection of such relations forms a definable class of sets. Second, in passing to the quotient, we remark that while the $\sim$-equivalence classes of morphisms in $\mathcal{P} G$ might be large (hence themselves could not be elements of a class), we can always use Scott's trick and consider instead those representatives in each equivalence class that have minimal ZF rank (using crucially the axiom of foundation); this collection of representatives is a set. Thus the collection of morphisms in the quotient may be identified with a definable class of sets.

The rub is that the localization might not be locally small, even if $C$ is.

Construction when there is a calculus of fractions

If the class $W$ admits a calculus of fractions, then there is a simpler description of $C[W^{-1}]$ in terms of spans instead of zig-zags. The idea is that any morphism $f: x \to z$ in $C[W^{-1}]$ is built from a morphism $f_2 : y \to z$ in $C$ and a morphism $f_1 : y \to x$ in $W$:

$$x \stackrel{f_1}{\longleftarrow} y \stackrel{f_2}{\longrightarrow} z$$

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.

Properties

• The localization of the product category of two categories with weak equivalences is (if it exists) the product of their localizations

  $$(\mathcal{C} \times \mathcal{D})[ (W_{\mathcal{C}} \times W_{\mathcal{D}})^{-1} ] \;\; \simeq \;\; ( \mathcal{C}[W_{\mathcal{C}}^{-1}] ) \times ( \mathcal{D}[W_{\mathcal{D}}^{-1}] ) \,.$$

  This is because localization is a reflector into the exponential ideal of minimal categories with weak equivalences.

Examples

• examples of localizations of locally small categories that are not themselves locally small:

  • Krause 2008, Ex. 4.15

  • math.SE.a/703354

In higher category theory

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

For 2-categories:

• 2-localization

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

• localization of an (∞,1)-category.

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

• left Bousfield localization of model categories

See also localization of a simplicial model category?.

Related concepts

• localizing subcategory, localizer

• localization at an object

• reflective localization

  essential localization

  quintessential localization

  reflective bicategory?

• monoidal localization

• simplicial localization

• localization of a simplicial model category?

• localization of a ring, localization of a commutative ring

  • localization of a module

• localization of model categories

  • localization of simplicial model categories?

  • Bousfield localization of model categories

    • Bousfield localization of spectra

    • p-localization

• localization of a type at a family of functions

• colocalization

• ring of fractions

References

Authors who say “category of fractions” for the result of inverting any class of morphisms:

• Pierre Gabriel, Michel Zisman, §I.1 in: Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 35, Springer (1967) [doi:10.1007/978-3-642-85844-4, pdf]

• Francis Borceux, §5.2 of: Handbook of Categorical Algebra vol. 1, Cambridge UP (1994)

  (See also vols. 2 and 3 for examples of the theory in action in abelian categories, sheaf theory etc.)

Authors who say “localization” for the result of inverting any class of morphisms:

• M. Kashiwara, P. Schapira, Section 7 of: Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften 332, Springer (2006) [ doi:10.1007/3-540-27950-4, pdf]

• Wikipedia, Localization of a category

and reserve “category of fractions” for the case that $W$ admits a calculus of fractions:

• Tobias Fritz, Categories of Fractions Revisited, Morfismos 15 2 (2011) 19-38 [arXiv:0803.2587, morfismos:vol15-n2-3]

See also:

• Amnon Yekutieli, Chapter 6 of: Derived Categories, Cambridge University Press (2019) [doi:10.1017/9781108292825, arXiv:1610.09640]

Authors who say “localization” for inverting with respect to a calculus of fractions:

• Jean-Louis Verdier, Def. 2.2.5 in: Des Catégories Dérivées des Catégories Abéliennes, Astérisque 239 (1996) [numdam:AST_1996__239__R1_0, pdf]

• Stacks Project, 4.27 Localization in categories [tag:04VB]

With focus on localization of abelian categories and of triangulated categories

• Pierre Gabriel, Des catégories abéliennes, Bulletin de la Société Mathématique de France, 90 (1962) 323-448 [numdam:BSMF_1962__90__323_0]

• Verdier 1996

Review of localization in homotopy theory:

• William Dwyer, Localizations, in: Axiomatic, enriched and motivic homotopy theory, NATO Sci. Ser. II 131, Kluwer Acad. Publ. (2004) 3-28 [doi:10.1007/978-94-007-0948-5]

For triangulated categories:

• Henning Krause, Localization theory for triangulated categories, in proceedings of Workshop on Triangulated Categories, Leeds 2006 [arXiv:0806.1324]

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

A formal implementation of Gabriel-Zisman localization in ZFC, which in turn is implemented in the proof assistant Coq is in

• Carlos Simpson, Explaining Gabriel-Zisman localization to the computer (arXiv:math/0506471, web)

A HoTT-Coq-formalization of left-exact reflective sub-(∞,1)-categories (localization of an (∞,1)-category) in homotopy type theory is in

• Mike Shulman, HoTT/Coq/Subcategories/LexReflectiveSubcategory.v

An original account of localization of commutative rings and of p-local homotopy theory is

• Dennis Sullivan, Localization, Periodicity and Galois Symmetry (The 1970 MIT notes) edited by Andrew Ranicki, K-Monographs in Mathematics, Dordrecht: Springer (pdf)