Exact weights, path metrics, and algebraic Wasserstein distances

Understand

We use weights on objects in an abelian category to define what we call a path metric.

• We introduce three special classes of weight: those compatible with short exact sequences; those induced by their path metric; and those which bound their path metric.
• We prove that these conditions are in fact equivalent, and call such weights exact.
• As a special case of a path metric, we obtain a distance for generalized persistence modules whose indexing category is a measure space.