  The quantale of distance distributions is of fundamental importance for
understanding probabilistic metric spaces as enriched categories. Motivated by
the categorical interpretation of partial metric spaces, we are led to
investigate the quantaloid of diagonals between distance distributions, which
is expected to establish the categorical foundation of probabilistic partial
metric spaces. Observing that the quantale of distance distributions w.r.t. an
arbitrary continuous t-norm is non-divisible, we precisely characterize
diagonals between distance distributions, and prove that one-step functions are
the only distance distributions on which the set of diagonals coincides with
the generated down set.

General Topology;Category Theory