Compact Hausdorff Spaces with Relations and Gleason Spaces

Bezhanishvili, Guram; Gabelaia, David; Harding, John; Jibladze, Mamuka

doi:10.1007/s10485-019-09573-x

Cited by 3 publications

(14 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Thus, in a certain sense, the compact Hausdorff space X can be identifed with the pair ( X, E). This was made precise in [5], where an equivalence is exhibited between KHaus R and a full subcategory of StoneE R . We recall some relevant definitions and results.…”

Section: Various Equivalences For Khaus Rmentioning

confidence: 97%

“…For our purpose, it is more natural to start with the category KHaus R whose objects are compact Hausdorff spaces and morphisms are closed relations (i.e., relations R : X → Y such that R is a closed subset of X × Y ). This category was studied in [5], and earlier in [19] in the more general setting of stably compact spaces. The latter paper establishes a duality for KHaus R that generalizes Isbell duality [16] between KHaus and the category of compact regular frames and frame homomorphisms.…”

mentioning

confidence: 99%

“…This is obtained by generalizing the notion of a frame homomorphism to that of a preframe homomorphism. However, a similar duality in the language of de Vries algebras remained problematic (see [5,Rem. 3.14]).…”

mentioning

confidence: 99%

“…We denote the resulting categories by Gle R and DeV S , respectively. Since Gle R is equivalent to KHaus R (see [5,Thm. 3.13]), we obtain our desired result that KHaus R is equivalent to DeV S , thus resolving the problem raised in [5,Rem.…”

mentioning

confidence: 99%

“…3.14]. We then use the equivalence of KHaus with the wide subcategory Gle of Gle R given in [5,Thm. 6.7] to obtain the wide subcategory DeV F of DeV S that is equivalent to KHaus, thus yielding an alternative to de Vries duality.…”

mentioning

confidence: 99%

See 4 more Smart Citations

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

Abbadini¹,

Bezhanishvili²,

Carai³

2022

Preprint

View full text Add to dashboard Cite

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 Stone R of Stone spaces and closed relations and the category BA S of boolean algebras and subordination relations. We apply the Karoubi envelope construction to this equivalence to obtain that the category KHaus R of compact Hausdorff spaces and closed relations is equivalent to the category 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 KHaus R and DeV S further restricts to an equivalence between the category KHaus of compact Hausdorff spaces and continuous functions and the wide subcategory DeV F of 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.

show abstract

Section: Various Equivalences For Khaus Rmentioning

confidence: 97%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

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

Abbadini¹,

Bezhanishvili²,

Carai³

2022

Preprint

View full text Add to dashboard Cite

show abstract

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

Abbadini

Bezhanishvili

Carai

2023

Topology and its Applications

View full text Add to dashboard Cite

Ideal and MacNeille completions of subordination algebras

Abbadini¹,

Bezhanishvili²,

Carai³

2022

Preprint

View full text Add to dashboard Cite

S5-subordination algebras were recently introduced as a generalization of de Vries algebras, and it was proved that the category SubS5 S of S5-subordination algebras and compatible subordination relations between them is equivalent to the category of compact Hausdorff spaces and closed relations. We generalize MacNeille completions of boolean algebras to the setting of S5-subordination algebras, and utilize the relational nature of the morphisms in SubS5 S to prove that the MacNeille completion functor establishes an equivalence between SubS5 S and its full subcategory consisting of de Vries algebras. We also generalize ideal completions of boolean algebras to the setting of S5-subordination algebras and prove that the ideal completion functor establishes a dual equivalence between SubS5 S and the category of compact regular frames and preframe homomorphisms. Our results are choice-free and provide further insight into Stone-like dualities for compact Hausdorff spaces with various morphisms between them. In particular, we show how they restrict to the wide subcategories of SubS5 S corresponding to continuous relations and continuous functions between compact Hausdorff spaces.

show abstract

Compact Hausdorff Spaces with Relations and Gleason Spaces

Cited by 3 publications

References 16 publications

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

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

Ideal and MacNeille completions of subordination algebras

Contact Info

Product

Resources

About