locale

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### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
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#### Topology
+-- {: .hide}
[[!include topology - contents]]
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# Contents
* table of contents
{: toc}

## Idea and motivation

Locale theory is one particular formulation of [[point-free topology]].

A _locale_ is, intuitively, like a [[topological space]] that may or may not have enough [[points]] (or even any points at all).  It contains things we call [[open subspaces]] but there may or may not be enough points to distinguish between open subspaces.  An open subspace in a locale can be regarded as conveying a bounded amount of information about the (hypothetical) points that it contains.

For example, there is a locale of all [[surjections]] from [[natural numbers]] (thought of as forming the [[discrete space]] $N$) to [[real numbers]] (forming the [[real line]] $R$, the _[[locale of real numbers]]_).  This locale has no points, since there are no such surjections, but it contains many nontrivial open subspaces.  (These open subspaces are generated by a family parametrised by $n\colon N$ and $x\colon R$; the basic open associated to $n$ and $x$ may be described as $\{f\colon N \twoheadrightarrow R \;|\; f(n) = x\}$.  Together with relations internalizing the statements ‘$\forall\, n,\; \exists!\, x,\; f(n) = x$’ and ‘$\forall\, x,\; \exists\, n,\; f(n) = x$’, which identify some opens but do not identify all of them, this specifies the locale. This construction is example 1.2.8 from section C1.2 of the [[Elephant]].)

Every topological space can be regarded as a locale (with some lost information if the space is not [[sober space|sober]]). The locales arising this way are the _[[topological locale|topological]]_ or _spatial_ locales.  Conversely, every locale induces a topology on its set of points, but sometimes a great deal of information is lost; in particular, there are many different locales whose set of points is [[empty set|empty]].

One motivation for locales is that since they take the notion of open subspace as basic, with the points (if any) being a derived notion, they are exactly what is needed to define [[sheaves]].  The notion of sheaf on a topological space only refers to the open subspaces, rather than the points, so it carries over word-for-word to a definition of sheaves on locales.  Moreover, passage from locales to their [[toposes]] of sheaves is a [[full and faithful functor]], unlike for topological spaces.

Another advantage of locales is that they are better-behaved than topological spaces in alternative [[foundations of mathematics]], including mathematics without the [[axiom of choice]], more generally [[constructive mathematics]], or mathematics [[internalization|internal]] to an arbitrary [[topos]].  For example, constructively the topological space $[0,1]$ need not be [[compact space|compact]], but the _locale_ $[0,1]$ is always compact (in a suitable sense).  It follows that the locale $[0,1]$, and hence also the locale $R$ of real numbers, is not necessarily spatial.  When it fails to be spatial, because there are "not enough real numbers," the locale of real numbers is generally a better-behaved object than the topological space of real numbers.

## Definitions

Recall that a **[[frame]]** $A$ is a [[partial order|poset]] with all [[join|joins]] and all finite [[meet|meets]] which satisfies the _infinite distributive law_:
$$ x \wedge (\bigvee_i y_i) = \bigvee_i (x\wedge y_i) .$$
A **frame [[homomorphism]]** $\phi\colon A\to B$ is a function which preserves finite meets and arbitrary joins.  Frames and frame homomorphisms form a [[category]] [[Frm]].

Note: By the [[adjoint functor theorem]] (AFT) for posets, a frame also has all meets (at least assuming that it is a [[small set]], or more generally that we allow [[predicative mathematics|impredicative quantification]] over it), but a frame homomorphism need not preserve them.  Similarly, by the AFT, a frame is automatically a [[Heyting algebra]], but again a frame homomorphism need not preserve the [[Heyting implication]].

By definition, the [[category]] [[Locale]] of **locales** is the [[opposite category|opposite]] of the category of frames
$$
  Locale \coloneqq Frm^{op}
  \,.
$$
That is, a locale $X$ "is" a frame, which we often write as $O(X)$ and call "the frame of open subspaces of $X$", and a **[[continuous map]]** $f\colon X \to Y$ of locales is a frame homomorphism $f^*\colon O(Y) \to O(X)$. If you think of a frame as an algebraic structure (a [[lattice]] satisfying a completeness condition), then this is an example of the [[duality]] of [[space and quantity]].

It is also possible to think of a continuous map $f\colon X \to Y$ as a map of [[posets]] $f_*\colon O(X) \to O(Y)$: a [[function]] that preserves all [[meets]] (and therefore is [[monotone function|monotone]] and has a [[left adjoint]] $f^*\colon O(Y) \to O(X)$) and such that the left adjoint $f^*$ preserves all finite meets.  This has the arrow pointing in the "right" direction, at the cost of a less direct definition.  (In [[predicative mathematics]], we would have to explicitly add into this definition that $f_*$ has a left adjoint; although since few locales even exist predicatively, we usually turn attention to [[base of a locale|bases]] of locales anyway.)

+-- {: .num_remark}
###### Remark
The map $f_\ast: O(X) \to O(Y)$ is of course *not* the "direct image" along $f$, rather it is a kind of dual to direct image, taking an open $u \in O(X)$ to the join
$$\bigvee_{v \in O(Y): f^\ast(v) \leq u} v$$

For [[topological spaces]] in [[classical mathematics]], denoting the [[complementation]] operator by $\neg$ and the [[interior]] operator by $int$, we have $f_\ast(u) = int(\neg f(\neg u))$ where $f$ on the right denotes the ordinary set-theoretic direct image.

Alternatively, $f_*(u)$ can also be described as the largest open subset of $Y$
whose preimage in $X$ is a subset of $u$.
=--

The category $Locale$ is naturally enhanced to a [[2-category]]:
+-- {: .num_defn}
###### Definition

The [[2-category]] [[Locale]] has

* as [[objects]] $X$ [[frames]] $O(X)$;

* as [[morphisms]] $f\colon X \to Y$ frame homomorphisms $f^*\colon O(Y) \to O(X)$;

* a unique [[2-morphism]] $f \Rightarrow g$ whenever for all $U \in O(Y)$ we have $f^*(U) \leq g^*(U)$.
=--
(See for instance [Johnstone, C1.4, p. 514](#Johnstone).)

Since [[parallel morphisms|parallel]] $2$-morphisms are equal, this [[2-category]] is in fact a [[(1,2)-category]]: a [[Poset]]-[[enriched category]].

## Subsidiary notions
{#Subsidiaries}

### Open and closed subspaces

Given a locale $X$, the elements of the frame $O(X)$ are traditionally thought of as being the [[open subspaces]] of $X$ and are therefore called the __opens__ (or __open subspaces__, __open parts__, or __open sublocales__) of $X$.  However, one may equally well view them as the [[closed subspaces]] of $X$ and call them the __closeds__ (or __closed subspaces__, __closed parts__, or __closed sublocales__) of $X$.  When viewed as closed subspaces, the [[opposite relation|opposite]] containment relation is used; thus $O(X)$ is the __frame of opens__ of $X$, while the [[opposite poset]] $O(X)^{op}$ is the __[[coframe]] of closeds__ of $X$.

An open subspace may be thought of as a potentially *verifiable* property of a hypothetical point of the space $X$.  Thus we may verify the [[intersection]] of two open subspaces by verifying both properties, and we may verify the [[union]] of any family of open subspaces by verifying at least one of them; but it may not necessarily make sense to verify the intersection of infinitely many open subspaces, because this would require us to do an infinite amount of work.  (The [[meet]] of an arbitrary family of open subspaces does exist, but the construction is impredicative, and it does not match the meet within the lattice of all subspaces.)

[[de Morgan duality|Dually]], a closed subspace may be thought of as a potentially *refutable* property.  Thus we may refute the union of two closed subspaces by refuting both of them, and we may refute the intersection of any family of closed subspaces by refuting at least one of them; but it may not necessarily make sense to refute the union of infinitely many closed subspaces.  (Again, the [[join]] of an arbitrary family of closed subspaces does not work right.)

If one views an element of $O(X)$ as a [[subspace]] of $X$, one usually means to view it as an open subspace, but we have seen that one may also view it as a closed subspace.  This is given by two different maps (one covariant and a frame homomorphism, one contravariant and a coframe homomorphism) from $O(X)$ to the lattice of all subspaces of $X$.  See [[sublocale]] for further discussion.

### Points

As a locale, the __abstract [[point]]__ is the locale $1$ whose frame of opens is the frame of [[truth values]] (classically $\{\bot \lt \top\}$).  This is the [[terminal object]] in $Locale$, since we must have (for $f\colon X \to 1$) $f^*(\top) = \top_X$ and $f^*(\bot) = \bot_X$ (and most generally $f^*(p) = \bigvee \{\top_X \;|\; p\}$, since $p = \bigvee \{\top \;|\; p\}$).

Given a locale $X$, a __concrete [[point]]__ of $X$ may be defined in any of the following equivalent ways:

1.  A point of $X$ is a [[continuous map]] $f\colon 1 \to X$;
2.  Unravelling this in terms of $f^*\colon O(X) \to O(1)$ and viewing this as a [[characteristic function]], a point of $X$ is a [[completely prime filter]] in $O(X)$;
3.  Unravelling (1) in terms of $f_*\colon O(1) \to O(X)$, which classically (using [[excluded middle]]) is determined by $f_*(\bot)$, a point of $X$ is a [[prime element]] of $O(X)$.

Definition (3) is simpler than (2), being an element of $O(X)$ satisfying a finitary condition rather than a subset of $O(X)$ satisfying an infinitary condition.  However, it doesn\'t work in [[constructive mathematics]], which provides much (but by no means all) of the motivation for studying locales.

## Properties

### Categories of internal locales
 {#CategoryOfLocales}

+-- {: .num_prop}
###### Proposition

The category [[Locale]]$(E)$ of locales [[internalization|internal to]] a [[topos]] $E$ is equivalent to the category of [[localic geometric morphisms]] $Sh_{E}(\Sigma) \to E$ in [[Topos]].

$$
  Locale(E) \simeq (Topos/E)_{loc}
  \,.
$$
=--

See [[localic geometric morphism]] for more.

+-- {: .num_prop}
###### Proposition

For every locale $X$, the category $Locale(Sh(X))$ of locales internal to the [[sheaf topos]] over $X$ is equivalent to the [[overcategory]] $Locale/X$

$$
  \mathcal{I}\colon Locale/X \stackrel{\simeq}{\to} Locale(Sh(X))
  \,.
$$
=--

This appears as [Johnstone, theorem C1.6.3](#Johnstone).

+-- {: .num_prop}
###### Proposition

For every morphism of locales $f\colon Y \to X$ the [[sheaf topos]] $Sh(Y)$ is equivalent as a topos over $Sh(X)$ to the topos $Sh_{Sh(X)}(\mathcal{I}(Y))$ of internal sheaves over the internal locale $\mathcal{I}(Y) \in Sh(X)$

$$
  Sh(Y) \simeq Sh_{Sh(X)}(\mathcal{I}(Y))
  \,.
$$
=--

This appears as [Johnstone, scholium C1.6.4](#Johnstone).

### Relation to topological spaces
 {#RelationToTopologicalSpaces}

Every [[topological space]] $X$ has a [[frame of opens]] $O(X)$, and therefore gives rise to a locale $X_L$.  For every [[continuous function]] $f\colon X \to Y$ between topological spaces, the [[preimage|inverse image]] map $f^{-1}\colon O(Y) \to O(X)$ is a frame homomorphism, so $f$ induces a continuous map $f_L\colon X_L \to Y_L$ of locales.  Thus we have a [[functor]]

$(-)_L\colon$ [[Top]] $\to$ [[Loc|Locale]].

Conversely, if $X$ is any locale, we define a **[[point]]** of $X$ to be a [[continuous map]] $1 \to X$.  Here $1$ is the [[terminal object|terminal]] locale, which can be defined as the locale $1_L$ corresponding to the terminal space.  Explicitly, we have $O(1) = P(1)$, the [[powerset]] of $1$ (the initial frame, the set of [[truth values]], which is $2$ classically or in a [[Boolean topos]]).  Since a frame homomorphism $O(X) \to P(1)$ is determined by the preimage of $1$ (the [[true]] truth value), a point can also be described more explicitly as a _[[completely prime filter]]_: an upwards-closed subset $F$ of $O(X)$ such that $X \in F$ ($X$ denotes the [[top element]] of $O(X)$), if $U,V \in F$ then $U \cap V \in F$, and if $\bigcup_i U_i \in F$ then $U_i \in F$ for some $i$.

