  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.
