Categorifications of the real numbers For the purposes of this question, a categorification of the real numbers is a pair $(\mathcal{C},r)$ consisting of:


*

*a symmetric monoidal category $\mathcal{C}$

*a function $r\colon \mathrm{ob}(\mathcal{C})\to\mathbb{R}$


such that:


*

*$r(X\otimes Y) = r(X) r(Y)$ for all objects $X$ and $Y$ of $\mathcal{C}$ 

*$r(\mathbb{1}) = 1$, where $\mathbb{1}$ is the monoidal unit

*$X\cong X'\implies r(X)=r(X')$


Some examples of categorifications of $\mathbb{R}$ are: finite sets and cardinality; finite-dimensional vector spaces and dimension; topological spaces (with the homotopy type of a CW-complex, say) and Euler characteristic.  However, in these examples the map $r$ factors through $\mathbb{N}$ or $\mathbb{Z}$.  I am interested in examples where the values of $r$ are not so restricted.

Question: What categorifications of $\mathbb{R}$ are there where $r$ can take all values in $\mathbb{R}$, or perhaps all values in $(0,\infty)$ or $(1,\infty)$?

I am especially interested in examples that already appear somewhere in the mathematical literature.  I am also especially interested in examples where $\mathcal{C}$ is symmetric monoidal abelian and r(A)+r(C) = r(B) for every short exact sequence $0\to A\to B\to C\to 0$.
 A: Via the notion of groupoid cardinality (generalizing ordinary set-theoretic cardinality), finite non-empty groupoids can be seen as a categorification of the positive rationals, and more general groupoids as a categorification of the non-negative reals:


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*John Baez and James Dolan, From Finite Sets to Feynman Diagrams, in Mathematics Unlimited—2001 and Beyond, Springer, 2001. (arXiv link)


An example that Baez and Dolan discuss is the groupoid of finite sets and bijections, which has cardinality $e$. (The notion of Euler characteristic of a category, mentioned in Gregory Arone's answer, can be seen as a further generalization of groupoid cardinality, see Example 2.7 of Leinster's article.)
A: Can we do an example along these lines:
An appropriate collection of metric spaces, with $\otimes$ the Cartesian product, and $r$ a fractal dimension?  Or more precisely, $r(X) = \exp(\dim(X))$.
Perhaps the arrows are weakly contracting $d(f(x),f(y) \le d(x,y)$.  And perhaps we want the finite-dimenaional (so that $\dim(X) = \infty$ is disallowed) fractals in the sense of Taylor (so that $\dim(X \times Y) = \dim(X)+\dim(Y)$).  
A: Janelidze and Street have described such a symmetric monoidal category over the nonnegative real numbers in 


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*George Janelidze and Ross Street, Real sets, Tbilisi Math. J., Volume 10, Issue 3 (2017), 23-49. (link)


The arXiv version is here. 
A: If you drop the condition that your category is symmetric monoidal, and content yourself with a category that's merely monoidal, then the tensor category of $R$-$R$-bimodules for $R$ a factor (factor = von Neumann algebra with trivial center) is an example .
The "quantum dimension" or "statistical dimension" of an $R$-$R$-bimodule is an $\mathbb R_{\ge 0}\cup\{\infty\}$-valued invariant with all the properties that you want.
See, e.g., my paper: https://arxiv.org/abs/1110.5671, specifically Proposition 5.2.
A: An example might be provided by various genera. A genus is not only multiplicative with respect to the Cartesian product of manifolds but also additive with respect to their disjoint sum, and is not only isomorphism but also cobordism invariant. Some genera do not factor through $\mathbb Z$ -- for example the $\hat A$-genus takes non-integer values on some non-spin manifolds.
I don't know though whether there is a genus which attains all real values. The $\Gamma$-genus of Morava (arXiv:1101.1647) seems to have lots of non-rational values but I am too ignorant to say anything trustworthy about it.
A: In a somewhat similar spirit to მამუკა ჯიბლაძე's answer: generalisations of Euler characteristic. As mentioned by the OP, classical Euler characteristic associates an integer to a finite complex in a way that is additive with respect to unions and multiplicative with respect to products. There are variations that apply not just to finite complexes (some are described in The Euler characteristic of a category (arXiv:math/0610260), by Tom Leinster). The extended Euler characteristic may have values that are not integer or even rational (for example, the Euler characteristic of the symmetric groupoid is $e$). But it is still a "ring map".
