# Categorical Extension of Dualities: From Stone to de Vries and Beyond, I

@article{Dimov2021CategoricalEO, title={Categorical Extension of Dualities: From Stone to de Vries and Beyond, I}, author={Georgi D. Dimov and Elza Ivanova-Dimova and Walter Tholen}, journal={Applied Categorical Structures}, year={2021} }

Propounding a general categorical framework for the extension of dualities, we present a new proof of the de Vries Duality Theorem for the category $\bf KHaus$ of compact Hausdorff spaces and their continuous maps, as an extension of a restricted Stone duality. Then, applying a dualization of the categorical framework to the de Vries duality, we give an alternative proof of the extension of the de Vries duality to the category $\bf Tych$ of Tychonoff spaces that was provided by Bezhanishvili… Expand

#### One Citation

G N ] 1 A ug 2 02 0 Categorical Extension of Dualities : From Stone to de Vries and Beyond

- 2021

Propounding a general categorical framework for the extension of dualities, we present a new proof of the de Vries Duality Theorem for the category KHaus of compact Hausdorff spaces and their… Expand

#### References

SHOWING 1-10 OF 77 REFERENCES

Two extensions of the Stone Duality to the category of zero-dimensional Hausdorff spaces, Filomat

- (preprint:
- 1901

Categorical extension of dualities: From Stone to de Vries and beyond, I

- Applied Categorical Structures,
- 2021

An extension of de Vries duality to completely regular spaces and compactifications

- Mathematics
- Topology and its Applications
- 2019

Abstract De Vries duality yields a dual equivalence between the category of compact Hausdorff spaces and a category of complete Boolean algebras with a proximity relation on them, known as de Vries… Expand

Region-Based Topology

- Mathematics, Computer Science
- J. Philos. Log.
- 1997

A topological description of space is given, based on the relation of connection among regions and the property of being limited. A minimal set of 10 constraints is shown to permit definitions of… Expand

The theory of representations for Boolean algebras

- Mathematics
- 1936

Boolean algebras are those mathematical systems first developed by George Boole in the treatment of logic by symbolic methodsf and since extensively investigated by other students of logic, including… Expand

Irreducible Equivalence Relations, Gleason Spaces, and de Vries Duality

- Mathematics, Computer Science
- Appl. Categorical Struct.
- 2017

This work provides an alternative “modal-like” duality by introducing the concept of a Gleason space, which is a pair (X,R), where X is an extremally disconnected compact Hausdorff space and R is an irreducible equivalence relation on X. Expand

Region–based theory of discrete spaces: A proximity approach

- Mathematics, Computer Science
- Annals of Mathematics and Artificial Intelligence
- 2007

This work introduces Boolean proximity algebras as a generalization of Efremovič proximities which are suitable in reasoning about discrete regions and shows that each such algebra is isomorphic to a substructure of a complete and atomic Boolean proximity algebra. Expand

Local proximity spaces Math

- Annalen 169,
- 1967