arXiv:1303.6576math.HO
The Foundations of Analysis
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Abstract
This is a detailed and self-contained introduction to the real number system from a categorical perspective. We begin with the categorical definition of the natural numbers, review the Eudoxus theory of ratios as presented in Book V of Euclid, and then use these classical results to define the positive real numbers categorically.
Along a similar vein you may also enjoy http://arxiv.org/pdf/1303.6576 The foundations of analysis by Larry Clifton. I always enjoy checking out the references in his papers as they are often hundreds of years old or more.
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