We introduce a categorical framework for higher-order probability theory based on quasi-M-spaces and Frölicher M-spaces. Besides Giry's and Forré's probability monads, we introduce our own hybrid pushforward probability monad generated by randomizations of a fixed reference measure. These constructions are canonically transported to saturated Frölicher M-spaces, yielding, in the measurable setting, Giry and quasi-probability monads on the corresponding Frölicher spaces. In the smooth setting, the resulting spaces of probability measures carry a natural smooth structure, which can be related to the continuity equation, Otto calculus and Lions' derivative.
This is joint work with Philipp Harms, Peter W. Michor and Josef Teichmann.