In mathematics, there is an ample supply of categorical dualities between certain 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 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.
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In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra that is complete as a lattice. Complete Heyting algebras are the of three different ; the category CHey, the category Loc of locales, and its , the category Frm of frames. Although these three categories contain the same objects, they differ in their morphisms, and thus get distinct names. Only the morphisms of CHey are homomorphisms of complete Heyting algebras.
In mathematics, there is an ample supply of categorical dualities between certain 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.
In mathematics, a spectral space is a topological space that is homeomorphic to the spectrum of a commutative ring. It is sometimes also called a coherent space because of the connection to coherent topos. Let X be a topological space and let K(X) be the set of all compact open subsets of X. Then X is said to be spectral if it satisfies all of the following conditions: X is compact and T0. K(X) is a basis of open subsets of X. K(X) is closed under finite intersections. X is sober, i.e.