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

Marco Abbadini, Guram Bezhanishvili, Luca Carai

Published 2022-06-12Version 1

Halmos duality generalizes Stone duality to the category of Stone spaces and continuous relations. This further generalizes to an equivalence (as well as to a dual equivalence) between the category $\mathsf{Stone}^{\mathsf{R}}$ of Stone spaces and closed relations and the category $\mathsf{BA}^\mathsf{S}$ of boolean algebras and subordination relations. We apply the Karoubi envelope construction to this equivalence to obtain that the category $\mathsf{KHaus}^\mathsf{R}$ of compact Hausdorff spaces and closed relations is equivalent to the category $\mathsf{DeV^S}$ of de Vries algebras and compatible subordination relations. This resolves a problem recently raised in the literature. We prove that 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 our approach is that composition of morphisms is usual relation composition.