Markov categories have recently gained prominence as a categorical approach to probability and statistics. In this talk, I will argue that Markov categories provide a very general theory of information flow, and that this theory generalizes probability theory in a manner analogous to how topos theory generalizes set theory.

In the first part, I will sketch some theorems of probability and statistics which have already been developed synthetically in terms of Markov categories, including a version of the Blackwell-Sherman-Stein theorem which seems to be new even when instantiated in the traditional measure-theoretic framework. In the second part, I will sketch the vast and largely unexplored landscape of Markov categories on which these synthetic results can be instantiated. Some basic knowledge of monoidal categories and discrete probability should be enough to follow the talk.