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 $2=\lbrace 0,1\rbrace$.

翻译：近年来，概率计算日益通过范畴代数的视角进行研究，特别是借助弦图演算。尽管离散和高斯概率过程的范畴已被深入研究并取得多种公理化结果，但更具表达力的连续概率类别由于代数方法描述无限行为的固有困难而理解较少。本文建立了一种通用构造，通过引入无限张量积使得可以从离散环境研究连续概率。我们的主要结果将该构造应用于有限集与随机矩阵范畴$\mathsf{FinStoch}$，得到了一个局部常数马尔可夫核范畴，其中对象为有限集加上康托尔空间$2^{\mathbb{N}}$。任何实数上的概率测度均可在此范畴中推理。此外，我们展示了如何通过无限张量积构造提升公理化结果，从而获得可数无穷次$2=\lbrace 0,1\rbrace$幂上连续概率的公理化表述。