  We use a generic notion of flatness in the enriched context to define various
completions of metric spaces -- enrichments over [0,\infty] -- and preorders --
enrichments over 2. We characterize the weights of colimits commuting in
[0,\infty] with the terminal object and cotensors. These weights can be
intrepreted in metric terms as peculiar filters, the so-called filters of type
1. This generalizes Lawvere's correspondence between minimal Cauchy filters and
adjoint modules. We obtain a metric completion based on the filters of type 1
as an instance of the free cocompletion under a class of weights defined by
G.M. Kelly. Another class of flat presheaves is considered both in the metric
and the preorder context. The corresponding completion for preorders is the
so-called dcpo-completion.
