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

Published online by Cambridge University Press:  08 March 2022

Tobias Fritz ,
Paolo Perrone  and
Sharwin Rezagholi Open the ORCID record for Sharwin Rezagholi [Opens in a new window]
Tobias Fritz
Affiliation:
Department of Mathematics, University of Innsbruck, Innsbruck, Austria
Paolo Perrone
Affiliation:
Department of Computer Science, University of Oxford, Oxford OX1 2JD, UK
Sharwin Rezagholi*
Affiliation:
Faculty of Applied Mathematics and Computer Science, University of Applied Sciences Technikum Wien, Austria Department of Informatics, Higher School of Economics St. Petersburg, Russia
*
*Corresponding author. Email: sharwin.rezagholi@emergentec.com, sharwin.rezagholi@technikum-wien.at
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Abstract

We consider three monads on $\mathsf{Top}$, 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 one 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 $V \to H$. In particular, this implies that every H-algebra (topological complete semilattice) is also a V-algebra. We show that V can be restricted to a submonad of $\tau$-smooth probability measures on $\mathsf{Top}$. By composing these morphisms of monads, we obtain that taking the supports of $\tau$-smooth probability measures is also a morphism of monads.

Keywords

Monads on topological spacesvaluationsBorel measuresprobability measureshyperspacesmorphism of monads

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Paper
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Mathematical Structures in Computer Science , Volume 31 , Issue 8 , September 2021 , pp. 850 - 897
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© The Author(s), 2022. Published by Cambridge University Press

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