Abstract: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 across another standard Borel space, and we derive this as an instance of a general categorical criterion for idempotent splitting in Markov categories.
From: Tobias Fritz [view email]
[v1]
Tue, 1 Aug 2023 16:36:01 UTC (115 KB)
[v2]
Wed, 6 Sep 2023 11:53:01 UTC (116 KB)
[v3]
Mon, 15 Sep 2025 17:32:56 UTC (111 KB)
[v4]
Wed, 11 Feb 2026 14:35:53 UTC (135 KB)
[v5]
Tue, 16 Jun 2026 05:57:22 UTC (118 KB)
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