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We have decided to discontinue the publication of preprints on our preprint server end of 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
Probability, valuations, hyperspaces: Three monads on Top and the support as a morphism
Tobias Fritz, Paolo Perrone and Sharwin RezagholiAbstract
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.
- Received:
- 25.03.19
- Published:
- 25.03.19