Probability monads with submonads of deterministic states - Extended version

TL;DR

This paper investigates the structure of probability monads in Markov categories, identifying sober objects where deterministic and pure morphisms coincide, and extends de Finetti's theorem within this framework.

Contribution

It introduces conditions for probability monads to identify sober objects and defines an idempotent sobrification functor, enhancing the understanding of deterministic states in probabilistic effects.

Findings

Identified conditions for sober objects in probability monads.

Defined an idempotent sobrification functor for these monads.

Extended de Finetti's theorem within the Markov category framework.

Abstract

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.