Universality of persistence diagrams and the bottleneck and Wasserstein distances

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. For example, they can also be applied to multiparameter persistence modules.

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