I understand that in the natural numbers, the sum of two numbers can be readily thought of as the disjoint union of two finite sets.
John Baez even spent a week talking about how you can extend this idea to thinking about the integers here: TWF 102. This led into a discussion of the homotopy groups of spheres.
But then you have to pass to a certain colimit to get the rationals, and take a certain completion to get the reals. It all gets very complicated.
One place where we rely on a correspondence between a sum of real numbers and a certain coproduct is in measure theory-- perhaps in analogy to the relation between finite sets and natural numbers, we should think of some measure space as the categorification of the real numbers. But this sounds unpromising-- what space would be in any sense a canonical categorification? Moreover, what I was really hoping for originally was a precise sense in which the $\sigma$-additivity of a measure states that it preserves coproducts or something, so I was hoping there might be more to it than sigma-algebras.