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

Abstract

Stone duality generalizes to an equivalence between the categories StoneR of Stone spaces and closed relations and BAS of boolean algebras and subordination relations. Splitting equivalences in StoneR yields a category that is equivalent to the category KHausR of compact Hausdorff spaces and closed relations. Similarly, splitting equivalences in BAS yields a category that is equivalent to the category DeVS of de Vries algebras and compatible subordination relations. Applying the machinery of allegories then gives that KHausR is equivalent to DeVS, thus resolving a problem recently raised in the literature. The equivalence between KHausR and DeVS further restricts to an equivalence between the category KHaus of compact Hausdorff spaces and continuous functions and the wide subcategory DeVF of DeVS 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.