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.
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 $W$.
Let $C$ be a category and $W \subset Mor(C)$ a collection of morphisms.
A localization of $C$ by $W$ (or “at $W$”) is
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
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.
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
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.
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.
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 $C[W^{-1}]$.
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 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
and
(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.
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 $W$:
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.
Localization is especially well developed in abelian setup where several competing formalisms and input data are used. See localization of an abelian category.
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
See also localization of a simplicial model category.
The classical reference to localization for categories is the book by Gabriel and Zisman:
A more modern account of localization with a calculus of fractions is section 7 of
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