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得到深入研究，并获得了多种公理化结果，但更具表达力的连续概率类别却因用代数方法描述无限行为的固有困难而较少被理解。在本工作中，我们建立了一种通用构造，通过添加无限张量积，使得可以从离散设置出发研究连续概率。我们的主要结果将此构造应用于 $\mathsf{FinStoch}$（有限集与随机矩阵的范畴），得到了一个局部常数马尔可夫核的范畴，其中对象是有限集加上康托空间 $2^{\mathbb{N}}$。任何实数上的概率测度都可以在此范畴中进行推理。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。通过这种方式，我们获得了在 $2=\lbrace 0,1\rbrace$ 的可数幂上连续概率的一个公理化表述。