Markov categories have recently turned out to be a powerful high-level framework for probability and statistics. They accommodate purely categorical definitions of notions like conditional probability and almost sure equality, as well as proofs of fundamental results such as the Hewitt-Savage 0/1 Law, the de Finetti Theorem and the Ergodic Decomposition Theorem. In this work, we develop additional relevant notions from probability theory in the setting of Markov categories. This comprises improved versions of previously introduced definitions of absolute continuity and supports, as well as a detailed study of idempotents and idempotent splitting in Markov categories. Our main result on idempotent splitting is that every idempotent measurable Markov kernel between standard Borel spaces splits through another standard Borel space, and we derive this as an instance of a general categorical criterion for idempotent splitting in Markov categories.

翻译：马尔可夫范畴近来已成为概率论与统计学的强大高层框架。它们能以纯范畴化方式定义条件概率、几乎必然相等性等概念，并证明诸如休伊特-萨维奇0/1律、德·芬内蒂定理和遍历分解定理等基础结论。本文在马尔可夫范畴框架下进一步发展了概率论中的若干相关概念，包括对先前引入的绝对连续性和支撑定义的改进版本，以及对马尔可夫范畴中幂等元与幂等分裂的详细研究。关于幂等分裂的主要结论是：标准波莱尔空间之间的任何幂等可测马尔可夫核均可通过另一标准波莱尔空间实现分裂，并且我们将其推导为马尔可夫范畴中幂等分裂的一般范畴化判据的一个实例。