# Contents

## Idea

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

### Motivation

A classic example is the localization of a commutative ring: we can ‘localize the ring $\mathbb{Z}$ at the prime $2$’ and obtain the ring $\mathbb{Z}[\frac{1}{2}]$, 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 $\mathbb{R}[x]$ 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 $\{a\}$.

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 written $C[W^{-1}]$ or $W^{-1}C$. 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 $Q : C \to C[W^{-1}]$;

• 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 $C ^{\mathbf{2}}$.

• size issues: If $C$ is large, then the existence of $C[W^{-1}]$ may depend on foundations, and it will not necessarily be locally small even if $C$ 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 $C \to L_W C$ 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 subcategory and reflective sub-(∞,1)-category, and also 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

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

Let $G$ be the following directed graph:

• the vertices of $G$ are the objects of $C$,
• the arrows 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 arrows are given by the empty path beginning and ending at a given object. We introduce a relation on the arrows of $\mathcal{P}G$ and quotient by the equivalence relation $\sim$ generated by it to get $C[W^{-1}]$.

The equivalence relation $\sim$ is 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

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

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

## 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 $(\infty,1)$-categories in terms of these models. This is called

## 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 $n$Forum here.

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