ScalingStacks

Given $I,J\subset{\mathbf{Z}}/n$, we define $\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}^{\geq-r}}(I,J)=\{\sigma\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)\ |\ l(\sigma)\leq r\}$ for $r\in{\mathbf{Z}}_{\geq 0}$. It follows from Lemma 6.2.1 that this defines a structure of ${\mathbf{Z}}_{\leq 0}$-filtered category on ${\mathcal{S}}_{n}$. We put ${\mathcal{H}}_{n}=\mathrm{gr}{\mathcal{S}}_{n}^{\bullet}$, a pointed ${\mathbf{Z}}_{\leq 0}$-graded category.

Note that a map $\sigma$ of length $0$ is invertible in ${\mathcal{H}}_{n}$. Note also that $F_{I}$ induces an isomorphism of graded pointed monoids $\hat{{\mathfrak{S}}}_{|I|}^{\operatorname{nil}\nolimits}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{End}\nolimits_{{\mathcal{H}}_{n}}(I)$.