Probability, valuations, hyperspace: Three monads on top and the support as a morphism

We consider three monads on Top, the category of topological spaces, which formalize aspects probability and possibility in categorical terms. The first one is Hoare hyperspace monad H, assigns to every space its closed subsets equipped with lower Vietoris topology. second V continuous valuations, also known as extended probabilistic powerdomain. construct both a unified way terms double dualization. This reveals close analogy between them, allows us prove that operation taking support valuation morphism from H. In particular, this implies H-algebra (topological complete semilattice) V-algebra. Third, we show can be restricted submonad tau-smooth measures Top. By composing these two morphisms monads, obtain measure monads.