Universality of persistence diagrams and the bottleneck and Wasserstein distances

We prove that persistence diagrams with the p-Wasserstein distance is the universal p-subadditive commutative monoid on an underlying metric space with a distinguished subset. This result applies to persistence diagrams, to barcodes, and to multiparameter persistence modules. In addition, the 1-Wasserstein distance satisfies Kantorovich-Rubinstein duality.