What category theory can do for probability theory

The theory of Markov categories offers a synthetic approach to probability theory and statistics, in which measure-theoretic reasoning can often be replaced or reorganized by categorical arguments. In recent years, this perspective has led to purely categorical proofs of classical results in probability and statistics. In this talk, I will focus on another aspect of this story: how the categorical approach provides fresh insights into the nature of probability, its axiomatics, and its distinction from other theories of uncertainty. Particular highlights include simple and intuitive axioms for information flow, and a recent axiomatization of the formation of empirical distributions from infinite sequences of outcomes.