The elements of $O(X)$ induce a [[topological structure|topology]] on the set of points of $X$ in an obvious way, thereby giving rise to a topological space $X_P$.  Any continuous map $f\colon X \to Y$ of locales induces a continuous map $f_P\colon X_P \to Y_P$ of spaces, so we have another functor

$(-)_P\colon Locale \to Top$.

One finds that $(-)_L$ is [[left adjoint]] to $(-)_P$.

In fact, this is an [[idempotent adjunction]]:

+-- {: .num_prop #TopLocAdjunction}
###### Proposition

The categories [[Top]] of [[topological spaces]] and [[Loc|Locale]] of locales are related by an [[idempotent adjunction]].

$$
  (
    (-)_L \dashv (-)_P
  )
  \colon
  Top
  \;
    \underoverset
      {\underset{(-)_P}{\longleftarrow}}
      {\overset{(-)_L}{\longrightarrow}}
      {\bot}
  \;
  Locale
  \,.
$$

=--

This appears for instance as ([MacLaneMoerdijk, theorem IX.3 1](#MacLaneMoerdijk)) or as ([Johnstone, lemma C.1.2.2](#Johnstone)).

Therefore the [[adjunction]] restricts to an [[equivalence of categories|equivalence]] between the fixed subcategories on either side.

+-- {: .num_defn}
###### Definition

A [[topological space]] $X$ with $X \cong X_{L P}$ is called **[[sober space|sober]]**.

A locale with $X \cong X_{P L}$ is called **[[spatial locale|spatial]]** or **topological**; one also says that it has **enough points**.
=--

see also MO [here](http://math.stackexchange.com/a/1904289/58526)

+-- {: .num_cor}
###### Corollary

The [[adjunction]] from prop. \ref{TopLocAdjunction} exhibits [[sober topological spaces]] as a [[coreflective subcategory]] of [[Locale]]

$$
  (
    (-)_L \dashv (-)_P
  )
  \colon
  SoberTop
    \underoverset
      {\underset{(-)_P}{\leftarrow}}
      {\overset{(-)_L}{\hookrightarrow}}
      {\bot}
  Locale
$$

and this restricts to an
[[equivalence of categories]] between the [[full subcategory]] of locales with enough points, and that of [[sober topological spaces]].
=--

This appears for instance as ([MacLaneMoerdijk, corollary IX.3 4](#MacLaneMoerdijk)).

Consequently, we often identify a sober topological space and the corresponding topological locale.

### Relation to toposes -- localic reflection
 {#RelationToToposes}

The notion of [[Grothendieck topos]] can be seen as a [[categorification]] of the notion of locale, by relating both notions to the notion of [[total category|lex totality]]:

+-- {: .num_prop}
###### Proposition

A [[poset]] $P$ is a [[frame]] if and only if the [[Yoneda embedding]]

$$
  y \colon P \to \mathbf{2}^{P^{op}}
$$
has a [[exact functor|left exact]] [[left adjoint]]. (Here the poset $\mathbf{2}$ of [[truth values]] is the base of enrichment for posets seen as [[enriched categories]].)
=--

The analogous result for toposes involves a bit of
[[set theory]]: under [[ZFC]] plus the existence of a [[strong inaccessible cardinal]] $\kappa$, the foundational assumption of MacLane in _[[Categories for the Working Mathematician]]_, call a category _[[moderate category|moderate]]_ if its set of morphisms has size $\kappa$. For example, $Set$ is moderate.

+-- {: .num_prop}
###### Proposition
([Street](#Street))

A moderate [[locally small category]] $C$ is a [[Grothendieck topos]] if and only if the [[Yoneda embedding]]
$$y \colon C \to Set^{C^{op}}$$
has a [[left exact functor|left exact]] [[left adjoint]].
=--

These results emphasize frames/toposes as algebras, where the morphisms are left exact left adjoints. The [[right adjoints]] to such morphisms are called [[geometric morphism|geometric morphisms]], and emphasize locales/toposes as [[spaces]]. This analogy, which plays an important but mostly tacit role in [Joyal and Tierney](#JT), can be developed further along the following lines.

The [[frame of opens]] $O(X)$ corresponding to a locale $X$ is naturally a [[site]]:

+-- {: .num_defn}
###### Definition

Given a locale $X$, with [[frame of opens]] $O(X)$, say that a family of [[morphisms]] $\{U_i \to U\}$ in $O(X)$ is a [[cover]] if $U$ is the [[join]] of the $U_i$:

$$
  U = \vee_i U_i
  \,.
$$
=--

+-- {: .num_prop}
###### Proposition

This defines a [[coverage]] on $O(X)$ and hence makes it a [[site]].
=--

See for instance ([MacLaneMoerdijk, section 5](#MacLaneMoerdijk)).

+-- {: .num_defn}
###### Definition

For $X$ a locale, write
$$
  Sh(X) \coloneqq Sh(O(X))
$$
for the [[sheaf topos]] over the category $O(X)$ equipped with the above canonical structure of a [[site]].
=--

Write [[Topos]] for the category of [[Grothendieck toposes]] and [[geometric morphisms]].

+-- {: .num_prop}
###### Proposition

This construction defines a [[full and faithful functor]] $Sh(-) : $ [[Locale]] $\to$ [[Topos]].
=--

This appears for instance as [MacLaneMoerdijk, section IX.5 prop 2](#MacLaneMoerdijk).

+-- {: .num_defn}
###### Definition

A topos in the [[image]] of $Sh(-)\colon Locale \to Topos$ is called a _[[localic topos]]_.
=--

+-- {: .num_prop}
###### Proposition

The functor $Sh(-)\colon Locale \to Topos$ has a [[left adjoint]]
$$
  L\colon Topos \to Locale
$$
given by sending a [[topos]] $\mathcal{E}$ to the locale that is formally dual to the [[frame]] of [[subobjects]] of the [[terminal object]] of $\mathcal{E}$:
$$
  O(L(\mathcal{E})) \coloneqq Sub_{\mathcal{E}}(*)
  \,.
$$
=--

This appears for instance as [MacLaneMoerdijk, section IX.5 prop 3](#MacLaneMoerdijk).

The functor $L$ here is also called **localic reflection**.

In summary this means that locales form a [[reflective subcategory]] of [[Topos]]:
$$
  Locale \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow}
  Topos
  \,.
$$

In fact this is even a genuine [[full sub-2-category]]:
+-- {: .num_prop}
###### Proposition

For all $X,Y \in $ [[Locale]] the [[2-functor]] $Sh\colon$ [[Locale]] $\to$ [[Topos]] induces an [[equivalence of categories]]
$$
  Sh\colon Locale(X,Y) \stackrel{\simeq}{\to} Topos(Sh(X), Sh(Y))
  \,.
$$
=--

This appears as ([Johnstone, prop. C1.4.5](#Johnstone)).

+-- {: .num_prop}
###### Proposition

The [[poset of subobjects]] $Sub_{\mathcal{E}}(*)$ of the terminal object of $\mathcal{E}$ is equivalent to the full [[subcategory]] $\tau_{\leq -1}(\mathcal{E})$ of $\mathcal{E}$ on the $(-1)$-[[truncated]] objects of $E$:
$$
  Sub_{\mathcal{E}}(*) \simeq \tau_{\leq -1} \mathcal{E}
  \,.
$$
=--

+-- {: .proof}
###### Proof

A [[(-1)-truncated]] sheaf $X$ is one whose values over any object are either the [[(-2)-groupoid|singleton set]], or the [[empty set]]

$$
  X(X) \in \{*, \emptyset\} = \subset Set
  \,.
$$

A [[monomorphism]] of [[sheaves]] is a [[natural transformation]] that is degreewise a monomorphism of sets (an [[injection]]). Therefore the [[subobjects]] of the [[terminal object|terminal]] sheaf (that assigns the singleton set to every object) are precisely the sheaves of this form.
=--

We may think of a [[frame]] as a [[Grothendieck (0,1)-topos]]. Then localic reflection is reflection of [[Grothendieck toposes]] onto $(0,1)$-toposes and is given by $(-1)$-truncation: for $X$ a locale, $Sh(X)$ the corresponding [[localic topos]] and $\mathcal{E}$ any Grothendieck topos we have a natural equivalence
$$
  Gr1Topos( \mathcal{E},  Sh X) \simeq Gr(0,1)Topos(\tau_{\leq -1} \mathcal{E}, O(X))
$$
which is
$$
  \cdots \simeq Frame(O(X), Sub_{\mathcal{E}}(*))
  \simeq
  Locale(L \mathcal{E} , X)
  \,.
$$

This is a small part of a pattern in [[higher topos theory]], described at [[n-localic (∞,1)-topos]].

## Examples
 {#Examples}

### General topology

As explained above, the [[functor]] from [[topological spaces]] to locales
provides a very large collection of examples.
When restricted to [[sober topological spaces]], this functor becomes [[fully faithful functor|fully faithful]]
and its [[essential images]] consists of [[spatial locales]].

Without the [[axiom]] of [[excluded middle]],
many classical examples, such as [[the locale of real numbers]]
or [[Cantor space]],
can cease to be [[spatial locale|spatial]].

Naturally, one is interested in examples of locales that are not spatial,
and a few are given below.

### Measure theory
 {#ExamplesMeasureTheory}

An amazing feature of [[pointfree topology]] is that
it contains not only point-set [[general topology]],
but also point-set [[measure theory]], again as a [[full subcategory]].

Specifically, recall from the [[duality between geometry and algebra]]
that various categories of [[commutative algebras]] are [[opposite category|contravariantly]]
[[equivalence of categories|equivalent]] to certain corresponding categories of spaces.
The category of algebras relevant for [[measure theory]]
is the category of [[commutative von Neumann algebras]]
and ultraweakly continuous \*-homomorphisms;
it is widely accepted that dropping the commutativity
condition and passing to the [[opposite category]] yields
the correct category of [[noncommutative measurable spaces]].

The category of [[commutative von Neumann algebras]]
is contravariantly [[equivalence of categories|equivalent]] to several other categories:

* compact strictly localizable enhanced measurable spaces and measurable maps;

* [[hyperstonean topological spaces]] and [[open maps]];

* [[hyperstonean locales]] and [[open maps]];

* [[measurable locales]] and maps of locales.

The last category is particularly interesting:
it is a [[full subcategory]] of [[locales]].
Thus, [[measure theory]] embeds into pointfree topology,
which means that methods and results from pointfree topology
can be used right away in [[measure theory]].
This stands in contrast to the traditional
point-set treatments, where two rather different
formalisms must be developed for point-set [[topological spaces]]
and point-set [[measurable spaces]].

With the exception of [[discrete locales]],
[[measurable locales]] are *never* spatial,
and in fact do not have any points other than those
in its atomic (discrete) part, which splits off as a coproduct summand.

### Intersections of dense sublocales

Another important source of nonspatial sublocales
is given by [[intersections]] (of arbitrary cardinality)
of [[dense sublocales]].

Again, in contrast to the point-set setting,
where the [[Baire category theorem]] identifies
the rather restrictive conditions under which
the intersection of dense topological subspaces is again dense,
in the pointfree setting arbitrary intersections
of [[dense sublocales]] are always dense.

In particular, one can intersection all [[dense sublocales]]
of a given [[locale]], which always produces a nonspatial locale,
unless the original locale is discrete.
This is the [[double negation sublocale]].

But there are other interesting examples.
Connecting to [[measure theory]], we can consider
a [[valuation]] $\nu$ on a given [[locale]] $L$
and take the intersection of all [[sublocales]] $S$
such that $\nu(S)=\nu(1)$.
The resulting sublocale can be seen as the smallest
sublocale with a measure 0 complement.

## Related concepts

* The notion of locale may be identified with that of a [[Grothendieck topos|Grothendieck]] [[(0,1)-topos]]. See [[Heyting algebra]] for more on this.

* An [[ionad]] is to a [[topological space]] as a [[Grothendieck topos]] is to a [[locale]].

* A [[group object]] [[internalization|internal]] to locales or an [[internal groupoid]] in locales is a [[localic group]] or [[localic groupoid]], respectively.

* [[positive locale]], [[locally positive locale]]

* [[premetric locale]], [[metric locale]]

* [[preuniform locale]], [[uniform locale]]

There is also a notion of [[internal locale]], see also [[internal site]].

