Under a general categorical procedure for the extension of dual equivalences as presented in this papers predecessor, a new algebraically defined category is established that is dually equivalent to the category $bf LKHaus$ of locally compact Hausdor
ff 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.
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 Ston
e 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, Morandi and Olberding. In the process of doing so, we obtain new duality theorems for both categories, $bf{KHaus}$ and $bf Tych$.
For a composition-closed and pullback-stable class S of morphisms in a category C containing all isomorphisms, we form the category Span(C,S) of S-spans (s,f) in C with first leg s lying in S, and give an alternative construction of its quotient cate
gory C[S^{-1}] of S-fractions. Instead of trying to turn S-morphisms directly into isomorphisms, we turn them separately into retractions and into sections in a universal manner, thus obtaining the quotient categories Retr( C,S) and Sect(C,S). The fraction category C[S^{-1}] is their largest joint quotient category. Without confining S to be a class of monomorphisms of C, we show that Sect(C,S) admits a quotient category, Par(C,S), whose name is justified by two facts. On one hand, for S a class of monomorphisms in C, it returns the category of S-spans in C, also called S-partial maps in this case; on the other hand, we prove that Par(C,S) is a split restriction category (in the sense of Cockett and Lack). A further quotient construction produces even a range category (in the sense of Cockett, Guo and Hofstra), RaPar(C,S), which is still large enough to admit C[S^{-1}] as its quotient. Both, Par and RaPar, are the left adjoints of global 2-adjunctions. When restricting these to their fixed objects, one obtains precisely the 2-equivalences by which their name givers characterized restriction and range categories. Hence, both Par(C,S)$ and RaPar(C,S may be naturally presented as Par(D,T)$ and RaPa(D,T), respectively, where now T is a class of monomorphisms in D. In summary, while there is no {em a priori} need for the exclusive consideration of classes of monomorphisms, one may resort to them naturally
Applying a general categorical construction for the extension of dualities, we present a new proof of the Fedorchuk duality between the category of compact Hausdorff spaces with their quasi-open mappings and the category of complete normal contact al
gebras with suprema-preserving Boolean homomorphisms which reflect the contact relation.
