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

Abstract

Propounding a general categorical framework for the extension of dualities, we present a new proof of the de Vries Duality Theorem for the category $\mathbf{KHaus}$ 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 $\mathbf{Tych}$ of Tychonoff spaces that was provided by Bezhanishvili, Morandi and Olberding. In the process of doing so, we obtain new duality theorems for both categories, $\mathbf{KHaus}$ and $\mathbf{Tych}$.

47 pages. This is a revised and expanded version of our paper arXiv:1906.06177v1 "Extensions of dualities and a new approach to the de Vries duality"