# Stone duality

Stone duality This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please help to improve this article by introducing more precise citations. (March 2018) (Learn how and when to remove this template message) In mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. Today, these dualities are usually collected under the label Stone duality, since they form a natural generalization of Stone's representation theorem for Boolean algebras. These concepts are named in honor of Marshall Stone. Stone-type dualities also provide the foundation for pointless topology and are exploited in theoretical computer science for the study of formal semantics.

This article gives pointers to special cases of Stone duality and explains a very general instance thereof in detail.

Contents 1 Overview of Stone-type dualities 2 Duality of sober spaces and spatial locales 2.1 The lattice of open sets 2.2 Points of a locale 2.3 The functor pt 2.4 The adjunction of Top and Loc 2.5 The duality theorem 3 References Overview of Stone-type dualities Probably the most general duality that is classically referred to as "Stone duality" is the duality between the category Sob of sober spaces with continuous functions and the category SFrm of spatial frames with appropriate frame homomorphisms. The dual category of SFrm is the category of spatial locales denoted by SLoc. The categorical equivalence of Sob and SLoc is the basis for the mathematical area of pointless topology, which is devoted to the study of Loc—the category of all locales, of which SLoc is a full subcategory. The involved constructions are characteristic for this kind of duality, and are detailed below.

Now one can easily obtain a number of other dualities by restricting to certain special classes of sober spaces: The category CohSp of coherent sober spaces (and coherent maps) is equivalent to the category CohLoc of coherent (or spectral) locales (and coherent maps), on the assumption of the Boolean prime ideal theorem (in fact, this statement is equivalent to that assumption). The significance of this result stems from the fact that CohLoc in turn is dual to the category DLat01 of bounded distributive lattices. Hence, DLat01 is dual to CohSp—one obtains Stone's representation theorem for distributive lattices. When restricting further to coherent sober spaces that are Hausdorff, one obtains the category Stone of so-called Stone spaces. On the side of DLat01, the restriction yields the subcategory Bool of Boolean algebras. Thus one obtains Stone's representation theorem for Boolean algebras. Stone's representation for distributive lattices can be extended via an equivalence of coherent spaces and Priestley spaces (ordered topological spaces, that are compact and totally order-disconnected). One obtains a representation of distributive lattices via ordered topologies: Priestley's representation theorem for distributive lattices.

Many other Stone-type dualities could be added to these basic dualities.

Duality of sober spaces and spatial locales The lattice of open sets The starting point for the theory is the fact that every topological space is characterized by a set of points X and a system Ω(X) of open sets of elements from X, i.e. a subset of the powerset of X. It is known that Ω(X) has certain special properties: it is a complete lattice within which suprema and finite infima are given by set unions and finite set intersections, respectively. Furthermore, it contains both X and the empty set. Since the embedding of Ω(X) into the powerset lattice of X preserves finite infima and arbitrary suprema, Ω(X) inherits the following distributivity law: {displaystyle xwedge bigvee S=bigvee {,xwedge s:sin S,},} for every element (open set) x and every subset S of Ω(X). Hence Ω(X) is not an arbitrary complete lattice but a complete Heyting algebra (also called frame or locale – the various names are primarily used to distinguish several categories that have the same class of objects but different morphisms: frame morphisms, locale morphisms and homomorphisms of complete Heyting algebras). Now an obvious question is: To what extent is a topological space characterized by its locale of open sets?

As already hinted at above, one can go even further. The category Top of topological spaces has as morphisms the continuous functions, where a function f is continuous if the inverse image f −1(O) of any open set in the codomain of f is open in the domain of f. Thus any continuous function f from a space X to a space Y defines an inverse mapping f −1 from Ω(Y) to Ω(X). Furthermore, it is easy to check that f −1 (like any inverse image map) preserves finite intersections and arbitrary unions and therefore is a morphism of frames. If we define Ω(f) = f −1 then Ω becomes a contravariant functor from the category Top to the category Frm of frames and frame morphisms. Using the tools of category theory, the task of finding a characterization of topological spaces in terms of their open set lattices is equivalent to finding a functor from Frm to Top which is adjoint to Ω.

Points of a locale The goal of this section is to define a functor pt from Frm to Top that in a certain sense "inverts" the operation of Ω by assigning to each locale L a set of points pt(L) (hence the notation pt) with a suitable topology. But how can we recover the set of points just from the locale, though it is not given as a lattice of sets? It is certain that one cannot expect in general that pt can reproduce all of the original elements of a topological space just from its lattice of open sets – for example all sets with the indiscrete topology yield (up to isomorphism) the same locale, such that the information on the specific set is no longer present. However, there is still a reasonable technique for obtaining "points" from a locale, which indeed gives an example of a central construction for Stone-type duality theorems.

Let us first look at the points of a topological space X. One is usually tempted to consider a point of X as an element x of the set X, but there is in fact a more useful description for our current investigation. Any point x gives rise to a continuous function px from the one element topological space 1 (all subsets of which are open) to the space X by defining px(1) = x. Conversely, any function from 1 to X clearly determines one point: the element that it "points" to. Therefore, the set of points of a topological space is equivalently characterized as the set of functions from 1 to X.