Flatness, preorders and general metric spaces (revised)

Details Flatness, preorders and general metric spaces (revised) Flatness, preorders and general metric spaces (revised)

2006-02-21

arxiv.org/abs/math/0602463v1

Mathematics

Category Theory

This a much improved version of the earlier drafts math.CT/0309209 and math.CT/0403164. It is now merely an application to met

Scientific paper

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.

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