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Suraaj K S (Oct 22 2024 at 22:02):I was reading https://people.mpi-sws.org/~dreyer/tor/papers/reynolds.pdf, which is supposed to be a seminal paper in computer science. In section 1, there is a 'fable' about two different implementations of Complex Numbers - one using cartesian coordinates, and the other using polar coordinates. And if I understand the paper correctly, I think that the paper argues that one can think of "Complex Numbers" as the set of all possible implementations.
This reminded me a little bit about universal mapping properties, which, actually does define a set of objects (which are going to be isomorphic in the category of course). For instance, we don't care how we define the natural numbers, as long as the definition satisfies that the natural numbers satisfy the UMP of natural numbers. The "Complex number" example given in the paper seems, at least to me, quite similar in spirit.
I was wondering if there is a connection, and whether we could characterize such types as those satisfying some universal mapping property?