ScalingStacks

Let $W^{\operatorname{nil}\nolimits}$ be the pointed ${\mathbf{Z}}_{\leq 0}$-graded monoid with underlying pointed set $\{T_{w}\}_{w\in W}\coprod\{0\}$ and multiplication given by (3.1.1). This is the pointed monoid ${\operatorname{gr}\nolimits}W$ associated to the filtration on $W$ given by $W^{\geq-i}=\{w\in W\ |\ \ell(w)\leq i\}$ and there is an identification ${\mathbf{F}}_{2}[W^{\operatorname{nil}\nolimits}]=H^{\operatorname{nil}\nolimits}(W)$ making $W^{\operatorname{nil}\nolimits}$ into a differential graded pointed monoid.