* [[topological space]], [[topological locale]]

* [[ionad]]

* [[topos]]

* [[profinite space]], [[profinite reflection]]

* [[localic homotopy theory]]

* [[geometric hyperdoctrine]]

## References

An early paper, containing much of the basic theory, and in which the term "locale" is introduced:

* [[John R. Isbell]], *Atomless Parts of Spaces*, Mathematica Scandinavica **31** (1972) 5–32 [[doi:10.7146/math.scand.a-11409](https://doi.org/10.7146/math.scand.a-11409)]

An introduction to and survey of the use of locales instead of topological spaces ("[[point-free topology]]"):

* {#Johnstone83} [[Peter Johnstone]], _The point of pointless topology_ ,  Bull. Amer. Math. Soc. (N.S.) **8** 1 (1983), 41-53 [[Bulletin AMS](http://www.ams.org/bull/1983-08-01/S0273-0979-1983-15080-2/home.html), [Euclid](http://projecteuclid.org/euclid.bams/1183550014)]

This is, in its own words, to be read as the trailer for the book

*  [[Peter Johnstone]], _[[Stone Spaces]]_, Cambridge Studies in Advanced Mathematics __3__, Cambridge University Press (1982, 1986) xxi+370 pp. [[MR85f:54002](http://www.ams.org/mathscinet-getitem?mr=698074)]

that develops, among other things, much of [[general topology]] entirely with the notion of locales used in place of that of [[topological spaces]]. See also *[[Stone Spaces]]* for more.

Other textbook accounts:

* [[Francis Borceux]], volume 3, chapter 1 of _[[Handbook of Categorical Algebra]]_,

* {#Johnstone} [[Peter Johnstone]], part C (volume 2) _[[Elephant|Sketches of an elephant: a topos theory compendium]]_.

* {#MacLaneMoerdijk} [[Saunders MacLane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_

* Jorge Picado & Aleš Pultr (2012). _Frames and Locales:  Topology without points_. Springer. [Web page](http://www.mat.uc.pt/%7Epicado/publicat/bookpage.html) (with preface, contents, and errata).

and

* [[Steve Vickers]], _Topology via logic_, Cambridge University Press (1989)

where the latter develops topology from the point of view of its [[internal logic]].

An introductory survey is

* Jorge Picado, Aleš Pultr, Anna Tozzi, _Locales_, pages 49--101 in Pedicchio&Tholen (eds.), _Categorical Foundations_ , CUP 2004. ([draft](http://www.mat.uc.pt/~picado/publicat/Chapter2.pdf))

See also

* [[eom]], _[Locale](https://www.encyclopediaofmath.org/index.php/Locale)_

Lex totality is the subject of an article of Street,

* {#Street} [[Ross Street]], _Notions of Topos_, Bull. Australian Math. Soc. 23 (1981), 199--208.

The connection between locales and toposes via lex totality plays a tacit role throughout the influential monograph

* {#JT} [[André Joyal]] and [[Myles Tierney]], _An Extension of the Galois Theory of Grothendieck_, Memoirs of the American Mathematical Society 51 (1984).

A basic introduction to locale theory can be found in section 1 of

* [[Ingo Blechschmidt]], *Generalized spaces for constructive algebra*, ([arXiv:2012.13850](https://arxiv.org/abs/2012.13850))

A [[model category]] of [[locales]] which makes the [[reflective subcategory|reflection]] of [[sober topological spaces]] a [[Quillen adjunction]] to the sober-restriction of the [[classical model structure on topological spaces]]:

* {#SterlingKapulkin23} [[Sterling Ebel]], [[Chris Kapulkin]], Thm. 4.45 in: *Synthetic approach to the Quillen model structure on topological spaces* [[arXiv:2310.14235](https://arxiv.org/abs/2310.14235)]

[[!redirects locale]]
[[!redirects locales]]

[[!redirects internal locale]]
[[!redirects internal locales]]

[[!redirects localic reflection]]
[[!redirects localic reflections]]

[[!redirects locale theory]]

[[!redirects map of locales]]
[[!redirects maps of locales]]
[[!redirects morphism of locales]]
[[!redirects morphisms of locales]]
[[!redirects map of frames]]
[[!redirects maps of frames]]
[[!redirects morphism of frames]]
[[!redirects morphisms of frames]]
[[!redirects homomorphism of frames]]
[[!redirects homomorphisms of frames]]


locale of real numbers

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
=--
=--

# The locale of real numbers
* table of contents
{: toc}

## Summary

This page gives an elementary description of the [[locale]] of [[real numbers]], that is the localic [[real line]]. The development is manifestly [[constructive mathematics|constructive]] and even [[predicative mathematics|predicative]] over the [[natural numbers]] (although we are somewhat careless with the language and do not always point out when a set may predicatively be a [[proper class]]).  Ideally, we will show that our construction satisfies the seven 'headline properties' of the real line described by [Bauer & Taylor](http://www.paultaylor.eu/ASD/dedras/intro) (although so far we cover only the [[Heine–Borel theorem]]).

The exposition here is pretty much [[Toby Bartels]]\'s own work, although of course the basic ideas are well known to many.  In particular, the Zigzag Lemma \ref{zigzaglemma} is Bartels\'s as far as they know (but it\'s not very deep, just bookkeeping).  The version of [[Cousin's theorem|Cousin's Theorem]] that appears here may also be original.

## Idea
{#idea}

To describe the locale of the real numbers, we need first of all to describe what an _open_ (in the axiomatic sense, equivalent to an [[open sublocale]]) in the real line is.  The key insight is that an open subspace is determined by the [[open intervals]] of [[rational numbers]] that it contains.  This is analogous to the key insight of [[Richard Dedekind]]\'s [definition of real number](real+number#Dedekind): that a point in the real line is determined by which open intervals of [[rational numbers]] that it belongs to.  (Once one is talking about rational numbers, things are manageable.)

## Opens

An __open__ in the real line is a [[binary relation]] ${\sim}$ on the [[rational numbers]] that satisfies the four properties listed below.  Intuitively, we have $a \sim b$ iff the open [[interval]] $(a,b)$ is contained in the corresponding [[open subspace|open set]].

1. If $a \geq b$, then $a \sim b$.

2. If $a \geq b \sim c \geq d$, then $a \sim d$.

3. If $a \sim b \gt c \sim d$, then $a \sim d$.

4. If $b \sim c$ whenever $a \lt b$ and $c \lt d$, then $a \sim d$.

In practice, rather than using relation-like symbols for opens, we mimic the set-theoretic notation and terminology from [[point-set topology]].  So if $G$ is an open in the real line, then we write $(a,b) \subseteq G$ to mean that $a$ is related to $b$ through $G$ and say that $G$ __contains__ the interval from $a$ to $b$ and that this interval is __contained in__ $G$.

In this notation, and writing $(a, b) \subseteq (c, d)$ for $c \leq a$ and $b \leq d$, and $(a, b) \Subset (c, d)$ for $c \lt a$ and $b \lt d$, these requirements say:

1. If $a \geq b$, then $(a, b) \subseteq G$.

2. If $(a, d) \subseteq (b, c)$ and $(b, c) \subseteq G$, then $(a, d) \subseteq G$.

3. If $c \lt b$ and $(a, b), (c, d) \subseteq G$, then $(a, d) \subseteq G$.

4. If $(b, c) \subseteq G$ for all $(b, c) \Subset (a, d)$, then $(a, d) \subseteq G$.

Property (1) is motivated because $(a,b)$ is [[empty subset|empty]] whenever $a \geq b$. Property (2) is motivated because inclusion is [[transitive relation|transitive]]. Property (3) is motivated because if $c \lt b$, then $(a, d) = (a, b) \cup (c, d)$. Property (4) is motivated because $\bigcup_{(b, c) \Subset (a, d)} (b, c) = (a, d)$.

The really interesting property is property (3). It immediately generalizes as follows:

* If
  \[
     a_1 \sim b_1 \gt a_2 \sim b_2 \gt \cdots \gt a_n \sim b_n
  ,\]
  then $a_1 \sim b_n$.

We call the combined hypothesis of this property a __zigzag__; each hypothesis $a_i \sim b_i$ is a __zig__, and each hypothesis $b_i \gt a_{i+}$ is a __zag__.  To indicate the length of a zigzag, we will count the zigs; the zigzag above has $n$ zigs (and $n - 1$ zags).  A typical nondegenerate zigzag with $3$ zigs is shown below; it consists of $3$ overlapping open intervals, each of which belongs to a given open set; we are motivated to conclude that the entire interval from $a_1$ to $b_3$ belongs to that open set.

+-- {: style="text-align:center"}
[[!include zigzag of real numbers - SVG]]
=--

Thus the zigzag property for $n = 2$ is property (3), the zigzag property for $n = 1$ is trivial (if $a \sim b$, then $a \sim b$), and the zigzag property for $n \gt 2$ may be proved by [[induction]].

We can incorporate property (2) into this by saying:

* If
  \[
     a \geq a_1 \sim b_1 \gt a_2 \sim b_2 \gt \cdots \gt a_n \sim b_n \geq b
  ,\]
  then $a \sim b$.

Then you can think of property (1) as the zigzag property for $n = 0$.  Alternatively, we can incorporate property (4) by saying:

* If, whenever $a \lt a_1$ and $b_n \lt b$, there is a zigzag
  \[
     a_1 \sim b_1 \gt a_2 \sim b_2 \gt \cdots \gt a_n \sim b_n
  ,\]
  then $a \sim b$.

Or by making both modifications, we can stuff the entire definition into a single statement.

## The Zigzag Lemma

A zigzag may not look like the picture above; instead of an orderly progression, the zigs and zags may be wild swings that are undone by other zags and zigs.  It is sometimes convenient to replace a zigzag by a more orderly one.  Of course, this is easy if all of the zigs belong to a single open, since (by property (3) of the definition of an open), the whole zigzag can be replaced by a single zig.  But we\'ll also want to consider zigzags where the zigs are taken from different opens.  The following definitions make precise what kind of zigzag we\'ll want, and then there will be a lemma that we can indeed have this.

+-- {: .num_defn #zigzagdefn}
###### Definitions

Given rational numbers $a$ and $b$ and a positive natural number $n$, a $2n$-tuple $(a_1,b_1,\ldots,a_n,b_n)$ of rational numbers forms a __zigzag of length $n$ from $a$ to $b$__ if we have:

* $a = a_1$, $b = b_n$, and
* $b_i \gt a_{i+}$ for $i = 1, \ldots, n - 1$.

A zigzag of positive length is __orderly__ if we additionally have:

* $a_i \lt a_{i+}$ for $i = 1, \ldots, n - 1$,
* $a_i \lt b_i$ for $i = 1, \ldots, n$, and
* $b_i \lt b_{i+}$ for $i = 1, \ldots, n - 1$.

As a technicality, if $a \geq b$, then we also count the empty $0$-tuple as an __orderly zigzag of length $0$__ from $a$ to $b$.  (Notice that no zigzag of positive length can be orderly from $a$ to $b$ when $a \geq b$.)

Given additionally a collection $\mathcal{U}$ of opens, a zigzag (orderly or not) has __zigs from $\mathcal{U}$__ if every pair $(a_i,b_i)$ is contained in at least one member of $\mathcal{U}$.
=--

Now we can replace any zigzag with an orderly zigzag with zigs from the same opens:

+-- {: .num_lemma #zigzaglemma}
###### Lemma
(Zigzag Lemma)

Given rational numbers $a$ and $b$, a natural number $n$, a collection $\mathcal{U}$ of opens, and a zigzag from $a$ to $b$ of length $n$ with zigs from $\mathcal{U}$, there exists an orderly zigzag from $a$ to $b$ of length at most $n$ with zigs from $\mathcal{U}$.
=--

+-- {: .proof}
###### Proof

If $a \geq b$, then we use a zigzag of length $0$.  Otherwise, we assume that $a \lt b$ and prove the lemma by induction on $n$.  If $n = 1$, then the original zigzag is orderly since $a \lt b$.

Now assume (as an inductive hypothesis) that the lemma holds for zigzags of length $k$ for some $k \geq 1$.  A zigzag of length $k+$ consists of a zig $(a,b_1)$ from some open in $\mathcal{U}$, a zag $b_1 \gt a_2$, and a zigzag of length $k$ from $a_2$ to $b$ with zigs from $\mathcal{U}$.  Using the inductive hypothesis, replace the zigzag of length $k$ with an orderly zigzag of length $l \leq k$ from $a_2$ to $b = b_{l+}$ with zigs from $\mathcal{U}$.  We now have a zigzag of length $l+ \leq k+$ from $a$ to $b$ with zigs from $\mathcal{U}$.

If $a \lt a_2$ and $b_1 \lt b_2$, then also $a \lt a_2 \lt b_1$, so this zigzag of length $l+$ from $a$ to $b$ is orderly.

