Kleisli category of the probability-sheaves monad is Markov and contains C

Establish that the Kleisli category Kl(ℳ) of the monad ℳ on the category of probability sheaves Sh(S(C)) is a Markov category and that the original Markov category C embeds into Kl(ℳ), acknowledging that conditionals may fail to exist in Kl(ℳ).

Background

The paper develops categories of sample spaces S(C) for any Markov category C with conditionals and builds a theory of probability presheaves and atomic sheaves over S(C). A monad ℳ on probability sheaves is introduced to model allocation of fresh random variables, generalizing the name-generation monad N on the Schanuel topos.

Determining whether the Kleisli category of this monad inherits the full structure of a Markov category, and whether the original C embeds into it, would clarify how probabilistic generativity interacts with the categorical structures studied in the paper. The authors anticipate that conditionals may not be preserved in the larger Kleisli category.