Distributing Retractions, Weak Distributive Laws and Applications to Monads of Hyperspaces, Continuous Valuations and Measures
Jean Goubault-Larrecq
TL;DR
This work develops distributing retractions as a practical bridge to weak distributive laws between monads in a 2-category, enabling the identification and construction of a combined monad U from S and T via a splitting of an idempotent. It proves a precise one-to-one correspondence between distributing retractions and weak distributive laws, and applies this to three hyperspace-valuation configurations: Smyth, Hoare, and Plotkin hyperspaces with continuous valuations, yielding algebras in terms of previsions and forks. The paper explicitly constructs the distributing retractions and weak distributive laws for Smyth-valuation (P_{D P}^•), Hoare-valuation (P_{A P}^•), and lenses-valuation (P_{A D P}^•) cases, and derives the corresponding algebras, along with topological characterizations in Top, Kontorshev-style and KHaus settings. As a culmination, it connects Vietoris and Radon monads on KHaus and shows the monotone weak distributive law between them, identifying the resulting combined monad as P_{A D P}^1, with implications for the algebraic structure of superlinear, sublinear previsions and their interplay with non-deterministic and probabilistic effects.
Abstract
Given two monads $S$, $T$ on a category where idempotents split, and a weak distributive law between them, one can build a combined monad $U$. Making explicit what this monad $U$ is requires some effort. When we already have an idea what $U$ should be, we show how to recognize that $U$ is indeed the combined monad obtained from $S$ and $T$: it suffices to exhibit what we call a distributing retraction of $ST$ onto $U$. We show that distributing retractions and weak distributive laws are in one-to-one correspondence, in a 2-categorical setting. We give three applications, where $S$ is the Smyth, Hoare or Plotkin hyperspace monad, $T$ is a monad of continuous valuations, and $U$ is a monad of previsions or of forks, depending on the case. As a byproduct, this allows us to describe the algebras of monads of superlinear, resp. sublinear previsions. In the category of compact Hausdorff spaces, the Plotkin hyperspace monad is sometimes known as the Vietoris monad, the monad of probability valuations coincides with the Radon monad, and we infer that the associated combined monad is the monad of normalized forks.