A generalization of de Vries duality to closed relations between compact Hausdorff spaces
Marco Abbadini, Guram Bezhanishvili, Luca Carai
TL;DR
The paper generalizes Stone duality to closed relations between compact Hausdorff spaces by embedding KHaus$^{\mathsf{R}}$ into allegories and splitting equivalences, obtaining an equivalence with DeV$^{\mathsf{S}}$. It builds explicit bridges via Gleason spaces and regular open algebras, showing KHaus$^{\mathsf{R}} \simeq Gle^{\mathsf{R}} \simeq DeV^{\mathsf{S}}$, and further derives an alternative duality through the DeV$^{\mathsf{F}}$ subcategory where morphisms are ordinary relations. The paper then provides a direct, contravariant translation between DeV and DeV$^{\mathsf{F}}$, recovering classical de Vries duality as a special case while enabling standard composition of morphisms. Overall, it unifies topological and algebraic perspectives via relational morphisms, resolving open questions and offering a robust, composition-friendly framework for dualities in the KHaus setting.
Abstract
Stone duality generalizes to an equivalence between the categories $\mathsf{Stone}^{\mathsf{R}}$ of Stone spaces and closed relations and $\mathsf{BA}^\mathsf{S}$ of boolean algebras and subordination relations. Splitting equivalences in $\mathsf{Stone}^{\mathsf{R}}$ yields a category that is equivalent to the category $\mathsf{KHaus}^\mathsf{R}$ of compact Hausdorff spaces and closed relations. Similarly, splitting equivalences in $\mathsf{BA}^\mathsf{S}$ yields a category that is equivalent to the category $\mathsf{DeV^S}$ of de Vries algebras and compatible subordination relations. Applying the machinery of allegories then yields that $\mathsf{KHaus}^\mathsf{R}$ is equivalent to $\mathsf{DeV^S}$, thus resolving a problem recently raised in the literature. The equivalence between $\mathsf{KHaus}^\mathsf{R}$ and $\mathsf{DeV^S}$ further restricts to an equivalence between the category ${\mathsf{KHaus}}$ of compact Hausdorff spaces and continuous functions and the wide subcategory $\mathsf{DeV^F}$ of $\mathsf{DeV^S}$ whose morphisms satisfy additional conditions. This yields an alternative to de Vries duality. One advantage of this approach is that composition of morphisms is usual relation composition.