M ar 2 00 4 Flatness , preorders and general metric spaces
نویسنده
چکیده
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 F latP(A) of P-flat presheaves over A. This completion is related to the free cocompletion under a class of colimits defined by Kelly. We define a notion of Q-accessible categories for a family Q of indexes. Our F latP (A) for small A's are exactly the Q-accessible categories where Q is the class of P-flat indexes. For a category A, for P = P0 the class of all presheaves, F latP 0 (A) is the Cauchy-completion of A. Two classes P1 and P2 of interest for general metric spaces are considered. The P1-and P2-flatness are investigated and the associated completions are characterized for general metric spaces (enrichments over ¯ IR+) 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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ar X iv : m at h / 06 02 46 3 v 1 [ m at h . C T ] 2 1 Fe b 20 06 Flatness , preorders and general metric spaces
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