  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.
