{
  "id": "2206.05711",
  "version": "v1",
  "published": "2022-06-12T10:08:10.000Z",
  "updated": "2022-06-12T10:08:10.000Z",
  "title": "A generalization of de Vries duality to closed relations between compact Hausdorff spaces",
  "authors": [
    "Marco Abbadini",
    "Guram Bezhanishvili",
    "Luca Carai"
  ],
  "comment": "31 pages",
  "categories": [
    "math.GN",
    "math.LO"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-12T10:08:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "compact hausdorff spaces",
      "closed relations",
      "vries duality",
      "halmos duality generalizes stone duality",
      "equivalence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}