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Probability, valuations, hyperspaces: Three monads on Top and the support as a morphism
We deﬁne a monad V of continuous subprobability valuations on the category Top of topological spaces and continuous maps, analogous to the extended probabilistic powerdomain. This monad can be restricted to a submonad of τ-smooth probability measures on Top. We also study the hyperspace monad H on Top, which assigns to every space its space of closed subsets equipped with the lower Vietoris topology. We show that the operation of taking the support of a valuation induces a morphism of monads from V to H. To do so, we use duality results for valuations and for closed subsets that are naturally compatible. As far as we know, this work is the ﬁrst to provide a morphism from a probabilistic to a possibilistic powerspace. We show that the V-algebras are topological convex spaces, and that every H-algebra (i.e. every topological semilattice) is canonically a V-algebra too.