Next suppose that $a \lt a_2$ but $b_1 \geq b_2$.  In this case, consider the largest value of $i \geq 2$ such that $b_1 \geq b_i$; if $i = l+$, then we can use the orderly zigzag of length $1$ from $a$ to $b$, which is contained in the same open as $(a,b_1)$ was, since $b_1 \geq b_i = b$.  If $i \leq l$ instead, then take the orderly zigzag from $a_{i+}$ to $b$, and precede it with the zig $(a,b_i)$, to get a zigzag of length $l - i + 2 \leq k$ from $a$ to $b$.  This zigzag is orderly, because $a \lt a_2 \lt a_{i+}$, and its zigs are from $\mathcal{U}$ since $(a,b_i)$ is contained in the same open as $(a,b_1)$ was (since $b_1 \geq b_i$).  So either way, we have an orderly zigzag of length at most $k$ from $a$ to $b$ with zigs from $\mathcal{U}$.

Finally, suppose that $a \geq a_2$.  In this case, consider the largest value of $i \geq 2$ such that $a \geq a_i$, take the orderly zigzag from $a_i$ to $b$, and change $a_i$ to $a$ to get a zigzag from $a$ to $b$.  If $i \leq l$, then this is orderly since $a \lt a_{i+} \lt b_i$; if $i = l+$, then this is orderly since $a \lt b = b_i$.  Either way, $(a,b_i)$ belongs to the same open as $(a_i,b_i)$ did (since $a \geq a_i$), so this orderly zigzag of length $l - i + 2 \leq k$ from $a$ to $b$ still has zigs from $\mathcal{U}$.

In any case, we have a zigzag of length at most $k+$ from $a$ to $b$ with zigs from $\mathcal{U}$, and the induction is complete.
=--

This lemma is used in the proofs of the infinite distributivity law and some results related to measure.

## The frame of opens

The opens form a [[subobject|sub]]-[[poset]] of the [[power set]] $\mathcal{P}(\mathbb{Q} \times \mathbb{Q})$.  This poset is in fact a [[frame]], as we will now show.  (It is *not* a subframe of the power set, since the [[joins]] are different.  It *is* a sub-[[inflattice]] of the power set, although this seems to be a red herring at least for infinitary [[meets]], since those are not part of the frame structure that we need.)

The [[top element|top]] open, denoted $\mathbb{R}$, is the binary relation which is always true.  Given two opens $G$ and $H$, their [[meet]] in the poset of opens, denoted $G \cap H$, is simply their [[logical conjunction|conjunction]], that is their [[intersection]] as [[subsets]] of $\mathbb{Q} \times \mathbb{Q}$.  (In fact, given any collection of opens, their meet is their conjunction.)  It is straightforward to check that $\mathbb{R}$ and $G \cap H$ are opens and to prove that these are the desired meets.  Intuitively, this all works because an open interval will be contained in the intersection of a family of open sets if and only if it is contained in each individual open set.

The [[bottom element|bottom]] open, denoted $\empty$, is the binary relation $\geq$.  That is, $(a,b) \subseteq \empty$ iff $a \geq b$.  It is easy to check that this is an open; it precedes every open by property (1).  Intuitively, this corresponds to the [[empty subset]] of the real line; $(a,b)$ is empty if and only if $a \geq b$.  However, note that $\empty$ is *not* the empty subset of $\mathbb{Q} \times \mathbb{Q}$; the notation follows our topological intuition rather than the [[relation theory|algebra of relations]].

Given two opens $G$ and $H$, their [[join]] in the poset of opens, denoted $G \cup H$, is defined as follows:  $(a,b) \subseteq G \cup H$ if, whenever $a \lt a_1$ and $b_n \lt b$, there exists a zigzag from $a_1$ to $b_n$ with zigs from $G$ and $H$.  It is immediate that this is an open in which $G$ and $H$ are both contained.  Conversely, any open in which $G$ and $H$ are contained must contain this open $G \cup H$, by properties (3) and (4).

More generally, given any collection $\mathcal{U}$ (or family $(G_k)_k$) of opens, their [[join]] in the poset of opens, denoted $\bigcup \mathcal{U}$ (or $\bigcup_k G_k$), is defined as follows:  $(a,b) \subseteq \mathcal{U}$ if and only if, whenever $a \lt a_1$ and $b_n \lt b$, there exists a zigzag from $a_1$ to $b_n$ with zigs from $\mathcal{U}$.  (Notice that $a \geq b$ counts even when $\mathcal{U}$ is empty, using zigzags of length $0$.)  The same argument applies as before.  Note that each individual zigzag has finitely many zigs, and therefore involves finitely many of the opens $G_k$, even when taking the join of an infinite collection.

Finally, we must check the distributive law $G \cap \bigcup_k H_k \subseteq \bigcup_k (G \cap H_k)$.  That is, if $a \sim b$ directly through $G$ and $a_1 \sim b_n$ through a zigzag of $H$s for $(a_1,b_n) \Subset (a,b)$, then we need that $a_1 \sim b_n$ through a zigzag in which each zig is related both through $G$ and through some $H$.  To prove this, start with the zigzag of $H$s, and apply the Zigzag Lemma \ref{zigzaglemma} to get an orderly zigzag of $H$s, so that each zig is bounded by $a$ and $b$.  Then these zigs hold for $G$ as well, by property (2).  Therefore, we may interpret each zig using $G \cap H_k$ for some $k$, proving the desired result.

This frame of opens, interpreted as a [[locale]], is __the locale of real numbers__.  As usual, we denote this locale with the same symbol as the top element of its frame, in this case $\mathbb{R}$.  (Of course, the true etymology of the symbols runs in the other order.)

## Open intervals as opens

Given rational numbers $a$ and $b$, the open interval $(a,b)$ may itself be interpreted as an open in the real line, also denoted $(a,b)$, as follows: let $(c,d) \subseteq (a,b)$ hold if every rational number strictly between $c$ and $d$ (in that order) is also strictly between $a$ and $b$ (in that order).  In other words, we interpret '${\subseteq}$' literally as comparing subsets of $\mathbb{Q}$.  It is straightforward to check that this condition does indeed define an open.  There is now a third way to interpret '$(c,d) \subseteq (a,b)$'; interpreting both intervals as opens in the real line, this states that the first is contained in the second.  But again, it is easy to check that this is equivalent; $(c,d) \subseteq (a,b)$ (in the set-theoretic sense) if and only if $(e,f) \subseteq (a,b)$ whenever $(e,f) \subseteq (c,d)$.  Notice that $(a,b) = \empty$ whenever $a \geq b$.

We can actually generalize this somewhat.  Given any set $L$ of rational numbers and any set $U$ of rational numbers, we may define the open $(\inf U, \sup L)$ as follows: let $(a,b) \subseteq (\inf U, \sup L)$ hold if every rational number strictly between $a$ and $b$ (in that order) is greater than some element of $U$ and less than some element of $L$.  If the [[infimum]] of $U$ and the [[supremum]] of $L$ exist in the usual sense as rational numbers, then this agrees with the previous paragraph.  If instead $U$ or $L$ is the set of all rational numbers, then we write $-\infty$ for $\inf U$ or $\infty$ for $\sup L$.  In general, we may interpret $\inf U$ as an [[extended real number|extended]] [[upper real number|upper real]] and $\sup L$ as an extended [[lower real number|lower real]].  Classically, every extended upper or lower real is either a real number, $-\infty$, or $\infty$; only the converse holds constructively.  Notice that $(-\infty,\infty) = \mathbb{R}$.

Since we will refer to them below, we will state for the record the complete definitions of $(-\infty,a)$ and $(a,\infty)$ for a rational number $a$.  We have $(b,c) \subseteq (-\infty,a)$ iff every rational number strictly between $b$ and $c$ is less than $a$, that is iff $b \geq c$ or $c \leq a$.  Similarly, we have $(b,c) \subseteq (a,\infty)$ iff every rational number strictly between $b$ and $c$ is greater than $a$, that is iff $b \geq c$ or $b \geq a$.  A fortiori, $(b,c) \subseteq (-\infty,0)$ if $c = 0$, and $(b,c) \subseteq (1, \infty)$ if $b = 1$.

## Open sets, closed sets, and points

We think of each open as defining an [[open subspace|open subset]] of the real line, but we can equally well think of it as defining a [[closed subspace|closed subset]].  The difference between these perspectives is reflected in [[complement]]ary criteria for when a real number belongs to the set.  So to make sense of this, we must identify the [[point|points]] of the real line.

Recall that a [[real number]] may be defined as a pair $(L,U)$ of [[inhabited set|inhabited]] [[subsets]] of $\mathbb{Q}$ satisfying the following properties:

1. If $a \in L$, then $a \lt b$ for some $b \in L$.
2. If $b \in U$, then $a \lt b$ for some $a \in U$.
3. If $a \in L$ and $b \in U$, then $a \lt b$.
4. If $a \lt b$, then $a \in L$ or $b \in U$.

We define a __point__ of the real line to be a real number in this sense.  Given such a point $x = (L,U)$ and a rational number $a$, we write $a \lt x$ to mean that $a \in L$ and $a \gt x$ to mean that $a \in U$.  If we wish to refer to $L$ and $U$ directly, we may call $L$ the __lower set__ of $x$ and $U$ the __upper set__.

Given a point $x$ and an open $G$, we say that $x$ __belongs__ to $G$, written $x \in G$, if $(a,b) \subseteq G$ for some $a \lt x$ and $b \gt x$; that is, $G$ contains an interval from some element of the lower set to some element of the upper set of $x$.  We have $x \in \mathbb{R}$ since its lower and upper sets are inhabited.  If $x \in G$ and $x \in H$, with $(a,b) \subseteq G$ and $(c,d) \subseteq H$, then $\big(\max(a,c), \min(b,d)\big) \subseteq G \cap H$, so $x \in G \cap H$; note that this argument fails for infinitary intersections.  (The converse, that $x \in G$ and $x \in H$ if $x \in G \cap H$, is immediate.)  Dually, suppose that $x \in \bigcup_k G_k$, as shown by some zigzag (since $a \geq b$ is impossible when $a \lt x$ and $b \gt x$, by 3).  Applying condition (4) of the definition of real number to each zag, we have $a_{i+} \lt x$ or $b_i \gt x$, for each $i \lt n$.  Checking all $2^{n-1}$ possibilities, and knowing that $a_1 \lt x$ and $b_n \gt x$ in any case, we must have $a_i \lt x$ and $b_i \gt x$ for some $i$.  Then we have $x \in G_k$ for some $k$, whichever corresponds to the $i$th zig.  (The converse, that $x \in \bigcup_k G_k$ if $x \in G_k$ for some $k$, is immediate.)

Therefore, each point defines a [[completely prime filter]] on the frame of all opens, which is the definition of a [[point of a locale|point]] in general locale theory.  Conversely, given such a completely prime filter $P$, let $L$ be the set of all rational numbers $a$ such that the open interval $(a, \infty)$ (when interpreted as an open in the real line as defined above) is in $P$, and symmetrically let $U$ be the set of all $a$ such that $(-\infty, a) \in P$.

Given a point $x$ and an open $G$, we say that $x$ __co-belongs__ to $G$, written $x \notin G$, if we never have $a \lt x$, $b \gt x$, and $(a,b) \subseteq G$, which is precisely the [[negation]] of the property that $x \in G$.  We think of this condition as defining a *[[closed subspace|closed]]* set to which $x$ *does* belong.  Notice that $x \notin \bigcup_k G_k$ if and only if $x \notin G_k$ for each $k$, giving the desired behaviour for an arbitrary intersection of closed sets (which corresponds to union of open sets under [[de Morgan duality]]).  We also have that $x \notin \mathbb{R}$ always fails, and $x \notin G \cap H$ if $x \notin G$ or $x \notin H$.  To prove that $x \notin G$ or $x \notin H$ whenever $x \notin G \cap H$, however, we must use [[excluded middle]]; constructively, closed sets don\'t behave well under union.

A related question is whether we can reconstruct $G$ from the set of points which belong to it.  This should be equivalent to the [[fan theorem]], which is classically true and also accepted by [[Jan Brouwer|Brouwer]]\'s school of intuitionism, but refuted in the Russian school in which all real numbers are assumed to be [[computable number|computable]].  (I should check this.)  Arguably, the real lesson of these logical technicalities is that we should remember that opens, not points, are the fundamental concept in a locale.

## The Heine--Borel theorem

The classical [[Heine-Borel theorem|Heine--Borel theorem]], as a statement about sets of real numbers, may fail constructively; this is related to the comments above about the [[fan theorem]].  But the beauty of the localic approach is that Heine--Borel necessarily holds when interpreted as a statement about opens in the locale of real numbers.  To state the theorem, we must define what it means for a collection of opens to [[cover]] the [[unit interval]].  We will give an ad-hoc definition, but this may also be derived from the general theory of [[closed sublocale]]s which allows us to interpret the unit interval as a [[compact space|compact]] locale in its own right.

