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 $0\to A\to B\to C\to 0$.
Best Answer
Janelidze and Street have described such a symmetric monoidal category over the nonnegative real numbers in
The arXiv version is here.