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

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 $\mathcal{P}{\mathrm{\ell }}_{\mathsf{V}}^{\mathrm{q}}$ of quasi-lenses, resp. the monad $\mathcal{P}{\mathrm{\ell }}_{\mathsf{V}}$ of lenses) over the continuous valuation monad $\mathbf{V}$, as well as over the subprobability valuation monad ${\mathbf{V}}_{\le 1}$ and the probability valuation monad ${\mathbf{V}}_1$, on the whole category $\mathbf{T}\mathbf{o}\mathbf{p}$ 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 ${\mathbf{V}}_{\le 1}$ or ${\mathbf{V}}_1$ instead of $\mathbf{V}$. As a special case, we obtain a weak distributive law of the monad $\mathcal{P}{\mathrm{\ell }}_{\mathsf{V}}^{\mathrm{q}}\cong \mathcal{P}{\mathrm{\ell }}_{\mathsf{V}}$ 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.