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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?

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In a sense we already have a great categorification of the complex numbers, and that's given by FdHilb, the *-category of finite-dimensional Hilbert spaces and continuous maps. Results like the Doplicher-Roberts theorem give us good reasons to believe this. So from this perspective, we don't need to go as far as looking at fancy von Neumann algebras to get what you want.

One point I'm implicitly making here is that categorification isn't necessarily the inverse process to taking isomorphism classes!

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Could you expand on this a bit? How does one get a complex number out of a finite dimensional Hilbert space? –  Charles Rezk Oct 22 '09 at 17:08
    
You don't! That's what I mean by "categorification isn't necessarily the inverse process to taking isomorphism classes". The point is that the category FdHilb has lots of properties which are formal categorifications of properties of C, and so it's arguable that FdHilb is a good categorification of C. John Baez's paper on 2--Hilbert spaces is all about this, which you can find at arxiv.org/abs/q-alg/9609018 . –  Jamie Vicary Oct 28 '09 at 15:50
    
I'm asking about categorification in the inverse process sense. –  Chris Schommer-Pries Dec 29 '09 at 21:10
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One categorification of the reals that I know of is via groupoids; see Aleks Kissinger's comment on the groupoid question as well as the accompanying link.

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Hi Chris,

that's a really good question, I think. When you find out anything stable, I'd be grateful if you could drop me a note somehow.

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Will do. Of course you can always mark it as a favorite question so you can easily return to it latter... –  Chris Schommer-Pries Oct 22 '09 at 18:55
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