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I would like to know a natural categorification of the rig of integers $\mathbb{Z}$. This should be a $2$-rig. Among the various notions of 2-rigs, we obviously have to exclude those where $+$ is a coproduct (since otherwise $a+b \cong 0 \Rightarrow a \cong b \cong 0$). So let's take rig categories instead.

Question. What is a natural and non-discrete rig category whose rig of isomorphism classes is $\mathbb{Z}$?

I've read MO/3476, but the answers are not really satisfactory. Both the category of tangles and Schnauel's categories of polyhedra don't qualify.

Here is my approach. Notice that $\mathbb{N}$ is initial rig, so its categorification is the initial rig category, which turns out to be the groupoid of finite sets. The rig $\mathbb{Z}$ is the free rig on one generator $x$ subject to the relations $x^2=1$ and $x+1=0$. For short, $\mathbb{Z} = \mathbb{N}[x]/(x^2=1,x+1=0)$. This suggests that our categorification is $\mathsf{FinSet}[X]/(X^2 \cong 1,X+1 \cong 0)$, where these isomorphisms probably should satisfy some coherence condition (which are not visible in the decategorification $\mathbb{Z}$) in order to "flatten" the rig category. Namely, if $e : X^2 \to 1$ and $f : X+1 \to 0$ are the isomorphisms, we could require (here I omit the of "rig axiom" isomorphisms of the rig category) that $eX = Xe : X^3 \to X$ and $fX = Xf =f \circ (e+X): X^2+X \to 0$.

I think that $\mathsf{FinSet}[X,X^{-1}]$ should be the rig category of vector bundles on $\mathbb{P}^1_{\mathbb{F}_0}$, where $\mathbb{F}_0$ is the field with no element in the sense of Durov. The Serre twist $\mathcal{O}(1)$ is not inverse to itsself, but this changes when we consider the Möbius strip $H$ on $S^1$. But vector bundles in topology have too much morphisms (for the same reason we cannot take $\mathbb{P}^1_{\mathbb{C}}$).

Question. What is a natural realization of the rig category $\mathsf{FinSet}[X]/(X^2 \cong 1,X+1 \cong 0)$?

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Very disappointing that this question did not get an answer... –  Jacques Carette May 1 at 2:17
    
Has someone access to Joyal's paper "Règle des signes en algèbre combinatoire"? It seems to deal with this question. –  Martin Brandenburg May 28 at 13:12

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