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

TL;DR

This paper extends duality theories for locally compact Hausdorff spaces by introducing a new algebraic category of complete local contact algebras, providing a more natural and accessible duality framework.

Contribution

It establishes a new algebraically defined category dually equivalent to locally compact Hausdorff spaces, extending previous Stone-type dualities with more natural morphisms.

Findings

New duality between locally compact Hausdorff spaces and complete local contact algebras

Simplified and natural morphisms in the algebraic category

Alternative proof of the classical duality result

Abstract

Under a general categorical procedure for the extension of dual equivalences as presented in this paper's predecessor, a new algebraically defined category is established that is dually equivalent to the category $\bf LKHaus$ of locally compact Hausdorff spaces and continuous maps, with the dual equivalence extending a Stone-type duality for the category of extremally disconnected locally compact Hausdorff spaces and continuous maps. The new category is then shown to be isomorphic to the category $\bf CLCA$ of complete local contact algebras and suitable morphisms. Thereby, a new proof is presented for the equivalence ${\bf LKHaus}\simeq{\bf CLCA}^{\rm op}$ that was obtained by the first author more than a decade ago. Unlike the morphisms of $\bf CLCA$, the morphisms of the new category and their composition law are very natural and easy to handle.