Weak Distributive Laws between Monads of Continuous Valuations and of Non-Deterministic Choice

TL;DR

This paper establishes weak distributive laws between various monads related to continuous valuations, non-deterministic choice, and measures, expanding the theoretical framework for combining probabilistic and non-deterministic effects in topology.

Contribution

It introduces new weak distributive laws between monads of valuations, hyperspaces, and lenses over topological spaces, generalizing the composition of probabilistic and non-deterministic monads.

Findings

Existence of weak distributive laws over the category of topological spaces.

Construction of the monad of superlinear and sublinear previsions as weak composites.

Special case of weak distributive law between hyperspace monads and Radon measures.

Abstract

We show that there is weak distributive law of the Smyth hyperspace monad $\mathcal Q_{\mathsf V}$ (resp., the Hoare hyperspace monad $\mathcal H_{\mathsf V}$, resp. the monad $\mathcal P\ell^{\mathrm q}_{\mathsf V}$ of quasi-lenses, resp. the monad $\mathcal P\ell_{\mathsf V}$ of lenses) over the continuous valuation monad $\mathbf V$, as well as over the subprobability valuation monad $\mathbf V_{\leq 1}$ and the probability valuation monad $\mathbf V_1$, on the whole category $\mathbf{Top}$ of topological spaces (resp., on certain full subcategories such as the category of locally compact spaces or of stably compact spaces). We show that the resulting weak composite monad is the author's monad of superlinear previsions (resp., sublinear previsions, resp. forks), possibly subnormalized or normalized depending on whether we consider $\mathbf V_{\leq 1}$ or $\mathbf V_1$ instead of $\mathbf V$. As a special case, we obtain a weak distributive law of the monad $\mathcal P\ell^{\mathrm q}_{\mathsf V} \cong \mathcal P\ell_{\mathsf V}$ over the monad of (sub)probability Radon measures $\mathbf R_\bullet$ on the category of stably compact spaces, which specializes further to a weak distributive laws of the Vietoris monad over $\mathbf R_\bullet$. The associated weak composite monad is the monad of (sub)normalized forks.