ScalingStacks

Remark 7.3.10. Consider $\Pi_{o}(Z)$ the category with objects the points of $Z$ and arrows the oriented homotopy classes of paths, a subcategory of $\Pi(Z)$. We define a ${\mathbf{Z}}_{\geq 0}$-filtration on $\Pi_{o}(Z)$ by defining a class $\zeta$ to have degree $\leq d$ if it is the product of $d+1$ admissible homotopy classes of paths. The category ${\mathcal{S}}^{\bullet}(Z,1)$ is isomorphic to the degree $0$ part of ${\operatorname{gr}\nolimits}\Pi_{o}(Z)$.

Note finally that if $Z$ is non-singular, then ${\mathcal{S}}^{\bullet}(Z,1)$ is the pointed category associated to $\Pi_{o}(Z)$.