Exact weights, path metrics, and algebraic Wasserstein distances

We use weights on objects in an abelian category to define what we call a path metric. introduce three special classes of weight: those compatible with short exact sequences; induced by their metric; and which bound prove that these conditions are fact equivalent, such exact. As case metric, obtain distance for generalized persistence modules whose indexing is measure space. this Wasserstein distances, coincide the previously defined distances one-parameter modules. For modules, also describe maps from interval module, give matrix reduction monomorphisms epimorphisms.