Flatness, preorders and general metric spaces

TL;DR

This paper introduces a generalized notion of flatness called P-flatness within enriched categories, providing new completions for categories, metric spaces, and preorders, extending classical notions like Cauchy and ideal completions.

Contribution

It defines P-flatness for enriched categories, explores associated completions, and characterizes new non-symmetric completions for metric spaces and preorders.

Findings

Introduces P-flatness as a generalization of flatness in enriched categories.

Characterizes completions of categories, metric spaces, and preorders using P-flat presheaves.

Retrieves known completions like Cauchy and ideal completions as special cases.

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.