De Finetti's Theorem in Categorical Probability

We present a novel proof of de Finetti's Theorem characterizing permutation-invariant probability measures of infinite sequences of variables, so-called exchangeable measures. The proof is phrased in the language of Markov categories, which provide an abstract categorical framework for probability and information flow. This abstraction allows for multiple versions of the original theorem to arise as consequences merely by interpreting the categorical result in different Markov categories. Moreover, the diagrammatic and abstract nature of the arguments makes the proof intuitive and easy to follow.