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：近年来，马尔可夫范畴已被证明是概率论与统计学中一个强大的高层次框架。它能够容纳条件概率与几乎必然相等这类概念的纯范畴化定义，并为基本结果（如休伊特-萨维奇0/1律、德菲内蒂定理与遍历分解定理）提供证明。在本研究中，我们在马尔可夫范畴的框架下发展了概率论中更多相关概念。这包括对先前引入的绝对连续性与支集定义的改进版本，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果表明：标准Borel空间之间的每个幂等可测马尔可夫核均可通过另一个标准Borel空间分裂，我们将此结论推导为马尔可夫范畴中幂等分裂的一般范畴判别准则的一个实例。