ScalingStacks

Given $n\geq 0$, we put $H_{n}^{\bullet}=({\mathfrak{S}}_{n})^{\mathrm{nil}}$. This is the quotient of the free pointed monoid generated by $T_{1},\ldots,T_{n-1}$ by the relations (3.2.1). The differential is given by $d(T_{i})=1$. Note that $k[H_{n}^{\bullet}]=H_{n}$ and $H_{n}^{\bullet}=\{0\}\cup\{T_{w}\}_{w\in{\mathfrak{S}}_{n}}$.

We define $\hat{{\mathfrak{S}}}_{n}^{\operatorname{nil}\nolimits}$ to be the differential graded pointed monoid with underlying differential pointed set $\{T_{\sigma}\}_{\sigma\in\hat{{\mathfrak{S}}}_{n}}\coprod\{0\}$ and multiplication, grading and differential that of $\hat{H}_{n}$.

We define $\hat{{\mathfrak{S}}}_{n}^{+,\operatorname{nil}\nolimits}$ to be its differential graded pointed submonoid with non-zero elements those that stabilize ${\mathbf{Z}}_{>0}$.