#
A probability monad as the colimit of spaces of finite samples

##
Tobias Fritz and Paolo Perrone

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. 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.

Keywords:
Categorical probability, Giry monad, graded monad, optimal transport,
Wasserstein spaces, Kantorovich-Rubinstein distance, monoidal Kan
extension

2010 MSC:
60A05, 18C15, 52A01

*Theory and Applications of Categories,*
Vol. 34, 2019,
No. 7, pp 170-220.

Published 2019-03-11.

http://www.tac.mta.ca/tac/volumes/34/7/34-07.pdf

TAC Home