So one way to categorify the natural numbers is to replace them with vector spaces. Then the dimension of the vector space reproduces the natural number. More generally you can categorify integers to graded vector spaces. Also a linear monoidal category (assuming some finiteness conditions) will lead to an algebra defined over the natural numbers and so can be viewed as a categorification of the this algebra.
Now one thing that I found intriguing when I learned about factor von Neumann algebras is that the type II_1 factor has modules which have "dimensions" which land in the (positive) real numbers. Has anyone ever seen a categorification of some real number quantity by using these sorts of modules? It seems that graded modules (or complexes of modules) would then give all real numbers.
Is there some sort of categorification of real algebras to a monoidal category enriched over II_1-modules? Or some other type of categorification of the reals which I am not even guessing at?