2006/02/21 by Vincent Schmitt, Schmitt, Vincent
Computer Science · Mathematics · #Advanced Algebra and Logic #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Metric Geometry (math.MG) #Rings, Modules, and Algebras #math.CT #math.MG
paper · pdf · doi:10.48550/arxiv.math/0602463
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.)
arxiv created 2006/02/21 · openalex publication_date 2006/02/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We use a generic notion of flatness in the enriched context to define various completions of metric spaces -- enrichments over [0,∞] -- and preorders -- enrichments over 2. We characterize the weights of colimits commuting in [0,∞] 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.