Approaching the Continuous from the Discrete: an Infinite Tensor Product Construction
Antonio Lorenzin, Fabio Zanasi
TL;DR
The paper targets the problem of extending discrete categorical probability frameworks to the continuous domain. It introduces a universal construction ${\mathsf{C}}^{\otimes\infty}$ that adjoins infinite tensor products to a semicartesian category ${\mathsf{C}}$ with cancellative deletions, enabling a diagrammatic and limit-based treatment of continuous probability. In concrete terms, it characterises ${\mathsf{FinStoch}}^{\otimes\infty}$ via Stone spaces and locally constant Markov kernels, showing it embeds fully faithfully into ${\mathsf{StoneStoch}_{\mathsf{lc}}}$ with objects either finite sets or the Cantor space $2^{\mathbb{N}}$, and derives a corresponding axiomatic presentation ${\mathsf{CantorStoch}_{\mathsf{lc}}}$ as ${\mathsf{Free}^{\infty}}(\Sigma,E)$ for the causal-circuit theory ${\mathsf{CausCirc}}$. This yields a route to represent all probability measures on $\mathbb{R}$ as measures on Cantor space, thereby enabling a robust diagrammatic calculus for continuous probability via limits of discrete seeds. The work also discusses the limitations of recovering the full ${\mathsf{BorelStoch}}$ and points to future directions such as disintegration, broader Markov kernels, and connections to Stone duality. Overall, the framework provides a principled bridge from discrete probabilistic reasoning to a rich, continuous probabilistic setting amenable to diagrammatic methods and axiomatisations.
Abstract
Increasingly in recent years, probabilistic computation has been investigated through the lenses of categorical algebra, especially via string diagrammatic calculi. Whereas categories of discrete and Gaussian probabilistic processes have been thoroughly studied, with various axiomatisation results, more expressive classes of continuous probability are less understood, because of the intrinsic difficulty of describing infinite behaviour by algebraic means. In this work, we establish a universal construction that adjoins infinite tensor products, allowing continuous probability to be investigated from discrete settings. Our main result applies this construction to $\mathsf{FinStoch}$, the category of finite sets and stochastic matrices, obtaining a category of locally constant Markov kernels, where the objects are finite sets plus the Cantor space $2^{\mathbb{N}}$. Any probability measure on the reals can be reasoned about in this category. Furthermore, we show how to lift axiomatisation results through the infinite tensor product construction. This way we obtain an axiomatic presentation of continuous probability over countable powers of $2=\lbrace 0,1\rbrace$.