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 $\mathsf{De}{\mathsf{V}}^{\mathsf{S}}$ of de Vries algebras and compatible subordination relations. Applying the machinery of allegories then yields that ${\mathsf{KHaus}}^{\mathsf{R}}$ is equivalent to $\mathsf{De}{\mathsf{V}}^{\mathsf{S}}$, thus resolving a problem recently raised in the literature. The equivalence between ${\mathsf{KHaus}}^{\mathsf{R}}$ and $\mathsf{De}{\mathsf{V}}^{\mathsf{S}}$ further restricts to an equivalence between the category $\mathsf{KHaus}$ of compact Hausdorff spaces and continuous functions and the wide subcategory $\mathsf{De}{\mathsf{V}}^{\mathsf{F}}$ of $\mathsf{De}{\mathsf{V}}^{\mathsf{S}}$ 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.

18 pages