Title: A Probability Monad as the Colimit of Finite Powers,
Abstract: 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 powers, which formalizes the intuition that probability measures are
limits of finite samples. The proof relies on a criterion for when an ordinary
left Kan extension of lax monoidal functors is a monoidal Kan extension. The
colimit characterization allows the development of integration theory and the
treatment of measures on spaces of measures, without measure theory.
We also show that the category of algebras of the Kantorovich monad is
equivalent to the category of closed convex subsets of Banach spaces with short
affine maps as morphisms.