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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 sets” but there may or may not be enough points to distinguish between open sets. An “open set” 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 the natural numbers $N$ to the real numbers $R$. It has no points, since there are no such surjections, but it contains many nontrivial “open sets;” these open sets are generated by a family parametrised by $n: N$ and $x: R$ that may be described as $\{f:N\to R | f$ is a surjection and $f(n) = x\}$.

Every topological space can be regarded as a locale (with a little bit of lost information if the space is not sober). Conversely, every locale induces a topology on its set of points, but sometimes a great deal of information is lost; there are many different locales whose space of points is empty. We say that a locale is spatial if it can be recovered from its space of points.

One motivation for locales is that since they take the notion of “open set” 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 sets, 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 constructive mathematics or 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 always spatial. When it fails to be spatial, because there are “not enough real numbers,” the locale is generally a better-behaved object than the topological space of real numbers.

Definition

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

A frame homomorphism $\phi: A\to B$ is a function which preserves finite meets and arbitrary joins.

By the adjoint functor theorem for posets, a frame also has all meets, but a frame homomorphism need not preserve them. Again by the AFT, a frame is automatically a Heyting algebra, but again a frame homomorphism need not preserve the Heyting implication.

The category 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 sets in $X$”, and a continuous map $f:X\to Y$ of locales is a frame homomorphism $f^*:O(Y)\to O(X)$. 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.

Relation to topological spaces

Every topological space $X$ has a frame of open sets $O(X)$, and therefore gives rise to a locale $X_l$. For every continuous function $f:X\to Y$ between spaces, the inverse image map $f^{-1}:O(Y)\to O(X)$ is a frame homomorphism, so $f$ induces a continuous map $f_l:X_l\to Y_l$ of locales. Thus we have a functor $(-)_l:Top \to Loc$.

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 $1$, 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 $f:X\to Y$ of locales induces a continuous map $f_p:X_p\to Y_p$ of spaces, so we have another functor $(-)_P:Loc\to Top$.

It is not hard to check that $(-)_l$ is left adjoint to $(-)_p$. In fact, this is an idempotent adjunction, and therefore it restricts to an equivalence between the fixed subcategories on either side. A space with $X\cong X_{lp}$ is called sober, while a locale with $X\cong X_{pl}$ is called spatial.

References

P. T. Johnstone, Sketches of an elephant: a topos theory compendium. Part C (volume 2).

P. T. Johnstone, Stone spacesStone spaces