For any small quantaloid \Q, there is a new quantaloid \D(\Q) of
diagonals in \Q. If \Q is divisible then so is \D(\Q) (and vice versa),
and then it is particularly interesting to compare categories enriched in \Q
with categories enriched in \D(\Q). Taking Lawvere's quantale of extended
positive real numbers as base quantale, \Q-categories are generalised metric
spaces, and \D(\Q)-categories are generalised partial metric spaces, i.e.\
metric spaces in which self-distance need not be zero and with a suitably
modified triangular inequality. We show how every small quantaloid-enriched
category has a canonical closure operator on its set of objects: this makes for
a functor from quantaloid-enriched categories to closure spaces. Under mild
necessary-and-sufficient conditions on the base quantaloid, this functor lands
in the category of topological spaces; and an involutive quantaloid is
Cauchy-bilateral (a property discovered earlier in the context of distributive
laws) if and only if the closure on any enriched category is identical to the
closure on its symmetrisation. As this now applies to metric spaces and partial
metric spaces alike, we demonstrate how these general categorical constructions
produce the "correct" definitions of convergence and Cauchyness of sequences in
generalised partial metric spaces. Finally we describe the Cauchy-completion,
the Hausdorff contruction and exponentiability of a partial metric space, again
by application of general quantaloid-enriched category theory.Comment: Apart from some minor corrections, this second version contains a
revised section on Cauchy sequences in a partial metric spac