Classical Stone duality is the dual equivalence between the categories of Boolean algebras and totally disconnected compact Hausdorff spaces, that is, Stone spaces. In the late 1960s, Manes proved that the algebras for the ultrafilter monad are precisely compact Hausdorff spaces. The ultrafilter monad is induced by the contravariant powerset functor, regarded as a functor from the category of sets to Boolean algebras, and its adjoint, which takes a Boolean algebra to its set of ultrafilters.
In the first part of this talk, we will outline a proof of the equivalence between the categories of algebras for the ultrafilter monad and compact Hausdorff spaces. We will then use Dubuc's adjoint triangle theorem to construct a left adjoint to the induced comparison functor from Boolean algebras to compact Hausdorff spaces; by restricting this adjunction in a canonical way, we obtain classical Stone duality.
Priestley duality refers to the dual equivalence between the categories of distributive lattices and Priestley spaces, a category of partially ordered spaces with a separation condition reminiscient of that of Stone spaces. In 1997, Flagg proved that the category of (partially) ordered compact Hausdorff spaces is equivalent to the category of algebras for the prime filter monad, a monad on the category of ordered sets. In the second part of this talk, we will show how we extend the machinery developed in the first half to derive Priestley duality by way of the prime filter monad.