Actually you have just described the same thing twice. There are actually fewer distance-preserving maps than there are continuous ones, and restricting to distance-preserving maps removes all the isomorphisms between the sphere and the cube.
That is a very good point. Hmm. So it seems just plain false that you can break equivalence between two objects by enriching the number of maps between them?
Yes. If f and g are in the original category and are inverses of each other, the same will be true of any larger category (technically: any category which is the codomain of a functor whose domain is the original category).
That is a very good point. Hmm. So it seems just plain false that you can break equivalence between two objects by enriching the number of maps between them?
Yes. If f and g are in the original category and are inverses of each other, the same will be true of any larger category (technically: any category which is the codomain of a functor whose domain is the original category).