  This paper studies a notion of parameterized flatness in the enriched
context: p-flatness where the parameter p stands for a class of presheaves. One
obtains a completion of a category A by considering the category F_p(A) of
p-flat presheaves over A. The completion is related to the free cocompletion
under a class of colimits defined by Kelly. We define a notion of Q-accessible
categories where Q is the class of p-flat indexes. For a category A, for p = P0
the class of all presheaves, F_P0(A) is the Cauchy-completion of A. Two classes
P1 and P2 of interest for general metric spaces and prorders are considered.
The F_P1- and F_P2- flatess are characterized yielding non-symmetric
completions of metric spaces a la Cauchy involving non-symmetric filters.
