Dualities via ComPactness, Openness, and their Symmetry

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This project is positioned inside the study of Stone-like dualities and touches on themes in algebraic logic, domain theory andpointfree topology.The interplay between compact sets and open sets is a key tool in topology, and various conditions and constructions (such as localcompactness, the compact-open topology on function spaces, well-filtered spaces) hint at a symmetrical role between these twonotions. Indeed, a precise symmetry occurs in the spaces used by M.H. Stone to represent Boolean algebras (1936) and distributivelattices (1938). However, in some important generalisations of Stone dualities (such as the duality between sober spaces and spatialframes), the symmetry between compactness and openness is only partial.This project aims at demonstrating that the aforementioned partial symmetry is in fact a slice of a formal, complete symmetry in alarger framework. This framework will provide new explanations for the occurrence of certain conditions and constructions (such aslocal compactness, the compact-open topology, sobriety), will suggest new results (such as new dualities), and will allow goodstructural properties (such as Cartesian closure).