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I don't understand. The whole point of the manifold definition is that we want to ignore some information and preserve other information. If you want a notion of angle, you should be working in an inner product space.
Fair enough. I do like that you mentioned universal properties, since my own response to this question is basically "categorify until it becomes obvious what to do." For example, the product of zero things in a category is a terminal object and the coproduct of zero things is an initial object.
Hmm. I meant Ring, but I may have to take that back. It looks like the standard category-theoretic description of a ring is as a monoid object in Ab, so we need to require that the functors preserve the monoid structure.
Poonen claimed, and I agree, that the determinant of a 0x0 matrix should be equal to 1. Consider what happens when you try to expand the determinant of a 1x1 matrix by minors.
I would argue as follows. If you're a combinatorialist who accepts that 0! = 1, you accept that there is one bijection from the empty set to itself, so you accept that there is one function from the empty set to itself, so you should accept that 0^0 = 1.
Darn. You probably understand why I asked this, though: this seems to be true of the standard concrete categories students are first introduced to, such as Grp, Rng, Top, ... are these categories characterized by a stronger property than concreteness?