Probability monads with submonads of deterministic states

Probability theory can be studied synthetically as the computational effect embodied by a commutative monad. In the recently proposed Markov categories, one works with an abstraction of the Kleisli category and then defines deterministic morphisms equationally in terms of copying and discarding. The resulting difference between 'pure' and 'deterministic' leads us to investigate the 'sober' objects for a probability monad, for which the two concepts coincide. We propose natural conditions on a probability monad which allow us to identify the sober objects and define an idempotent sobrification functor. Our framework applies to many examples of interest, including the Giry monad on measurable spaces, and allows us to sharpen a previously given version of de Finetti's theorem for Markov categories. This is an extended version of the paper accepted for the Logic In Computer Science (LICS) conference 2022. In this document we include more mathematical details, including all the proofs, of the statements and constructions given in the published version.

About citing this work. All the definitions, propositions, and theorems appearing in the published version also appear here, with the same numbering as in the published version. There is one result here, Lemma 3.18, not present in the published version. The numbering of particular equations is however inevitably different between the two versions. Because of this, if future readers need to refer to any of the equations contained here, we recommend them to refer to the corresponding definition or theorem instead.