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I have often (on the math.se main page) encountered the idea that, "A structure must never be viewed in isolation, but always as an object of some category." This doesn't make sense to me, since:

  • We can develop a lot of the theory of metric spaces without mentioning morphisms between metric spaces, and
  • Metric spaces can naturally be viewed as the objects of more than one category, including the category whose morphisms are uniformly continuous functions, and the category whose morphisms are Lipschitz functions

What would be an acceptable way of opening a question about this topic?

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    $\begingroup$ It seems like you basically already have a very reasonable soft-question question inquiring about what is meant by the idea in the opening paragraph. I think finding an explicit statement of this idea would strengthen the question as a whole, but it may not be necessary. Find a place to insert a question mark, and you're probably good to go. $\endgroup$ – user642796 May 12 '13 at 16:03
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    $\begingroup$ I wish you'd phrase your question in a less "straw man-y" way. One must never do X is nicely teed up for a refutation, almost independently of what X is. In this case: sure, there are some parts of mathematics in which categorical ideas are absent and cannot obviously be imported in a useful way (e.g. a lot of classical analysis). If that's the point you're trying to make, I think you can stop now and claim it. If you're getting at something else, I think you should work harder at bringing out what you really want to know. $\endgroup$ – Pete L. Clark May 12 '13 at 17:00

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