Before revision: We undertake a formal study of persistence diagrams and their metrics. We show that barcodes and persistence diagrams together with the bottleneck distance and the Wasserstein distances are obtained via universal constructions and thus have corresponding universal properties. In addition, the 1-Wasserstein distance satisfies Kantorovich-Rubinstein duality. Our constructions and results apply to any metric space with a distinguished basepoint.
After revision: We prove that persistence diagrams with the p-Wasserstein distance form the universal p-subadditive commutative monoid on an underlying metric space with a distinguished basepoint.