Flatness, preorders and general metric spaces
Flatness, preorders and general metric spaces
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发表时间：
2003-09
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通讯作者：
V. Schmitt
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作者：
V. Schmitt
本文在丰富的背景下研究了平坦性的一般概念：P-平坦性，其中参数P代表一类预层。通过考虑A上P-平坦预层的范畴Flat_P（A），得到范畴A的完备化。这个完备化与A在Kelly定义的一类余极限下的自由余完备化有关。对于范畴A，P = P0，即所有预层的类，Flat_P0（A）是A的柯西完备化。考虑了一般度量空间的两类P1和P2。研究了一般度量空间（R+上的增集）和预序（Bool上的增集）的P1-平坦性和P2-平坦性，并刻画了相应的完备化.这样我们得到了度量空间的两个非对称完备化，并得到了前序的理想完备化。
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.