Probability, valuations, hyperspace: Three monads on Top and the support as a 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 Halgebra (topological complete semilattice) is also a Valgebra. Third, we show that V can be restricted to a submonad of tausmooth probability measures on Top. By composing these two morphisms of monads, we obtain that taking the support of a tausmooth probability measure is also a morphism of monads.
 Publication:

arXiv eprints
 Pub Date:
 October 2019
 arXiv:
 arXiv:1910.03752
 Bibcode:
 2019arXiv191003752F
 Keywords:

 Mathematics  General Topology;
 Computer Science  Logic in Computer Science;
 Mathematics  Category Theory;
 Mathematics  Functional Analysis;
 Mathematics  Probability;
 28B99;
 54C99;
 18C15;
 46M99
 EPrint:
 65 pages