Flatness, preorders and general metric spaces

2003/09/12 by Vincent Schmitt, Schmitt, Vincent · 1 citation
Mathematics · #Advanced Topology and Set Theory #Homotopy and Cohomology in Algebraic Topology #Rings, Modules, and Algebras #math.CT

paper · pdf · doi:10.48550/arxiv.math/0309209

arxiv created 2003/09/12 · arxiv updated 2009/12/01

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 general 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.

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