Multiple categorifications of structures I recently read about how the category of finite sets and the category of finite-dimensional vector spaces are both categorifications of the natural numbers. I was wondering if there are any other examples of structures being categorified by multiple categories and if this kind of thing is common.
 A: You could take a categorification of an abelian group $M$ to be an exact category $\mathscr{M}$ such that $K_0(\mathscr{M}) \cong M$, where $K_0(-)$ denotes the Grothendieck group of an exact category. Then the exact category of finitely generated projective modules $\mathscr{M}$ over any principal ideal domain is the categorification of the integers $\mathbb{Z}$ since any finitely generated projective module over a pid is free, so $K_0(\mathscr{M}) \cong\mathbb{Z}$.
To obtain more useful results, you could strengthen your definition of categorification in various ways. I suggest looking at Alistair Savage's notes, Introduction to Categorification.
A: There are lots and lots of examples of this phenomena. Here's a fairly standard example.
Let $\mathsf M_n(\Bbb C)$ be the category whose objects are $n\times n$ matrices over $\Bbb C$. A morphism $X:A\to B$ in $\mathsf M_n(\Bbb C)$ is an $n\times n$ matrix $X$ over $\Bbb C$ such that $XA=BX$. Note that the isomorphism classes of objects in $\mathsf M_n(\Bbb C)$ are the conjugacy classes.
Now, we have obvious set-maps
\begin{align*}
\det&:\DeclareMathOperator{Ob}{Ob}\Ob(\mathsf M_n(\Bbb C))\to\Bbb C
&
\DeclareMathOperator{trace}{trace}\trace&:\Ob(\mathsf M_n(\Bbb C))\to\Bbb C
\end{align*}
The fact that the determinant and trace are invariant under conjugation implies that these maps both categorify the complex numbers.
