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

TL;DR

This paper explores three monads on Top related to probability and possibility, showing how support operations form monad morphisms and revealing deep structural connections between them.

Contribution

It constructs and relates three monads on Top, demonstrating support as a monad morphism and unifying probabilistic and topological concepts through double dualization.

Findings

Support of continuous valuations is a monad morphism from V to H.

Every H-algebra is also a V-algebra.

Support of tau-smooth probability measures is a monad morphism.

Abstract

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