+-- {: .num_defn}
###### Definition

An __[[open cover]] of the unit interval__ is a collection $\mathcal{C}$ of opens in the real line such that $\mathbb{R}$ is the join of $(-\infty,0)$, $(1,\infty)$, and the members of $\mathcal{C}$.
=--

+-- {: .num_theorem #HeineBorel}
###### Theorem
(Heine--Borel)

Every open cover of the unit interval has a finite subcover.
=--

The proof is almost embarrassingly simple.  The key point is that the construction of joins in terms of zigzags involves only finite zigzags, even for an infinitary join.

+-- {: .proof}
###### Proof

Let $J$ be the join of $(-\infty,0)$, $(1,\infty)$, and the members of $\mathcal{C}$.  Since this equals $\mathbb{R}$, then in particular $(-2,3) \subseteq J$, and since $(-1,2) \Subset (-2,3)$, we get a corresponding zigzag $\zeta$ involving finitely many zigs using finitely many of the members of $\mathcal{C}$.  Let $\mathcal{D}$ be the collection of these members of $\mathcal{C}$, and let $K$ be the join of $(-\infty,0)$, $(1,\infty)$, and $\mathcal{D}$.  Now if $(a,b)$ is any pair of rational numbers, we construct a zigzag showing directly that $(a,b) \subseteq K$ as follows: the zig $(a,0) \subseteq (-\infty,0)$, the zag $0 \gt -1$, the zigzag $\zeta$ from $-1$ to $2$, the zag $2 \gt 1$, and the zig $(1,b) \subseteq (1,\infty)$.  This is always a valid zigzag, so $K = \mathbb{R}$.  Therefore, the finite collection $\mathcal{D}$ covers the unit interval.
=--

This proof generalizes to any closed interval $[a,b]$, for $a$ any upper real and $b$ any lower real.  But note that we do not say 'extended' here; we need to find some rational number (analogous to $-1$ in the proof above) smaller than $a$ and some rational number (analogous to $2$ above) larger than $b$.  So the Heine--Borel theorem applies only to *bounded* closed intervals.

Another generalization is [[Cousin's lemma|Cousin's Theorem]]:
+-- {: .num_theorem #Cousin}
###### Theorem
**(Cousin\'s)**

Given any open cover $\mathcal{C}$ of the unit interval, there is a partition $0 = a_0 \leq a_1 \leq \cdots \leq a_n = 1$ such that each subinterval $[a_i, a_{i+}]$ is covered by a single member $U$ of $\mathcal{C}$ (in that $(-\infty,a_i) \cup U \cup (a_{i+},\infty) = \mathbb{R}$).
=--

+-- {: .proof}
###### Proof

As in the previous proof, construct a zigzag $\zeta$ from $-1$ to $2$ with zigs from $\mathcal{C}$.  Using the Zigzag Lemma \ref{zigzaglemma}, replace this with an orderly zigzag.  Let $a_0$ be $0$, let $a_1$ be the smallest left endpoint of a zig from $\zeta$ satisfying $0 \lt a_1 \lt 1$, let $a_2$ be smallest left endpoint satisfying $a_1 \lt a_2 \lt 1$, and so on until none are left, and then finish with $a_n \coloneqq 1$.
=--

A corollary of this theorem, when the open cover is given by a function $\delta\colon [0,1] \to (0,\infty)$ (so that $\mathcal{C} \coloneqq \big\{\big(x - \delta(x), x + \delta(x)\big) \;|\; x \in [0,1]\big\}$), is used (classically) to prove the uniqueness (indeed, non-vacuity) of the [[Henstock–Kurzweil integral]], which could be used to define [[Lebesgue measure]] (see below).

## A result on measure

The set of [[computable real numbers]] has [[null set|measure zero]] in the sense that, given any [[positive number]] $\epsilon$, there\'s a collection $\mathcal{I}$ of open intervals (even with rational endpoints), such that every computable real number belongs to at least one member of $\mathcal{I}$, and the sum of the lengths of any finite list of distinct members of $\mathcal{I}$ is less than $\epsilon$.  (We can do this by enumerating [[Turing machines]].)  This is a problem for a theory of [[Lebesgue measure]] in [[Russian constructivism]], where every real number is computable.

From a localic perspective, however, while $\mathcal{I}$ might cover all the *points* of the real line, it cannot cover all of the pointless parts (which need not be empty).  Indeed, we have this result ruling out such shenanigans:
+-- {: .num_prop #not-null}
###### Proposition

Given any open interval $I = (a,b)$ with rational endpoints and any collection $\mathcal{J}$ of such intervals, if $I \subseteq \bigcup \mathcal{J}$, then for any length $L \lt b - a$, there must be a [[finite subset|finite]] (in the strictest sense) subcollection $\{K_1, \ldots, K_n\}$ of $\mathcal{J}$ such that
$$ L \leq \sum_{i=1}^n \len K_i .$$
=--
In other words, any cover of $I$ by intervals must have a total length (defined as the [[supremum]] of the lengths of finite subcollections) at least the length of $I$.

+-- {: .proof}
###### Proof

Let $(c,d) \Subset (a,b)$ with $d - c \geq L$ (say with $c \coloneqq (a + b - L)/2$ and $d \coloneqq (a + b + L)/2$).  Then there is a zigzag from $c$ to $d$ with zigs from $\mathcal{J}$; by the Zigzag Lemma \ref{zigzaglemma}, we may take this zigzag to be orderly, say of length $n$.  Each zig is from one of the intervals in $\mathcal{J}$, say $K_1, \ldots, K_n$.  If these intervals are all distinct (which we can decide, since the endpoints are rational), then we\'re done, since
$$ \sum_{i=1}^n \len K_i \geq \sum_{i=1}^n (b_i - a_i) = \sum_{i=1}^{n-1} (b_i - a_i) + (b_n - a_n) \geq \sum_{i=1}^{n-1} (a_{i+} - a_i) + (b_n - a_n) = b_n - a_1 = d - c \geq L ,$$
or $ 0 \geq L $ if $n = 0$.  If they\'re not distinct, then we instead use the smallest $a_i$ and largest $b_i$ that appears in a given interval, and the argument still works (possibly even with some overlap now).
=--

This result can be strengthened to intervals without necessarily rational endpoints, by using the rational intervals that they contain, but this is not the most general statement either, and I think that what we really need to do is to develop a theory of measures of open subspaces (or perhaps even more general subspaces) and state a result about that.

## As a classifying locale

The locale of real numbers is the [[classifying locale]] of the [[geometric theory]] of [[Dedekind real numbers]].

## Defining functions in the locale of real numbers

There are two general ways to define functions on the locale of real numbers, via the [[rational numbers]] and via the [[Dedekind real numbers]].

### Using the rational numbers

The first approach uses existing functions already defined on the [[rational numbers]] $\mathbb{Q}$. Suppose that one has a locally uniformly continuous function $f:\mathbb{Q} \to \mathbb{Q}$. Then one could extend $f$ to a continuous map $\widetilde{f}:\mathbb{R} \to \mathbb{R}$ on the locale of real numbers.

### Using the Dedekind real numbers

The other approach is via using existing functions defined on the [[Dedekind real numbers]] $\mathbb{R}$. In particular, given any well-defined continuous function $x \mapsto f(x)$ on the [[Dedekind real numbers]], and a [[geometric morphism]] $F:\mathrm{Set} \to \mathrm{Sh}(\mathbb{R}_D)$ from [[Set]] to the [[sheaf topos]] $\mathrm{Sh}(\mathbb{R}_D)$, if for all Dedekind real numbers $x \in \mathbb{R}_D$, $f(F^*(x)) = F^*(f(x))$, then the function $x \mapsto f(x)$ lifts to a function $x \mapsto \widetilde{f}(x)$ on the locale of real numbers.

This allows us to first define real-valued functions normally on the set of [[Dedekind real numbers]], and then lift them up to a real-valued function on the locale of real numbers, thus allowing us to define functions such as the [[exponential function]], the [[sine]], and the [[cosine]] which can\'t be directly defined using functions on the rational numbers.

### Arithmetic

Significantly, since the [[field]] operations on the discrete locale of [[rational numbers]] are all locally uniformly continuous on the $\mathbb{Q}$, and the field operations are well-defined and continuous on the Dedekind real numbers $\mathbb{R}_D$, the field operations in both cases lift up to a field structure on the locale of real numbers, allowing one to do basic [[arithmetic]] on $\mathbb{R}$.

## References

* [[Simon Henry]], *Localic Metric spaces and the localic Gelfand duality*, ([arXiv:1411.0898](https://arxiv.org/abs/1411.0898))

* Guillaume Raynaud, *Fibred Contextual Quantum Physics* ([etheses:1685](https://etheses.bham.ac.uk/id/eprint/1685/), [pdf](https://etheses.bham.ac.uk//id/eprint/1685/1/Raynaud14PhD.pdf))

The locale of real numbers as the [[classifying locale]] of the [[geometric theory]] of [[two-sided Dedekind cuts]]:

* [[Ingo Blechschmidt]], section 2.2 of: *Generalized spaces for constructive algebra*, ([arXiv:2012.13850](https://arxiv.org/abs/2012.13850))

An (impredicative) construction of the locale of real numbers can be found in section 5.3 of:

* Graham Manuell, *Uniform locales and their constructive aspects*, ([arXiv:2106.00678](https://arxiv.org/abs/2106.00678))

On lifting functions from the [[Dedekind real numbers]] to the locale of real numbers, see:

* [[Madeleine Birchfield]], [[Simon Henry]], *The field structure on the locale of real numbers* (2022) [[MO:q/434706](https://mathoverflow.net/q/434706/381)]

* [[Valery Isaev]], [[Simon Henry]], *Localic maps given by series* ([MathOverflow](https://mathoverflow.net/q/426682/))

[[!redirects the locale of real numbers]]
[[!redirects locale of real numbers]]
[[!redirects localic real numbers]]
[[!redirects localic real number]]
[[!redirects the localic real line]]
[[!redirects localic real line]]


localic completion

# Localic completion
* contents
{: toc}

## Idea

If $X$ is a [[metric space]], we can construct a version of its [[complete space|Cauchy completion]] as a [[locale]], by taking as basic opens formal symbols $B_\delta(x)$ representing [[open balls]], and imposing relations ensuring that the points are [[Cauchy filters]].  (We can equivalently regard $X$ as a [[Lawvere metric space]] and build a locale whose points are "Cauchy [[profunctors]]".)  This is the **localic completion** of $X$.

The construction can also be generalized in various ways.

## Definition

Consider the collection of "formal balls" $B(x,\delta)$, where $x\in X$ and $\delta\in\mathbb{Q}^+$ (technically this is just another notation for the ordered pair $(x,\delta)\in X\times \mathbb{Q}^+$).  We say:

* $B(x,\delta)\le B(y,\epsilon)$ if $d(x,y)+\delta\le\epsilon$
* $B(x,\delta)\lt B(y,\epsilon)$ if $d(x,y)+\delta\lt\epsilon$

The localic completion of $X$ is the [[locale]] presented by [[generators]] $B(x,\delta)$ and relations:

1. If $B(x,\delta)\le B(y,\epsilon)$ (as above) then $B(x,\delta)\le B(y,\epsilon)$ (in the locale)
2. $\top = \bigvee_{x\in X} B(x,\epsilon)$ for any $\epsilon$
3. $B(x,\delta)\cap B(y,\epsilon) = \bigvee \{ B(z,\eta) \mid B(z,\eta) \le B(x,\delta) \, \text{and} \, B(z,\eta) \le B(y,\epsilon) \}$
4. $B(x,\delta) = \bigvee \{ B(y,\epsilon) \mid B(y,\epsilon) \lt B(x,\delta) \}$

[Vickers](#VickersI) combines conditions (3) and (4) into:

* $B(x,\delta)\cap B(y,\epsilon) = \bigvee \{ B(z,\eta) \mid B(z,\eta) \lt B(x,\delta) \, \text{and} \, B(z,\eta) \lt B(y,\epsilon) \}$

On the other hand, condition (3) makes sense for any poset of generators replacing $\{B(x,\delta)\}$, and simply says that we regard the generators as a [[base]] rather than a [[subbase]].  Since the elements of a [[formal topology]] automatically form a base, [Palmgren](#Palmgren) (who works with formal topologies) can omit condition (3).

