Categorification of the integers 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 the initial rig, so its categorification should be the initial rig category, which turns out to be the groupoid of finite sets and bijections, which is equivalent to the  permutation groupoid $\mathbb{P}$. 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 $\mathbb{P}[X]/(X^2 \cong 1,X+1 \cong 0)$, 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 coherence 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 $\mathbb{P}[X,X^{-1}]$ should be the rig category of vector bundles on the projective space $\mathbb{P}^1_{\mathbb{F}_0}$, where $\mathbb{F}_0$ is the "field with no element" in the sense of Durov. Here, 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 $\mathbb{P}[X]/(X^2 \cong 1,X+1 \cong 0)$?
 A: I think the answer is given by the free Picard groupoid $\mathbb{S}$ on one object. This is the $1$-truncation of the sphere spectrum, keeping only its first two homotopy groups $\mathbb{Z}$ and $\mathbb{Z}_{2}$, which is defined and studied in Section 3 of

Niles Johnson, Angélica M. Osorno, Modeling stable one-types. Theory Appl. Categ. 26 (2012), No. 20, 520–537; arXiv:1201.2686.

The Picard groupoid $\mathbb{S}$ is defined as follows:

*

*The objects of $\mathbb{S}$ are the integers;

*For $m,n\in\mathrm{Obj}(\mathbb{S})$, we have
$$
\mathrm{Hom}_{\mathbb{S}}(m,n)
\overset{\mathrm{def}}{=}
\begin{cases}
    \emptyset         &\text{if $m\neq n$,}\\
    \mathbb{Z}_{2}    &\text{if $m=n$;}
\end{cases}
$$

*The monoidal structure $\oplus$ on $\mathbb{S}$ is given by addition of integers;

*The symmetry of $\mathbb{S}$ at $(A,B)$ is the morphism $\beta^{\mathbb{S},\oplus}_{m,n}\colon m\oplus n\to n\oplus m$ defined by
$$
\beta^{\mathbb{S},\oplus}_{m,n}
\overset{\mathrm{def}}{=}
\begin{cases}
    0             &\text{if $mn$ is even,}\\
    \eta_{m+n}    &\text{if $mn$ is odd,}
\end{cases}
$$
where $\eta_{m+n}$ is the unique non-zero element of $\mathrm{Hom}_{\mathbb{S}}(n,n)$.

There are at least three senses in which this is the right answer:

*

*The first one is that it truly is the $1$-truncation of the sphere spectrum:

Proposition 3.3 of Johnson–Osorno. There is a map of
$\mathbb{E}_{\infty}$-ring spaces
$$\overline{\mathrm{B}\xi}\colon QS^{0}\longrightarrow\mathrm{B}\mathbb{S}$$
witnessing $\mathrm{B}\mathbb{S}$ as the Postnikov $1$-truncation of $QS^{0}$.



*There second one relates to $2$-rigs and $2$-rings: the statement

Rigs and rings are monoids in $(\mathsf{CMon},\otimes_{\mathbb{N}},\mathbb{N})$ and $(\mathsf{Ab},\otimes_{\mathbb{Z}},\mathbb{Z})$.

categorifies  to

$2$-rigs and $2$-rings are pseudomonoids in $(\mathsf{SymMonCats},\otimes_{\mathbb{F}},\mathbb{F})$ and $(\mathsf{2Ab},\otimes_{\mathbb{S}},\mathbb{S})$,

where $\mathbb{F}$ is the symmetric monoidal category of finite sets and bijections, the monoidal unit of the tensor product $\otimes_{\mathbb{F}}$ of symmetric monoidal categories, and $\mathbb{S}$ is the Picard groupoid of Johnson–Osorno, which is again the monoidal unit of the tensor product $\otimes_{\mathbb{S}}$ of Picard groupoids.


*Finally, we have an analogue of Baez's conjecture for  $\mathbb{S}$. In Remark 3.2 of their paper, Johnson–Osorno note that $\mathbb{S}$ admits also a second monoidal structure given by
\begin{align*}
(m,n) &\mapsto mn,\\
(m\xrightarrow{f}m,n\xrightarrow{g}n) &\mapsto nf+mg,
\end{align*}
with symmetry isomorphism $\beta^{\mathbb{S},\otimes}_{m,n}\colon m\otimes n \to n\otimes m$ defined by
$$
\beta^{\mathbb{S},\otimes}_{m,n}
\overset{\mathrm{def}}{=}
\begin{cases}
0             &\text{if ${m\choose 2}{n\choose 2}$ is even,}\\
\eta_{nm}     &\text{if ${m\choose 2}{n\choose 2}$ is odd.}
\end{cases}
$$
We then have:

Just as $\mathbb{N}$ is the initial rig and $\mathbb{F}$ is the initial $2$-rig, the ring $\mathbb{Z}$ is the initial ring and $\mathbb{S}$ is the initial $2$-ring.

A: It should definitely be mentioned here that one often-used categorification of the integers is the sphere spectrum, 
$\mathbb S$, i.e. the infinite loop space $\mathbb S = \varinjlim \Omega^n S^n$. I've heard this idea attributed to Waldhausen. There's a fun exposition in TWF 102 by John Baez.
The idea is to think of $\mathbb Z$ is the (additive) group completion of the rig $\mathbb N$, and to categorify the group completion process. A natural categorification of the rig $\mathbb N$ is the 2-rig $\mathsf{FinBij}$ of finite sets and bijections, with $\times$ distributing over $\amalg$. So we group complete $\mathsf{FinBij}$ the $\amalg$ part. The thing is, rather than creating a 1-groupoid with two monoidal structures and the appropriate universal property, it's more natural to regard $\mathsf{FinBij}$ as an $\infty$-groupoid with two monoidal structures, and to construct a group completion in the world of $\infty$-groupoids. But by the homotopy hypothesis, an $\infty$-groupoid is the same thing as a space. So we end up with a space that has two monoidal structures, one distributing over the other. The "additive" monoidal structure is appropriately commutative, so this structure can be regarded as the structure of an infinite loop space, otherwise known as a connective spectrum. And the multiplicative structure makes it a "multiplicative infinite loop space", otherwise known as a ring spectrum.
Note that $\pi_0(\mathbb S) = \mathbb Z$, so this is a categorification in the most basic sense.
