Contents

Idea and motivation

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 $R$). 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 $\{f\colon N \twoheadrightarrow R \;|\; f(n) = x\}$.

Every topological space can be regarded as a locale (with some lost information if the space is not sober). The locales arising this way are the 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.

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 internal to an arbitrary topos. For example, constructively the topological space $[0,1]$ need not be 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 or real numbers is generally a better-behaved object than the topological space of real numbers.

Definition

Recall that a frame $A$ is a poset with all joins and all finite meets which satisfies the infinite distributive law:

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 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 of the category of frames

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 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 must explicitly add into this definition that $f$ has a left adjoint; compare the definition of geometric morphism one level up.)

This category is naturally enhanced to a 2-category:

(See for instance Johnstone, C1.4, p. 514.)

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

Properties

Categories of internal locales

See localic geometric morphism for more.

This appears as Johnstone, theorem C1.6.3.

This appears as Johnstone, scholium C1.6.4.

Relation to topological spaces

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 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 $\to$ 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 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 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

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

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

In fact, this is an idempotent adjunction:

This appears for instance as (MacLaneMoerdijk, theorem IX.3 1) or as (Johnston, lemma C.1.2.2).

Therefore the adjunction restricts to an equivalence between the fixed subcategories on either side.

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

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

Relation to toposes – localic reflection

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

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 if its set of morphisms has size $\kappa$. For example, $Set$ is moderate.

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

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

See for instance (MacLaneMoerdijk, section 5).

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

This appears for instance as MacLaneMoerdijk, section IX.5 prop 2.

This appears for instance as MacLaneMoerdijk, section IX.5 prop 3.

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

In summary this means that locales form a reflective subcategory of Topos:

In fact this is even a genuine full sub-2-category:

This appears as (Johnstone, prop. C1.4.5).

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

which is

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

Related concepts

• The notion of locale may be identified with that of a 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 internal to locales or an internal groupoid in locales is a localic group or localic groupoid, respectively.

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

• topological space, topological locale

• ionad

• topos

Examples

• the locale of real numbers

• Cantor space

References

An introduction to and survey of the use of locales instead of topological spaces is

• Peter Johnstone, The point of pointless topology , Bull. Amer. Math. Soc. (N.S.) Volume 8, Number 1 (1983), 41-53. (Bulletin AMS)

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

• Peter Johnstone, Stone Spaces

that develops, among other things, much of standard topology entirely with the notion of locale used in place of that of topological spaces. See Stone Spaces for details.

Another dedicated textbook is

• Jorge Picado, Aleš Pultr, Frames and Locales. Topology without points, Birkhäuser (2012)

Plenty of material is also in part C (volume 2) of

• Peter Johnstone, Sketches of an elephant: a topos theory compendium.

Locales are also discussed in section IX.1 of

• Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic

Lex totality is the subject of an article of 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

• André Joyal and Myles Tierney, An Extension of the Galois Theory of Grothendieck, Memoirs of the American Mathematical Society 51 (1984).