When interpreted as a [[geometric theory]], conditions (1)--(4) say that the localic completion of $X$ is the [[classifying locale]] for [[Cauchy filters]] in $X$ that are additionally *rounded* (condition (4)).

## Examples

* When $X$ is the rational numbers $\mathbb{Q}$, this yields the [[locale of real numbers]], which is in many respects better-behaved constructively than either its space of points (the [[Dedekind real numbers]]) or the completion of $\mathbb{Q}$ by Cauchy sequences (the [[Cauchy real numbers]]).  Thus, the localic completion provides a "good" notion of completion for arbitrary metric spaces which generalizes the good localic real numbers.

* By localically completing $\mathbb{Q}$ under a different metric, we obtain a "locale of [[p-adic numbers]]"; see [this MO question](http://mathoverflow.net/questions/214677/the-formal-p-adic-numbers).

## Generalizations

We can allow $X$ to be a [[pseudometric space]], a [[Lawvere metric space]], or even a "[[metric locale]]".  We can also allow $X$ to be a certain sort of [[uniform space]], with uniformity induced by a family of [[upper real number]] valued [[pseudometrics]].  See the references.

## References

* [[Steve Vickers]], "Localic Completion Of Generalized Metric Spaces I", [TAC](http://www.tac.mta.ca/tac/volumes/14/15/14-15abs.html)
{#VickersI}

* [[Steve Vickers]], "Localic Completion Of Generalized Metric Spaces II: Powerlocales", [pdf](http://www.cs.bham.ac.uk/~sjv/LocCompPower.pdf)

* [[Simon Henry]], *Localic Metric spaces and the localic Gelfand duality*, [arXiv](http://arxiv.org/abs/1411.0898)

* [[Erik Palmgren]], _Continuity on the real line and in formal spaces_. (2003). Revised version of U.U.D.M. Report 2003:32. [link](http://www2.math.uu.se/~palmgren/crealform-rvd2.pdf)
 {#Palmgren03}

* [[Erik Palmgren]], _From intuitionistic to point-free topology_ (2005) U.U.D.M. Report 2005:47. [link](http://www2.math.uu.se/~palmgren/homotopy.pdf)
 {#Palmgren05}

* [[Erik Palmgren]], _A constructive and functorial embedding of locally compact metric spaces into locales_. Topology and its Applications, 154
(2007), 1854--1880.
 {#Palmgren}

* [[Erik Palmgren]], _Resolution of the uniform lower bound problem in constructive analysis_, [link](http://onlinelibrary.wiley.com/doi/10.1002/malq.200710034/full)

* [[Bas Spitters]], _Locatedness and overt sublocales_, doi:10.1016/j.apal.2010.07.002 APAL

* [[Thierry Coquand]], [[Erik Palmgren]], [[Bas Spitters]], _Metric complements of overt closed sets_, Mathematical Logic Quarterly, 57(4), 373-378, 2011. [link](http://dx.doi.org/10.1002/malq.201010011)

* Tatsuji Kawai, _A point-free characterisation of Bishop locally compact metric spaces_, [link](http://www.logicandanalysis.com/index.php/jla/article/viewFile/239/120)

* Tatsuji Kawai. *Localic completion of uniform spaces*. Log. Methods Comput. Sci. 13 2017), no. 3, Paper No. 22, 39pp.  [arxiv](https://arxiv.org/abs/1703.02255)

[[!redirects localically complete]]


localic geometric morphism

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--

# Localic geometric morphisms
* table of contents
{: toc}

## Definition

A [[geometric morphism]] $f\colon E\to F$ between [[topoi]] is **localic** if every [[object]] of $E$ is a [[subquotient]] of an object in the [[inverse image]] of $f$: of the form $f^*(x)$.

## Examples

* Any geometric morphism between [[localic topoi]] is localic.

* Any [[geometric embedding]] is localic.

* Any [[étale geometric morphism]] is localic. From the [[internal language|point of view]] of a base topos $F$, an étale geometric morphism $F/A \to F$ looks like the unique geometric morphism $\Sh(A) \to Set$ attached to the topos of sheaves over the discrete locale $A$.

* If $g:C\to D$ is a [[faithful functor]] between [[small categories]], then the induced geometric morphism $Set^C \to Set^D$ is localic.

## Properties

+-- {: .un_prop}
###### Proposition

A [[Grothendieck topos]] is a [[localic topos]] if and only if its unique [[global section]] geometric morphism to [[Set]] is a localic geometric morphism.

=--

Thus, in general we regard a localic geometric morphism $E\to S$ as exhibiting $E$ as a "localic $S$-topos".

This is supported by the following fact.

+-- {: .un_prop}
###### Proposition

For any base $S$, the [[2-category]] of localic $S$-toposes (i.e. the full sub-2-category

  $$
    (Topos/S)_{loc} \subset Topos/S
  $$

of the [[over-category]] [[Topos]] over $S$ spanned by the localic morphisms into $S$) is equivalent to the 2-category of [[internalization|internal]] [[locales]] in $S$

$$
  Loc(S) \simeq (Topos/S)_{loc}
$$

Concretely, the internal locale in $\mathcal{E}$ defined by a localic geometric morphism $(f^* \dashv f_*) : \mathcal{F} \to \mathcal{E}$ is the formal dual to the [[direct image]] $f_*(\Omega_{\mathcal{F}})$ of the [[subobject classifier]] of $\mathcal{F}$, regarded as an internal poset (as described there) and $\mathcal{F}$ is equivalent to the internal [[category of sheaves]] over $f_*(\Omega_{\mathcal{F}})$.

=--

The last bit is lemma 1.2 in ([Johnstone](#Johnstone)).

+-- {: .un_prop}
###### Proposition

Localic geometric morphisms are the right class of a 2-categorical [[orthogonal factorization system]] on the 2-category [[Topos]] of topoi. The corresponding left class is the class of [[hyperconnected geometric morphism]]s.

([[(hyperconnected,localic) factorization system]])

=--

This is the main statement in ([Johnstone](#Johnstone)).

## References

Localic geometric morphisms are defined in def. 4.6.1 of

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_

The discussion there is based on

* [[Peter Johnstone]], _Factorization theorems for geometric morphisms_ Cahiers, 22, no1 (1981) ([numdam](http://www.numdam.org/item?id=CTGDC_1981__22_1_3_0))
{#Johnstone}

[[!redirects localic geometric morphism]]
[[!redirects localic geometric morphisms]]


localic group

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#### Group Theory
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#Contents#
* table of contents goes here
{:toc}

## Idea ##

A *localic group* is a [[group object]] [[internalization|internal]] to [[locales]].

As the notion of *[[locale]]* is a point-free version of that of *[[topological space]]*, a _localic group_ is much like a [[topological group]], but there are some differences.

The further generalization to [[groupoids]] is that of *[[localic groupoids]]*.

## Definition ##

A **localic group** is a [[group object]] [[internalization|in]] the [[category]] of [[locales]]. More explicitly, this means we have a locale $G$ with [[continuous maps]] $\epsilon:1 \to G$, $\iota:G \to G$, and $\mu:G \times G \to G$, such that $(\mathrm{id}_G, m) \circ m = (m, \mathrm{id}_G) \circ m$, $(\mathrm{id}_G, u_G^1 \circ \epsilon) \circ m = \mathrm{id}_G$, $(u_G^1 \circ \epsilon, \mathrm{id}_G) \circ m = \mathrm{id}_G$, $(\iota, \mathrm{id}_G) \circ m = u_G^1 \circ \epsilon$, and $(\mathrm{id}_G, \iota) \circ m = u_G^1 \circ \epsilon$, where $1$ is the terminal locale and $u_G^1:G \to 1$ is the unique map into $1$.

## Localic groups versus topological groups

Localic groups are similar to [[topological groups]], and many examples can be considered as either one.  For instance, the real numbers $\mathbb{R}$ under addition can be considered as either a topological group or a localic group.

Since the "space of points" [[functor]] $Loc \to Top$ is a [[right adjoint]], it preserves [[limits]] and hence [[group objects]], so every localic group has an underlying topological group.  However, this functor can discard information; for instance, [IKPR](#IKPR) constructs a nontrivial localic group with only one point.

Moreover, the "locale of opens" functor $Top\to Loc$ does not preserve [[product]]s, so not every topological group is a localic group---even if its underlying topological space is [[sober space|sober]] (hence is the space of points of some locale).  In particular, the locale $\mathbb{Q}$ of rational numbers (with [[topology]] induced from that of $\mathbb{R}$) is not a localic group under addition, because the locale product $\mathbb{Q}\times_l \mathbb{Q}$ is "bigger" than the topological-space product (and in particular is not spatial), and the addition map $\mathbb{Q}\times \mathbb{Q}\to \mathbb{Q}$ cannot be extended to the locale product.

But if $G$ is a [[locally compact space|locally compact]] topological group (such as $\mathbb{R}$), then the space product $G\times G$ does agree with the locale product (using the [[ultrafilter principle]] in the proof), and hence $G$ is also a localic group.

## Examples

* Another important source of localic groups is from [[progroups]]: cofiltered limits of discrete groups.
* The [[isotropy group of a topos]] is naturally regarded as a localic group inside the topos.

## Localic subgroups are closed

A remarkable fact about localic groups is the following (Corollary C5.3.2 of the [[Elephant]]; this also proves that $\mathbb{Q}$ cannot be a localic group):

+-- {: .num_theorem #LocalicSubgroupsAreClosed}
###### Theorem
Any [[overt space|overt]] localic subgroup of a localic group is weakly [[closed sublocale|closed]]. If the ambient localic group is in a [[Boolean topos]] then any localic subgroup is a [[closed subgroup]].
=--
+-- {: .proof}
###### Sketch of Proof
Details can be found in C5.3.1 of the [[Elephant]], in the more general case of localic [[groupoids]].  The basic idea of the proof is to use the fact that the intersection of any two [[dense sublocale]]s is again dense (a fact which very much fails for topological spaces).

If $H\rightarrowtail G$ is a localic subgroup, we construct its closure $\bar{H}$, which is also a localic subgroup in which $H$ is dense.  By [[pullback]], it follows that $H\times \bar{H} \to \bar{H} \times \bar{H}$ is [[fiberwise dense subspace|fiberwise dense]] over $\bar{H}$ via the second projection.  Applying the [[automorphism]] $(g,h) \mapsto (g,g^{-1}h)$ of $G\times G$, we conclude that $H\times \bar{H} \to \bar{H} \times \bar{H}$ is also [[fiber]]wise dense over $\bar{H}$ via the "composition" map.  Dually, $\bar{H}\times H \to \bar{H} \times \bar{H}$ is also fiberwise dense over $\bar{H}$ via the "composition" map, and thus (by the basic fact cited above), so is their intersection, which is $H\times H$.  Since $\bar{H}\times \bar{H}\to \bar{H}$ is an [[epimorphism]], so is $H\times H\to\bar{H}$.  But this map factors through $H\rightarrowtail \bar{H}$ (since $H$ is itself a subgroup of $G$), so that inclusion is also epic.  But it is also a [[regular monomorphism]], and hence an [[isomorphism]]; thus $H$ is closed.
=--

## See also

* [[topological group]]

* [[localic groupoid]]

## References

* The [[Elephant]], chapter C5.

* [[John Isbell]], [[Igor Křiž]], [[Aleš Pultr]], [[Jiři Rosický]], _Remarks on Localic Groups_, Lecture Notes in Mathematics 1348 (1988), pp. 154-172: [doi](https://doi.org/10.1007/bfb0081357)
{#IKPR}

* Graham Manuell, *Uniform locales and their constructive aspects*, ([arXiv:2106.00678](https://arxiv.org/abs/2106.00678))

An expository account of the closed subgroup theorem can be found in

* [[Gavin Wraith]], [_The Closed Subgroup Theorem_](http://www.wra1th.plus.com/gcw/math/closed.html)

[[!redirects localic groups]]


localic groupoid

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#### Higher geometry
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[[!include higher geometry - contents]]
=--
=--
=--

\tableofcontents

## Definition

A __localic groupoid__ is a [[internal category|internal]] [[groupoid]] in the category of locales. A special case is of [[localic group]]s.

Localic groupoids are important, among other reasons, because every [[Grothendieck topos]] can be presented as the [[topos]] of equivariant [[sheaves]] on some localic groupoid.  This fact is due to [[Andre Joyal|Joyal]] and Tierney. For more see [[classifying topos of a localic groupoid]].

## See also

* [[topological groupoid]]

* [[localic group]]

* [[localic stack]]

* [[localic infinity-groupoid]]

## References

* The [[Elephant]], chapter C5.

