Corollary [efr-S87B]

The category \mathsf {TychDiffStoch} of Tychonoff diffeological spaces and Kleisli maps for the monad P described in Proposition [efr-7RS2] is a pullback-positive Markov category. Its deterministic category is \mathsf {TychDiff}. There is a fully faithful functor \mathsf {SmMfd} \to \mathsf {TychDiff} which preserves transverse pullbacks. The maps between smooth manifolds are given by weakly continuous families of Radon probability measures, so that the expectation operator carries smooth maps to smooth maps.