Questions tagged [locales]

Locales are an algebrization of point-set topological spaces that mimic the properties of open set lattices. Classically, a topological space $X$ consists of a set of points together with a topology $\mathcal{O}(X),$ a collection of subsets called open sets that is closed under finite intersections and arbitrary unions. The set of open sets $\mathcal{O}(C)$ forms a sup-lattice and meet semilattice under the inclusion order. On the other hand, in lattice theory, a poset with finite meets and arbitrary joins which satisfies the infinite distributive law $$x\land\left(\bigvee_{i\in I}y_i\right)=\bigvee_{i\in I}(x\land y_i)$$ is known as a frame. This concept resembles most properties of topological spaces without referring to the underlying set of points.

Next, one would expect to lift the above construction into a functor between the category of topological spaces and the category of frames, where frame homomorphisms are monotone maps preserving finite meets and all joins. Given two continuous map $X\xrightarrow{f}Y\xrightarrow{g}Z$ between three topological spaces, we clearly have frame homomorphisms $\mathcal{O}(Z)\xrightarrow{\mathcal{O}(g)}\mathcal{O}(Y)\xrightarrow{\mathcal{O}(f)}\mathcal{O}(X)$ that sends an open set $U\in\mathcal{O}(Y)\mapsto f^{-1}(U)$ such that

1. $\mathcal{O}(gf)=\mathcal{O}(f)\mathcal{O}(g)$, and
2. $\mathcal{O}(\operatorname{Id}_X)=\operatorname{Id}_{\mathcal{O}(X)}$.

Hence the functor that we are expecting exists, but is contravariant. So, the correct analogue of topological spaces together with continuous maps in this lattice theoretic setting is not the category of frames, rather its opposite category, which are the category of locales. Note that there is no difference between the objects of both of these categories, but a localic continuous map is just a formally reversed frame homomorphism.

To make this association more meaningful, there is also a right adjoint functor to $\mathcal{O}$ called spectrum that produces a topological space for any given locale, which restricts to an equivalence of categories between the full subcategory of locales with enough points (called spatial), and that of sober topological spaces.

While spatial locales capture the essence of most of the interesting point set topology, non-spatial locales are particularly good for doing measure theory. One of the main advantages of the point-free approach to these subjects is the fact that some important classical theorems that depend on axiom of choice become choice-free in this new language. Other advantages include the much better behaviour of (topological) properties such as paracompactness, the fact that every locale has a smallest dense sublocale, subgroups of localic groups are always closed, and so on.