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?