Stone duality

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 locales denoted by SLoc. The categorical equivalence of Sob and SLoc is the basis for the mathematical area of pointless topology, that 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:

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