A duality of generalized metric spaces
In domain theory, Lawson duality is the result that the category \textbf{Dom} of continuous dcpos and open filter-reflecting maps is self-dual. A dcpo may be defined as a category \(X\) enriched in a two-element lattice (and is thus a poset) satisfying the condition that the lower closure functor \(\downarrow :X\to JX\) has a left adjoint which itself has a left adjoint (here \(JX\) is the set of filters in \(X\)). It is a classical result due to Lawvere that quasi-pseudometric spaces may be defined as categories enriched in the quantale \([0,\infty]\). As such it is natural to define a \([0,\infty]\)-domain to be a quasi-metric space which, seen as a \([0,\infty]\)-category, satisfies the condition that the left adjoint of the (enriched) Yoneda embedding has itself a left adjoint. Developing the necessary theory of \([0,\infty]\)-domains, the author shows that the category \([0,\infty]\)-\textbf{Dom} of \([0,\infty]\)-domains and open filters reflecting morphisms is self-dual. Lawson duality for domains is then a corollary by restriction along the embedding \(\{0,\infty\}\to [0,\infty]\). The article is quite self-contained and clearly written, finishing with a short discussion of the possible applications of the new duality.
scientific article; zbMATH DE number 5969488
scientific article