A generalization of de Vries duality to closed relations between compact Hausdorff spaces

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.