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Is there a standard/good reference text that does category of metric spaces?

Say, it seems that by looking at this category one can recover everything about particular metric space up to scaling --- is it written somewhere, or does it follow from something standard?

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    $\begingroup$ I had never seen "metric maps" but they're well often called "1-Lipschitz maps". $\endgroup$
    – YCor
    Commented Feb 11, 2020 at 9:05
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    $\begingroup$ The obvious choice of morphisms between metric space is continuous functions, but then you have the category of metrisable topological spaces. Elsewhere, morphisms are supposed to preserve structure, which is what 1-Lipschitz maps do. $\endgroup$
    – Paul Taylor
    Commented Feb 11, 2020 at 9:41
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    $\begingroup$ Bill Lawvere has a way to look at this as a particular case of enriched categories. See tac.mta.ca/tac/reprints/articles/1/tr1abs.html and the nLab entry on the subject. $\endgroup$
    – D.-C. Cisinski
    Commented Feb 11, 2020 at 12:53
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    $\begingroup$ @Denis-CharlesCisinski I don't think the Lawvere paper, seminal though it might be (waves to the Edinburgh magnitude crew) serves well as a reference for the category that the OP is asking for. I assume (but Anton Petrunin can correct me if I am wrong) that what is desired is a reference which will state the properties of this category (existence and nature of finite limits/colimits) and its relation with other categories via various adjoint functors $\endgroup$
    – Yemon Choi
    Commented Feb 11, 2020 at 15:46
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    $\begingroup$ The question needs more focus. $\endgroup$
    – Martin Brandenburg
    Commented Feb 14, 2020 at 13:14

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