Completing simple valuations in -categories

Abstract

We prove that Keimel and Lawson's K-completion Kc of the simple valuation monad Vs defines a monad Kc∘Vs on each ▪-category K. We also characterise the Eilenberg-Moore algebras of Kc∘Vs as the weakly locally convex K-cones, and its algebra morphisms as the continuous linear maps.In addition, we explicitly describe the distributive law of Vs over Kc, which allows us to show that the K-completion of any locally convex (resp., weakly locally convex, locally linear) topological cone is a locally convex (resp., weakly locally convex, locally linear) K-cone.We also give an example – the Cantor tree with a top – that shows the dcpo-completion of the simple valuations is not the D-completion of the simple valuations in general, where D is the category of monotone convergence spaces and continuous maps.