Getting Stone duality from the duality between sober spaces and spatial frames

Question

Here it says:

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.

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Now one can easily obtain a number of other dualities by restricting to certain special classes of sober spaces:

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• 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.

How can the duality between Stone spaces and Boolean algebras be a restriction of the duality between Sober spaces and spatial frames? I mean, every Stone space is a sober space, but not every Boolean algebra is a spatial frame, since spatial frames are complete, whereas Boolean algebras don't have to be complete.

Answer 1

Coherent frames are precisely the ideal completions of distributive lattices. There is an equivalence between coherent frames and distributive lattices given by the functor which sends a distributive lattice to the lattice of its ideals and conversely the functor which sends a coherent frame to the distributive lattice of its compact elements.

Stone duality between Boolean algebras and Stone spaces can, in a way, be seen as a restriction of the duality between spatial locales and sober spaces, but your point about Boolean algebras not necessarily being frames is completely valid. To construct the dual of a spatial frame, you take completely prime filters, while to construct the dual of a Boolean algebra you take prime filters, so what's going on here is not a restriction in the obvious sense of the word.

You should think of Stone duality as the end point of this sequence of dualities:

spatial frames – sober spaces

coherent frames – coherent spaces (by restricting both sides)

distributive lattices – coherent spaces (by composing with the above equivalence, distributive lattices – coherent frames)

Boolean algebras – Stone spaces (by restricting both sides)

Answer 2

The equivalence between spatial frames and sober spaces restricts to an equivalence between coherent spaces and coherent frames. The latter are in turn equivalent to distributive lattices, so you get an equivalence between distributive lattices and coherent spaces. This is the equivalence that restricts to Stone duality, for Boolean algebras are contained in distributive lattices and correspond precisely to Hausdorff coherent spaces.