Dear all, I put a recent paper (just submitted) in the math Arxiv at http://front.math.ucdavis.edu/math.CT/0309209 Your comments are most welcome. Thanks. Vincent. *Title:* Flatness, preorders and general metric spaces *Author:* Vincent Schmitt <http://front.math.ucdavis.edu/author/Schmitt-V*> *Categories:* CT Category Theory <http://front.math.ucdavis.edu/math.CT> *Abstract:* This paper studies a general notion of 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 Flat_P(A) of P-flat presheaves over A. This completion is related to the free cocompletion of A under a class of colimits defined by Kelly. For a category A, for P = P0 the class of all presheaves, Flat_P0(A) is the Cauchy-completion of A. Two classes P1 and P2 of interest for general metric spaces are considered. The P1- and P2-flatness are investigated and the associated completions are characterized for general metric spaces (enrichemnts over R+) and preorders (enrichments over Bool). We get this way two non-symmetric completions for metric spaces and retrieve the ideal completion for preorders.