Flatness, preorders and general metric spaces (revised)
Abstract
We use a generic notion of flatness in the enriched context to define various completions of metric spaces -- enrichments over [0,\infty] -- and preorders -- enrichments over 2. We characterize the weights of colimits commuting in [0,\infty] with the terminal object and cotensors. These weights can be intrepreted in metric terms as peculiar filters, the so-called filters of type 1. This generalizes Lawvere's correspondence between minimal Cauchy filters and adjoint modules. We obtain a metric completion based on the filters of type 1 as an instance of the free cocompletion under a class of weights defined by G.M. Kelly. Another class of flat presheaves is considered both in the metric and the preorder context. The corresponding completion for preorders is the so-called dcpo-completion.
Cite
@article{arxiv.math/0602463,
title = {Flatness, preorders and general metric spaces (revised)},
author = {Vincent Schmitt},
journal= {arXiv preprint arXiv:math/0602463},
year = {2007}
}
Comments
This a much improved version of the earlier drafts math.CT/0309209 and math.CT/0403164. It is now merely an application to metric spaces of the theory developed in math.CT/0501383 (that appeared in print in TAC 2005.)