  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. We use this distance to define Wasserstein distances, which
coincide with the previously defined Wasserstein distances for one-parameter
persistence modules. For one-parameter persistence modules, we also describe
maps to and from an interval module, and we give a matrix reduction for
monomorphisms and epimorphisms.
