Post #2729846

An interesting question here is whether there exists such a category where \(\mathcal{C}_\mathrm{det} \cong \mathsf{Set}\), or more generally a commutative monad on \(\mathsf{Set}\) so that the Kleisli category has Kolmogorov products. The Vitali sets prove that it can't be given by the ordinary distribution monad on countable sets, but doesn't rule out some more exotic construction. It is interesting to ask which, say, toposes, admit a probability monad with Kolmogorov products which behaves "as expected", maybe formalized by saying the distributions on the natural numbers object should all be discrete, i.e given by a point of the countable-dimensional simplex.