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From: "V. Schmitt" <vs27@mcs.le.ac.uk>
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Subject: Preprint: Flatness, preorders and general metric spaces
Date: Thu, 18 Sep 2003 11:56:49 +0100
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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.