De Vries duality yields a dual equivalence between the category of compact Hausdorff spaces and a category of complete Boolean algebras with a proximity relation on them, known as de Vries algebras. In a previous paper [Appl. Categ. Struct. 22, No. 1, 43–78 (2014; Zbl 1316.06009)], the authors extended de Vries duality to a duality between the categories of stably compact spaces and regular proximity frames. More recently, G. Bezhanishvili et al. [Topology Appl. 257, 85–105 (2019; Zbl 1412.54034)] extended de Vries duality also to completely regular spaces by replacing the category of de Vries algebras with certain extensions of de Vries algebras. This was done via a duality between compactifications of completely regular spaces and de Vries extensions.
In the present paper, the authors generalize this duality further to the category \(\mathbf{StComp}\) of stable compactifications \(e \colon X \to Y\) of \(T_0\)-spaces. The corresponding dually equivalent category is a category \(\mathbf{RE}\) of extensions of proximity frames, called Raney extensions, that generalize de Vries extensions. These are proximity frame embeddings of a regular proximity frame into a Raney lattice (i.e. the lattice of upsets of a poset).
This duality is then restricted to obtain a duality for \(T_0\)-spaces. This is done by considering certain Raney extensions, called maximal Raney extensions, which correspond to Smyth compactifications (i.e. largest stable compactifications). The duality between \(\mathbf{StComp}\) and \(\mathbf{RE}\) can also be restricted to a duality involving spectral compactifications of \(T_0\)-spaces. These are stable compactifications \(e \colon X \to Y\) where \(Y\) is a spectral space.