ScalingStacks

We now define a groupoid of $n$-periodic bijections.

Given $I$ a subset of ${\mathbf{Z}}/n$ we denote by $\tilde{I}$ its inverse image in ${\mathbf{Z}}$.

Let ${\mathcal{S}}_{n}$ be the category with objects the subsets of ${\mathbf{Z}}/n$ and where $\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)$ is the set of $n$-periodic bijections $\sigma:\tilde{I}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\tilde{J}$. The group $n{\mathbf{Z}}$ acts by translation on $\operatorname{Hom}\nolimits$-sets. Note that $\hat{{\mathfrak{S}}}_{n}=\operatorname{End}\nolimits_{{\mathcal{S}}_{n}}({\mathbf{Z}}/n)$.

Given $i,j\in\tilde{I}$ with $i-j{\not\in}n{\mathbf{Z}}$, the element $s_{ij}\in\hat{{\mathfrak{S}}}_{n}$ restricts to an $n$-periodic bijection $\tilde{I}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\tilde{I}$, which we also denote by $s_{ij}$.

Let $I$ be a subset of ${\mathbf{Z}}/n$. There is a unique increasing bijection $\beta_{I}:\{1,\ldots,|I|\}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\tilde{I}\cap\{1,\ldots,n\}$. We extend it to an increasing bijection ${\mathbf{Z}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\tilde{I}$ by $\beta_{I}(r+d|I|)=\beta_{I}(r)+dn$ for $r\in\{1,\ldots,|I|\}$ and $d\in{\mathbf{Z}}$. There is an isomorphism of groups