The Joyal–Tierney theorem appeared in

* [[Andre Joyal]], [[M. Tierney]], _An extension of the Galois theory of Grothendieck_ Mem. Amer. Math. Soc. no 309 (1984)

An expository account of the Joyal–Tierney theorem:

* Graham Manuell, Joshua L. Wrigley, _The representing localic groupoid for a geometric theory_, [arXiv:2305.15209v1](https://arxiv.org/abs/2305.15209v1).

[[!redirects localic groupoids]]


localic homotopy theory

#Contents#
* table of contents
{:toc}

## Idea

[[homotopy theory]] for [[locales]] (as opposed to, say [[topological homotopy theory]] or [[simplicial homotopy theory]]).

## References

The [[fundamental group]] for locales is considered in

* W. He, _Homotopy theory for locales_, Acta Math. Sinica 46 (2003), no. 5

* J. F. Kennison, _What is the fundamental group?_, J. Pure Appl. Algebra 59(1989)

See also

* {#Palmgren05} [[Erik Palmgren]], _From Intuitionistic to Point-Free Topology: On the Foundation of Homotopy Theory_, Logicism, Intuitionism, and Formalism Volume 341 of the series Synthese Library pp 237-253, 2005 ([pdf](http://www2.math.uu.se/~palmgren/homotopy_rev2.pdf))

[[!redirects homotopy theory for locales]]


localic stack

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=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A **localic stack** is a [[geometric stack]] on the [[site]] [[Loc]]: a [[localic groupoid]] regarded as presenting an object in the [[(2,1)-sheaf]] [[(2,1)-topos]] $Sh_{(2,1)}(Loc)$.

## See also

* **localic stack**, [[localic groupoid]]

* [[topological stack]]

[[!redirects localic stacks]]


localic topos

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=--

# Contents
* automatic table of contents goes here
{:toc}

## Definition

In intrinsic terms, a [[topos]] is _localic_ if it is generated under [[colimit]]s by the [[subobject]]s of its [[terminal object]] $1$.

In equivalent but extrinsic terms, a category is a localic topos if it is equivalent to the [[category of sheaves]] on a [[locale]] with respect to the [[coverage|topology]] of jointly epimorphic families (accordingly, every localic topos is a [[Grothendieck topos]]).

The [[frame]] of [[open set|opens]] specifying the locale may indeed be taken as the [[poset]] of [[subobject]]s of $1$ (i.e., internal [[truth value|truth values]]). From the perspective of logic, localic toposes are those categories which are equivalent to the category of [[partial equivalence relation]]s of the [[tripos]] given by a [[complete Heyting algebra]] (as before, the complete Heyting algebra may be taken as the poset of internal truth values).

## Properties

* A Grothendieck topos $E$ is a localic topos if and only if its unique [[global section]] [[geometric morphism]] to [[Set]] is a [[localic geometric morphism]].

  Thus, in general we regard a localic geometric morphism $E \to S$ as exhibiting E as a "localic S-topos".

* Moreover, just as localic topoi can be identified with locales, for any base topos $S$ the [[2-category]] of localic $S$-topoi is equivalent to the 2-category [[Loc]]$(S)$ of [[internalization|internal]] [[locale]]s in $S$.

  $$
    LocTopos(S) \simeq (Topos/S)_{loc} \simeq Loc(S)
    \,.
  $$

  Here $LocTopos(S)$ is the [[2-category]] whose

  * objects are localic toposes over $S$;

  * morphisms are [[geometric morphism]]s, i.e. [[adjunctions]] in which the left adjoint preserves [[finite limits]], considered as pointing in the direction of their right adjoint; and

  * 2-morphisms are [[mate]]-pairs of [[natural transformation]]s.

  Then the 2-category $LocTopos$ is [[equivalence of categories|equivalent]] to the 2-category $Loc$ of [[locales]] (see C1.4.5 in the [[Elephant]]).

  The 2-category $Loc$ is actually a [[(1,2)-category]]; its [[2-morphism]] are the pointwise ordering of [[frame]] homomorphisms.  Thus this equivalence implies that $LocTopos$ is also a (1,2)-category, and moreover that it is [[locally small category|locally essentially small]], in the sense that its hom-categories are essentially small.  (The 2-category $Topos$ of all toposes is not locally essentially small.)  Assuming sufficient separation axioms, the hom-posets of $Loc$, and hence $LocTopos$, become discrete.

## Examples

* Obviously, every [[Grothendieck topos]] that is a [[category of sheaves]] on (the [[category of open subsets]] of) a [[topological space]] is localic.

* Every [[sheaf topos]] over a [[posite]] is localic. (See there for details.)

Many familiar toposes $E$, even when they are not localic, can be covered by a localic [[over topos|slice]] $E/X$ ("covered" means the unique map $X \to 1$ is an epi). For example, if $G$ is a group, then $E = Set^G$ is not itself localic, but it has a localic slice $Set^G/G \simeq Set$ that covers it. Such a topos is called an [[étendue]] (cf. Lawvere's 1975 monograph _Variable Sets Etendu and Variable Structure in Topoi_).[^fine]

[^fine]: The 'etendu' in the title of Lawvere's monograph might not be a misspelled noun, but an adjective as part of a back translation of a (hypothetical) French expression 'ensembles étendus'. See this [nForum thread](http://nforum.mathforge.org/discussion/5805/etendue/) for some discussion and speculation on this point.

A significant result due to Joyal and Tierney is that for any Grothendieck topos $E$, there exists an open surjection $F \to E$ where $F$ is localic. This fact is reproduced in Mac Lane and Moerdijk's text _Sheaves in Geometry and Logic_ (section IX.9), where the localic cover taken is called the _Diaconescu cover_ of $E$.

Then, using methods of descent theory, Joyal and Tierney deduce that every Grothendieck topos is equivalent to the category $B G$ of continuous discrete representations of a [[localic groupoid]] $G$. (Their result is relativized so as to hold internally over any Grothendieck topos $S$ as base.) This should be regarded as a major extrapolation of Grothendieck's [[Galois theory]] (as in [[SGA]] 1), where it is shown that the [[etale topos]] of a field $k$ is equivalent to the category of continuous discrete representations of the fundamental [[pro-group]] $Gal(\bar{k}/k)$, where $\bar{k}$ denotes the separable closure of $k$. It was a watershed event for the penetration of localic methods in topos theory.

## The connection with propositional logic

Recall that a [[geometric theory]] $\mathbb{P}$ over a signature with no sort symbols is called [[propositional theory|propositional]]. Such a signature can contain at most 0-ary relation symbols but lacks variables and, accordingly, $\mathbb{P}$ admits only sequents over the empty context consisting of nested conjunctions or (infinitary) disjunctions of such relation symbols - in other words, logic boils down to [[propositional logic]].

Propositional theories have the peculiarity that their classifying toposes always exist regardless of the availability of a [[natural numbers object]] in the base topos (cf. for details&references see at [[classifying topos]]).

+-- {: .num_prop #localicclasifiespropositional}
###### Proposition

Localic toposes correspond exactly to [[classifying topos|classifying toposes]] of propositional theories.
=--

This appears in Johnstone (2002, D3.1.14, p.897f.).

Given a locale $L$, the **theory of completely prime filters** $\mathbb{P}_L$ has a 0-ary relation symbol $F_x$ for each $x\in L$, thought to express the proposition that $x$ is contained in the [[filter]] $F$, and the following sequents:

* $\top\vdash F_1$ ,

* $F_x\wedge F_y\vdash F_{x\wedge y}$ for all pairs $x,y\in L$,

* $F_{\big (\bigvee_{i\in I} y_i\big )}\vdash \bigvee_{i\in I} F_{y_i}$ .

In $Set$ models of $\mathbb{P}_L$ correspond precisely to completely prime filters i.e. the multiplicatively closed subsets of $L$ containing $1$ that are inaccessible by infinite joins ($(\bigvee y_i)\in F$ implies $y_i\in F$ for some $i$). Note in particular, that the properness $0\notin F$ of a completely prime filter $F$ is implicit in the third axiom schema with $I=\emptyset$.

The relation between $L$ and $\mathbb{P}_L$ is that $Sh(L)\simeq Set[\mathbb{P}_L]$.

Conversely, given a propositional theory $\mathbb{P}$, the [[Lindenbaum-Tarski algebra]] of classes of provably equivalent formulas over $\mathbb{P}$ together with the entailment order yields a locale $L_{\mathbb{P}}$ such that $Sh(L_\mathbb{P})$ classifies $\mathbb{P}$.

### The initial and terminal toposes as classifying toposes

For illustration let us consider the _empty theory_ $\mathbb{T}_1$ over the empty signature i.e $\mathbb{T}_1$ has no axioms. This is certainly propositional, its deductive closure consists of all tautologies using $\bot,\top,\wedge,\bigvee$. The Lindenbaum-Tarski algebra is simply $\mathbf{2}\simeq\{[\bot]\leq[\top]\}$ which corresponds to the frame of open sets of the one-point space whence $Set[\mathbb{T}_1]=Sh(\mathbf{2})=Set$. Similarly, $\mathbb{P}_{\mathbf{1}}$, the theory of completely prime filters of the one-point locale $\mathbf{1}$, is classified by $Sh(\emptyset)$ i.e. sheaves on the empty space.

Whereas $\mathbb{P}_{\mathbf{2}}$ has up to isomorphism exactly one model in every topos $\mathcal{E}$ namely the one interpreting $F_1$ as $id_{1_\mathcal{E}}$ and $F_0$ as $0_{\mathcal{E}}\hookrightarrow 1_{\mathcal{E}}$, $\mathbb{P}_{\mathbf{1}}$ has up to isomorphism exactly one model in up to isomorphism exactly one topos i.e. its model is the [[zero object]] of the (necessarily) degenerate topos.

Note that since $Set$ is the terminal Grothendieck topos, the empty theory $\mathbb{T}_1$ over the empty signature is [[Morita equivalence|Morita equivalent]] to any other geometric theory $\mathbb{T}$ over any signature whatsoever provided $\mathbb{T}$ has up to isomorphism exactly one model in every Grothendieck topos $\mathcal{E}$. E.g. a slight modification of the inconsistent theory $\{\top\vdash\bot\}$ over the empty signature, namely adding a sort symbol $O$ and "contextualising" $\mathbb{T}_1'=\{\top\vdash_{x:O}\bot\}$ has models exactly the initial objects but these are unique (and every topos has one) whence $Set[\mathbb{T}_1']\simeq Set$.

Note also the difference in behavior between the inconsistent and the empty theory with respect to enlargening the signature: adding a sort symbol does not change the categories of models of the inconsistent theory up to isomorphism whereas $\mathbb{T}_1'$ has entirely different categories of models from the _empty theory_ over the signature containing a sort symbol $O$ since the latter is the [[theory of objects]] $\mathbb{O}$; but $\mathbb{T}_1'$ is a quotient of $\mathbb{O}$ whence we can think of $\mathbb{T}_1'$ as an axiomatisation of $Set$ as subtopos of $Set[\mathbb{O}]$ (see at [[level]] for further information on the duality between quotient theories and [[subtopos|subtoposes]] of the classifying topos). To sum up "paraphrasing" Tolstoy: there is only one way to be inconsistent but an infinite number of ways of being empty!

## Generalizations

In the context of [[(∞,1)-topos]] [[Higher Topos Theory|theory]] there is a notion of [[n-localic (∞,1)-topos]].

Notice that a [[locale]] is itself a (Grothendieck) [[(0,1)-topos]]. Hence a localic topos is a 1-[[topos]] that behaves essentially like a [[(0,1)-topos]]. In the wider context this would be called a [[n-localic (infinity,1)-topos|1-localic (1,1)-topos]].

## Related concepts

* [[locale]]

* [[localic geometric morphism]]

* [[spatial topos]], [[category of sheaves on a topological space]]

* [[étendue]]

* [[n-localic (∞,1)-topos]]

* [[n-localic 2-topos]]

## References

* {#MacLaneMoerdijk} [[Saunders Mac Lane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_ , Springer Heidelberg 1994. (chapter IX, p.472ff)

* [[Peter Johnstone]], _[[Sketches of an Elephant]] 2 vols._ , Oxford UP 2002. (In general around C.1.4.5 p.515f, cp filters at D1.1.7 p.816f, as classifying topos B4.2.12. p.431f, D1.1.14 p.897f.)

[[!redirects Localic topos]]
[[!redirects localic toposes]]
[[!redirects localic topoi]]


#Contents#
* table of contents
{:toc}

## Idea

The localisation of a commutative ring $A$ away from an element $a\in A$ is a universal means to 'invert $a$'.  The resulting ring captures information that is relevant 'away from $a$', i.e. 'locally on the complement of $a$'.

