Free complete Wasserstein algebras

We present an algebraic account of the Wasserstein distances W_p on complete metric spaces, for p ≥ 1. This is part of a program of a quantitative algebraic theory of effects in programming languages. In particular, we give axioms, parametric in p, for algebras over metric spaces equipped with probabilistic choice operations. The axioms say that the operations form a barycentric algebra and that the metric satisfies a property typical of the Wasserstein distance W_p. We show that the free complete such algebra over a complete metric space is that of the Radon probability measures with finite moments of order p, equipped with the Wasserstein distance as metric and with the usual binary convex sums as operations.