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.

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:

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

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:

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.