nLab
localization

Idea

In general, localization is a process of adding formal inverses to an algebraic structure. A classic example is the localization of a 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 $ℝ[x]$ consists of polynomial functions on the real line. If we pick a point $a\in ℝ$ and localize $ℝ[x]$ 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 $ℝ[x]$ 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 $ℝ[x]$ is the whole real line. When we localize away from $a$, the resulting ring has spectrum $ℝ-\{a\}$. 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 $W$.

Localization as described here has more sophisticated variants which apply to model categories:

• Bousfield localization

• localization of a simplicial model category.

Definition

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

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

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

is full and faithful.

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 $C^2$.

• 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.

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 $W^{\mathrm{op}}$ be the set in $C^{\mathrm{op}}$ 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 $C(x,y)\coprod W^{\mathrm{op}}(x,y)$.

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

Let $𝒫G$ 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 $𝒫G$ 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 $C$,

  $$(x;{\mathrm{id}}_x;x)\sim (x;\varnothing ;x)$$(x;id_x;x) \sim (x;\emptyset;x)

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

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

• for all $f:x\to y$ in $W$,

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

  and

  $$(y;\overline{f},f;y)\sim (y;{\mathrm{id}}_y;y)$$(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 $C$ 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 $W$:

$$x\stackrel{f_1}{⟵}y\stackrel{f_2}{⟶}z$$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.

Relation to geometric embeddings

If $C$ is a topos then a class of localizations of $C$ are related to geometric embeddings $C[W^{-1}]\to C$.

Zoran: not every localization has an adjoint.

In particular, sheafification given by the geometric embedding $\mathrm{Sh}(S)\hookrightarrow \mathrm{PSh}(S)$ is the localization of a presheaf topos $\mathrm{PSh}(S)$ at local isomorphisms.

More generally:

Proposition

Every geometric embedding $C\prime \hookrightarrow C$ of topoi is of the form ${\mathrm{Sh}}_{j}C\hookrightarrow C$ for some Lawvere-Tierney topology $j$ on $C$. The objects of ${\mathrm{Sh}}_{j}C$ are the local objects with respect to the dense monomorphisms of $C$ determined by the Lawvere-Tierney topology. The class $W$ of morphism inverted this way is the saturation of the dense monomorphisms of $C$.

(…let me check that I said this precisely correctly…)

This perspective on localization is very useful for understanding the important case of localization of model categories.

These constructions are to be thought of as modelling or presenting the passage not just from categories to reflective subcategories but from (∞,1)-categories to reflective (∞,1)-subcategories, i.e. the localization of an (∞,1)-category:

with the right $\mathrm{\infty }$-version of left exact, left adjoint etc. understood, a localization of an (∞,1)-category $C$ is nothing but a fully faithful (∞,1)-functor

$$L\hookrightarrow C$$L \hookrightarrow C

with a left adjoint.

References

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

• P. Gabriel and 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

• Kashiwara-Schapira, Categories and Sheaves.

Discussion

This discussion started on the question in which sense “localization” is a descriptive term or not.

Zoran Skoda Mike, why do you say confusing? First of all localization of a ring induces localization of categories of 1-sided modules by tensoring with the localized ring over original ring and conversely applying localization functor to a ring itself produces the localized ring. The canonical morphism from the ring to its localization, sometimes also called localization, is the adjunction morphism indexed by the ring. The localization functor is just a natural extension of the localization from a ring to all modules over the ring not just the ring itself. The same for corresponding (components of) the adjunction morphisms in that case.

Second, if one does special case when localizations can be made via categories of fractions then Ore conditions are literally Ore conditions from the theory of localization of monoids or rings (besides a monoid is just a category with one object). Third, Bousfield localization in triangulated setup is a localization associated to an idempotent monad just like usual flat localization (or any localization having faithfully flat right adjoint), just the functors are triangulated, and the monad is Z-graded. Finally Cohn universal localization is just H_0 of Bousfield localization and in the matrix form one is essentially solving the Ore condition, as it is shown by Malcolmson and independently and earlier Gerasimov. In fact when one restricts the Cohn localization to finitely generated projectives one has a flat localization. So it is not just an analogy – these are all special cases of the same picture and mechanism.

Mike: The thing that is confusing to me is when one extends the use of “localization” beyond the context of localization of things like modules and sheaves. I do not see any “locality” involved in the process of inverting an arbitrary class of morphisms in a category. Of course there is just one concept here, but I do not like the choice of the word “localization” to describe it.

Tim: Might I suggest that a little historical note tracing the origin of the term (including local ring, as well) might be a good point. Sometimes such a look back to the origins of a term can show old light on new concepts and help one ‘create’ good new concepts or to view the concepts in a new light.

Mike: Just to clarify, I do definitely see the reason why inverting a multiplicative system on a ring is called “localization.” I still might prefer it if people had chosen a word that describes what happens to the ring itself, rather than its spectrum, but I understand the motivation behind the term.

Tim: My point was just that others (i.e. more ‘debutant’ in the area) might benefit from a few lines on the geometric origin of the term.

Actually I agree with what you, Mike, sort of imply namely that some ‘local’ example would be good to see. Is there something in the geometric function theory area that would provide a nice example say of a naturally occurring ‘prestack’ where the passage to the corresponding stack is clearly restricting to ‘germs’ of the categories involved? I have never thought about that point, any ideas?? It may be easy, I just don’t know.

Urs: sounds good. I have now tried to rework the entry a bit reflecting this discussion. We could/should still add a more detailed historical note.

Also, it would be good to arrange the points that Zoran mentions into a coherent bulleted list in an examples section. Maybe somebody can do that.