A Probability Monad as the Colimit of Spaces of Finite Samples

Understand

We define and study a probability monad on the category of complete metric spaces and short maps.

• It assigns to each space the space of Radon probability measures on it with finite first moment, equipped with the Kantorovich-Wasserstein distance.
• This monad is analogous to the Giry monad on the category of Polish spaces, and it extends a construction due to van Breugel for compact and for 1-bounded complete metric spaces.
• We prove that this Kantorovich monad arises from a colimit construction on finite power-like constructions, which formalizes the intuition that probability measures are limits of finite samples.