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We define and study a probability monad on the category of complete metricspaces and short maps. It assigns to each space the space of Radon probabilitymeasures on it with finite first moment, equipped with theKantorovich-Wasserstein distance. This monad is analogous to the Giry monad onthe category of Polish spaces, and it extends a construction due to van Breugelfor compact and for 1-bounded complete metric spaces.  We prove that this Kantorovich monad arises from a colimit construction onfinite power-like constructions, which formalizes the intuition thatprobability measures are limits of finite samples. The proof relies on acriterion for when an ordinary left Kan extension of lax monoidal functors is amonoidal Kan extension. The colimit characterization allows the development ofintegration theory and the treatment of measures on spaces of measures, withoutmeasure theory.  We also show that the category of algebras of the Kantorovich monad isequivalent to the category of closed convex subsets of Banach spaces with shortaffine maps as morphisms.

A Probability Monad as the Colimit of Spaces of Finite Samples