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

TL;DR

This paper extends de Vries duality to include closed relations between compact Hausdorff spaces, providing a new categorical equivalence that simplifies composition and offers an alternative perspective to classical duality.

Contribution

It generalizes de Vries duality to closed relations, establishing new categorical equivalences and simplifying morphism composition in the duality framework.

Findings

Establishes equivalence between categories of compact Hausdorff spaces with closed relations and de Vries algebras.

Provides an alternative to classical de Vries duality with usual relation composition.

Resolves a recent open problem in the literature.

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.