TITLE: Categorifications of the real numbers
QUESTION [27 upvotes]: 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 $0\to A\to B\to C\to 0$.

REPLY [5 votes]: 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:

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.)

REPLY [5 votes]: 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 $\dim(X \times Y) = \dim(X)+\dim(Y)$).