Two high-level "pictures" of probability theory have emerged: one that takes as central the notion of random variable, and one that focuses on distributions and probability channels (Markov kernels). While the channel-based picture has been successfully axiomatized, and widely generalized, using the notion of Markov category, the categorical semantics of the random variable picture remain less clear. Simpson's probability sheaves are a recent approach, in which probabilistic concepts like random variables are allowed vary over a site of sample spaces. Simpson has identified rich structure on these sites, most notably an abstract notion of conditional independence, and given examples ranging from probability over databases to nominal sets. We aim bring this development together with the generality and abstraction of Markov categories: We show that for any suitable Markov category, a category of sample spaces can be defined which satisfies Simpson's axioms, and that a theory of probability sheaves can be developed purely synthetically in this setting. We recover Simpson's examples in a uniform fashion from well-known Markov categories, and consider further generalizations.

翻译：概率论已形成两种高层次的"图景"：一种以随机变量概念为核心，另一种则聚焦于分布与概率通道（马尔可夫核）。虽然基于通道的图景已通过马尔可夫范畴的概念成功公理化并得到广泛推广，但随机变量图景的范畴语义仍不够清晰。辛普森的概率层是近期提出的一种方法，其中随机变量等概率概念被允许在样本空间基上变化。辛普森揭示了这些基上的丰富结构——最显著的是条件独立的抽象概念，并给出了从数据库概率到名义集等实例。本文旨在将这一进展与马尔可夫范畴的普遍性和抽象性相结合：我们证明对于任何合适的马尔可夫范畴，均可定义满足辛普森公理的样本空间范畴，且概率层理论可在此设定中纯综合地发展。我们以统一的方式从已知马尔可夫范畴中恢复辛普森的实例，并探讨进一步的推广。