Contents

# 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 $\mathbb{Z}$ 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 $Q : C \to C[W^{-1}]$;

• 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.

### 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

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

## In higher category theory

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

For 2-categories:

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

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

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

An excellent account emphasizing the interplay of the different notions (reflective subcategory, calculus of fractions, closure operator) can be found in ch. V of

• F. Borceux, Handbook of Categorical Algebra vol. 1 , Cambridge UP 1994.1

The pioneering work on abelian categories, with a large part on 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.

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

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

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)

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