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

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

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A generalization of de Vries duality to closed relations between compact Hausdorff spaces 2023 Topology and its Applications 10.1016/j.topol.2023.108641 abbadini2023-a-generalization-of-de-vries-duali abbadini2023-a-generalization-of-de-vries-duali.txt, the project's staged copy 4189a861cee1e529bd1b8248b032d51e32edeada open-access category-theory

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0 spans located by searching the stored text for the quote's exact words, allowing any whitespace between them existence Quotation withheld because this record is not cleared for redistribution. de Vries duality for compact Hausdorff spaces Quotation withheld because this record is not cleared for redistribution. mathematical-proof mathematical-proof Quotation withheld because this record is not cleared for redistribution. mechanisms Realising compact Hausdorff spaces as quotients of Stone spaces by closed equivalence relations, and then splitting those equivalences as idempotents inside the allegory, transports the Stone-space/boolean-algebra equivalence up to the compact Hausdorff/de Vries level. Quotation withheld because this record is not cleared for redistribution. One of the intermediate categories the construction produces has badly behaved isomorphisms. Quotation withheld because this record is not cleared for redistribution. In one of the categories used, two objects can be isomorphic while differing in cardinality, so the word does not carry its usual force there. what this limits: a statement that two objects of StoneER are isomorphic: isomorphism there is not a structure-preserving bijection, and two objects may be isomorphic even without sharing the same cardinality Quotation withheld because this record is not cleared for redistribution. The composition operation in the classical dual category is not the usual function composition, which is the drawback the paper exists to remove. what this limits: any argument that composes morphisms in de Vries' original dual category DeV: composition there is not usual function composition, whereas in the alternative proposed here it is usual relation composition Quotation withheld because this record is not cleared for redistribution. The direction chosen for the morphisms is declared a matter of taste, with the opposite convention giving a dual statement of the same result. what this limits: the direction in which this equivalence is stated: the authors call the choice of direction a matter of taste and note the result can equally be stated as a dual equivalence Quotation withheld because this record is not cleared for redistribution. de Vries algebras de Vries algebras Quotation withheld because this record is not cleared for redistribution. closed relation closed relation Quotation withheld because this record is not cleared for redistribution. isomorphism isomorphism Quotation withheld because this record is not cleared for redistribution. structure-preserving bijection structure-preserving bijection Quotation withheld because this record is not cleared for redistribution. subordination relations subordination relations Quotation withheld because this record is not cleared for redistribution. isomorphism - IN StoneER, NOT A BIJECTION: two objects may be isomorphic even when their underlying spaces differ in cardinality, so the word here does not imply the structure-preserving correspondence a reader would assume Quotation withheld because this record is not cleared for redistribution. isomorphism - IN DeVS AND GleR, RECOVERED: the paper proves that in these two categories isomorphisms ARE structure-preserving bijections, which is what makes them the better home for the duality Quotation withheld because this record is not cleared for redistribution. composition - THE ORDINARY OPERATION OR NOT: in de Vries' original category it is not usual function composition; in the approach proposed here it is usual relation composition, and that difference is given as the advantage Quotation withheld because this record is not cleared for redistribution. extends → de Vries duality [12] Quotation withheld because this record is not cleared for redistribution. extends → the open problem raised in [5, Rem. 3.14] Quotation withheld because this record is not cleared for redistribution. builds-on → Celani [11]'s quasi-semi-homomorphisms and the subordination relations of [4] Quotation withheld because this record is not cleared for redistribution. isomorphism —distinct-from→ structure-preserving bijection isomorphism structure-preserving bijection strength of the relation: 1 Quotation withheld because this record is not cleared for redistribution. Stone duality lifts to an equivalence between Stone spaces with closed relations and boolean algebras with subordination relations. Quotation withheld because this record is not cleared for redistribution. Splitting the equivalences (closed equivalence relations) in the allegory of Stone spaces and closed relations produces a category equivalent to compact Hausdorff spaces with closed relations. Quotation withheld because this record is not cleared for redistribution. Splitting the equivalences on the algebraic side produces a category equivalent to de Vries algebras with compatible subordination relations. Quotation withheld because this record is not cleared for redistribution. Compact Hausdorff spaces with closed relations are equivalent to de Vries algebras with compatible subordinations, resolving a problem recently raised in the literature. Quotation withheld because this record is not cleared for redistribution. That equivalence restricts to one between compact Hausdorff spaces with continuous maps and the wide subcategory DeVF of de Vries algebras whose morphisms satisfy additional conditions. Quotation withheld because this record is not cleared for redistribution. One advantage is that composition of morphisms is ordinary relation composition. Quotation withheld because this record is not cleared for redistribution. The equivalence is realised concretely by sending a closed relation R to the subordination on regular opens defined by U S V exactly when R maps the closure of U inside V. Quotation withheld because this record is not cleared for redistribution. Isomorphisms in the intermediate categories are badly behaved: two S5-subordination spaces can be isomorphic in StoneER while their underlying Stone spaces differ in cardinality. Quotation withheld because this record is not cleared for redistribution. In the de Vries algebra category DeVS, by contrast, every isomorphism is a structure-preserving bijection. Quotation withheld because this record is not cleared for redistribution. DeVF is dually isomorphic to the classical category DeV of de Vries algebras and de Vries morphisms. Quotation withheld because this record is not cleared for redistribution. The construction yields, as a by-product, a new proof that the category of compact Hausdorff spaces is the exact completion of the category of Stone spaces. Quotation withheld because this record is not cleared for redistribution. closed relation between Stone spaces closed relation between Stone spaces Quotation withheld because this record is not cleared for redistribution. quasi-semi-homomorphism quasi-semi-homomorphism Quotation withheld because this record is not cleared for redistribution. subordination relation subordination relation Quotation withheld because this record is not cleared for redistribution. allegory allegory Quotation withheld because this record is not cleared for redistribution. S5-subordination S5-subordination Quotation withheld because this record is not cleared for redistribution. S5-subordination space S5-subordination space Quotation withheld because this record is not cleared for redistribution. extremally disconnected extremally disconnected Quotation withheld because this record is not cleared for redistribution. Gleason space Gleason space Quotation withheld because this record is not cleared for redistribution. de Vries algebra de Vries algebra Quotation withheld because this record is not cleared for redistribution.

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