Algebraically, it might make more sense to call this localization *at* $a$, but in [[algebraic geometry]] it really does correspond to "local behavior on the complement of (the zero-set of) $a$", and the "away from" terminology is traditional.

## Definition

Let $A$ be a commutative ring, and let $a$ be an element of $A$.

+-- {: .num_defn}
###### Definition

The _localisation of $A$ away from $a$_, usually denoted $A_{a}$ or $A[1/a]$, is the commutative ring $A[x] / (a x - 1)$.

=--

Here (the equivalence class of) $x$ is to be thought of as $a^{-1}$.

Equivalently, we can define $A_{a}$ to be the localisation of $A$ with respect to the multiplicative system $S = { a, a^{2}, a^{3}, \ldots }$. This general notion of localisation is discussed at [[localisation of a commutative ring]].

+-- {: .num_remark}
###### Example

If $A$ is the ring of integers $\mathbb{Z}$, and $a$ is $10$, then $A_{a}$ is the ring of [[decimal rational numbers]] $\mathbb{Z}[1/10]$.

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+-- {: .num_remark}
###### Example

If $A$ is the polynomial ring $\mathbb{Z}[x]$, and $a$ is $x$, then $A_{a}$ is the ring of [[Laurent polynomials|Laurent polynomial]] $\mathbb{Z}[x, x^{-1}]$.

=--

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localizable Boolean algebra

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### Context
#### Measure and probability theory
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#Contents#
* table of contents
{:toc}

## Idea

A [[Boolean algebra]] is _localizable_ if it admits “sufficiently many” [[measures]].

## Definition

A __localizable Boolean algebra__ is a [[complete Boolean algebra]] $A$
such that $1\in A$ equals the [[supremum]] of all $a\in A$
such that the [[Boolean algebra]] $a A$ admits a faithful [[continuous valuation]] $\nu\colon A\to[0,1]$.
Here a [[valuation]] $\nu\colon A\to[0,\infty]$ is faithful if $\nu(a)=0$ implies $a=0$.

A __morphism of localizable Boolean algebras__ is a complete (i.e., suprema-preserving) [[homomorphism]] of [[Boolean algebras]].

## Properties

The [[category]] of localizable Boolean algebras admits all small [[limits]] and small [[colimits]].

It is [[equivalence of categories|equivalent]] to the category of [[commutative von Neumann algebras]].

The equivalence sends a [[commutative von Neumann algebra]]
to its localizable Boolean algebra of projections.
It sends a localizable Boolean algebra $A$
to the [[complexification]] of the [[completion]] of the free real algebra on $A$,
given by the left adjoint to the functor that takes idempotents.

## Related concepts

* [[commutative von Neumann algebra]]

* [[hyperstonean space]]

* [[enhanced measurable space]]

* [[duality between geometry and algebra]]

## References

* [[Dmitri Pavlov]], *Gelfand-type duality for commutative von Neumann algebras*.
Journal of Pure and Applied Algebra 226:4 (2022), 106884. [doi:10.1016/j.jpaa.2021.106884](https://doi.org/10.1016/j.jpaa.2021.106884), [arXiv:2005.05284](https://arxiv.org/abs/2005.05284).

[[!localizable Boolean algebras]]


localizable measure

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#### Measure and probability theory
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# Localizable measures
* table of contents
{: toc}

## Idea

Many of the important theorems of [[measure theory]] fail to hold in full generality.  (See below under Theorems for which theorems we\'re talking about.)  Often these theorems are stated for $\sigma$-[[sigma-finite measures|finite measures]], but they do hold a bit more generally than that.  In fact, they hold for _localizable_ measures, and this fact *characterizes* the localizable measures.

## Definition

The following definition is in elementary terms; but we will see that there are many other characterizations.

Let $\mu$ be a [[positive measure]] on an [[abstract set]] $X$.  (That is, certain [[subsets]] of $X$, forming a $\sigma$-[[sigma-algebra|algebra]], are [[measurable set|measurable]] by $\mu$, and $\mu$ maps these sets to the space $[0,\infty]$ of [[lower real numbers]] in a [[monotone function|monotone]] and [[countably additive measure|countably additive]] way.)

Given two measurable subsets $E$ and $F$, $E$ __essentially contains__ $F$ if the set
$$ \{ x\colon X \;|\; x \in F \;\Rightarrow\; x \in E \} $$
is [[full set|full]]; or equivalently (using [[excluded middle]]) if $F \setminus E$ is [[null set|null]].  (This is a [[preorder]] on the measurable sets.)

Then $\mu$ is __localizable__ if the following conditions both apply:

*  Semifiniteness:  Every measurable set $E$ with positive measure essentially contains a measurable set with finite positive measure.  (We may strengthen 'essentially contains' to 'contains' in this clause.)

*  Essential suprema:  Given any collection $\mathcal{C}$ of measurable sets, there is a measurable set $E$ such that:

   *  $E$ essentially contains each element of $\mathcal{C}$; and
   *  Given any measurable set $F$ that essentially contains every element of $\mathcal{C}$, $F$ essentially contains $E$.

   (This set $E$ is essentially unique, in that it essentially contains and is essentially contained in any other set with the same property; we call $E$ [[the]] __essential union__ $\ess \bigcup \mathcal{C}$ of $\mathcal{C}$.)

We generalize to [[measures]] taking place in some space other than $[0,\infty]$: a $\mu$ is __localizable__ if ${|\mu|}$ is, as long as $\mu$ has an [[absolute value]] (or [[total variation]]) ${|\mu|}$ that takes values in $[0,\infty]$.

Of course, a set equipped with a localizable measure is a __localizable measure space__.

## Examples

Every $\sigma$-[[sigma-finite measure|finite]] measure is localizable.  (Since this includes so many examples, the theorems below are often stated for $\sigma$-finite measures.)

Any [[counting measure]] is localizable (but $\sigma$-finite only on a [[countable set]]).

## Properties

The following [[sheaf]] condition is fundamental:  Given any [[cover]] $\mathcal{U}$ of $X$ (a localizable measure space) by [[measurable sets]] and a $\mathcal{U}$-indexed [[family]] $f$ of [[partial function|partial]] [[measurable functions]] (with $\dom f_A = A$), if always $f_A = f_B$ almost everywhere on $A \cap B$ (meaning that there is a [[full subset]] $E$ of $X$ such that $f_A = f_B$ on $A \cap B \cap E$), then there exists a (necessarily unique up to [[almost equality]]) measurable (and total) function $\ess \bigcup f$ such that always $\ess \bigcup f = f_A$ almost everywhere on $A$.  (This is Theorem 213N in [Fremlin](#Fremlin).  I don\'t know if it characterizes localizable measures.)

Slightly more generally, start with *any* family of partial measurable functions on $X$ and treat it as a cover of (any representative of) the essential union of its domains.

... other results such as the [[Radon–Nikodym theorem]] ...

These ideas are elaborated at [[measurable space]] [[Boolean toposes]] and [[Boolean valued model]]s and [[forcing]].

## Other localizable structures

Whether a semifinite measure $\mu$ is localizable depends only on which measurable sets are [[full set|full]] (or [[null set|null]]).  Sometimes it is convenient to equip a [[measurable space]] with a $\delta$-[[delta-filter|filter]] of full sets (or a $\sigma$-[[sigma-ideal|ideal]] of null sets), without equipping it with the measure of any other set.  A __localizable measurable space__ is a measurable space so equipped, such that every collection of measurable sets has an essential union.  (It is a theorem, I believe, that every localizable measurable space is capable of supporting a semifinite, hence localizable, measure with the same full/null sets.)

Knowing the full/null sets is also sufficient to define [[almost equality]] of [[measurable functions]] between measurable spaces; this gives us a [[category]] $Loc Meas$ of localizable measurable spaces.  This category is [[dual category|dual]] to the category of [[commutative ring|commutative]] [[von Neumann algebras]]; hence an arbitrary von Neumann algebra may be viewed as a _noncommutative_ localizable measurable space (in the sense of [[noncommutative geometry]]).

If $\mathcal{M}$ is the $\sigma$-[[sigma-algebra|algebra]] of measurable sets and $\mathcal{F}$ is the $\delta$-filter of full sets (or $\mathcal{N}$ is the $\sigma$-ideal of null sets), then we may form the [[quotient ring|quotient]] [[boolean algebra]] $\mathcal{M}/\mathcal{F}$ (or $\mathcal{M}/\mathcal{N}$) by identifying each full set with the [[improper subset|entire space]] (or identifying each null set with the [[empty subset|empty set]]).  Then a measurable space is localizable iff this quotient is [[complete boolean algebra|complete]]; this is the real point of the notion of localizability.

This boolean algebra is all the structure needed to specify a measure on the original space (at least one that is [[absolutely continuous measure|absolutely continuous]] in that it has at least the requisite full/null sets).  We can now abstract away the [[underlying set]] and simply look at the complete boolean algebra.  However, it is *not* true that every complete boolean algebra is capable of arising in this way from a measurable space.  Those which do so arise may be called __[[measurable locales]]__, since a complete boolean algebra is a [[frame]] and measurable functions (up to almost equality) between the measurable spaces correspond to [[continuous maps]] between the frames, thought of as [[locales]].  (That is, $Loc Meas$ is [[equivalence of categories|equivalent]] to $Meas Loc$, a sort of pun.)

Without requiring localizability, the boolean algebra $\mathcal{M}/\mathcal{F}$ (or $\mathcal{M}/\mathcal{N}$) is called a [[measurable algebra]]; equipped with a measure, we have a [[measure algebra]].  Thus a measurable locale is the same thing as a __localizable measurable algebra__, and one may also speak of __localizable measure algebras__ (with the category of these equivalent to the category localizable measure spaces, so long as morphisms in the latter are again only defined up to almost equality).

## In weak foundations

The [[axiom of choice]] is indispensable for the development above, as stated.  (The reason is that one constantly makes choices among essentially equivalent measurable sets, or among almost equal measurable functions.)  However, the [[category]] of localizable measurable spaces (and [[measurable functions]] up to [[almost equality]]) is (assuming choice) [[equivalence of categories|equivalent to]] the category of [[measurable locales]], which may prove to be more tractable without choice, even in [[constructive mathematics]].  That said, nobody has worked out a constructive development of this yet.  (In particular, identifying which boolean algebras we want is more difficult; perhaps surprisingly, they are still boolean, but they are no longer necessarily complete!)

## References

This giant treatise on all of [[measure theory]] is free (in both senses) online:

*  D.H. Fremlin, _Measure Theory_, [web](https://www1.essex.ac.uk/maths/people/fremlin/mt.htm)
   {#Fremlin}

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localization

## Idea

In mathematics, localization refers to the idea of formally adding inverses to some elements in an algebraic structure.

In particular, the structure itself can be a category, in which case localization adds formal inverses to morphisms.  This construction is known as a __[[category of fractions]]__ and also ambiguously as a __[[localization of a category]]__.  An important special case of this idea is the notion of a [[reflective localization]] / [[left Bousfield localization]], in which case the localization functor has a fully faithful right adjoint.  In some books (e.g., [[Higher Topos Theory]]) the term “localization” (when applied to categories or (∞,1)-categories) refers specifically to [[reflective localizations]].

## For categories and (∞,1)-categories

### General

* [[category of fractions]], [[homotopy category]]

* [[quasicategory of fractions]], [[simplicial localization]], [[hammock localization]], [[simplicial homotopy category]]

### Reflective

* [[reflective localization]], [[left Bousfield localization]]

### Coreflective

* [[coreflective localization]], [[right Bousfield localization]]

## For commutative rings

* [[localization of a commutative ring]]

* [[field of fractions]], [[ring of fractions]]

## For noncommutative rings

* [[localization of a ring]], [[noncommutative localization]]

* [[Cohn localization]], [[Ore localization]]

## For modules

* [[localization of an abelian group]]

* [[localization of a module]]

## For monoids

* [[localization of a monoid]]

* [[group completion]]

## For categories of spaces

* [[localization of a space]]

## For categories of spectra

* [[Bousfield localization of spectra]]


localization and completion -- table

**[[Bousfield localization of spectra|localization]] [[localization of a space|of spaces/spectra]] at $E$**

|  $E$  |  $E$-acyclic  |  $E$-local |
|-------|------------|-----------|
| $S \mathbb{Z}_{(p)}$ |   |  $p$-local |
| $S \mathbb{F}_p$ |  $\not\{p\}$-local |  $p$-complete |