Categorical Extension of Dualities: From Stone to de Vries and Beyond, II (2109.07172v1)

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.