Subsection 7.3.2. (0PHB) · ScalingStacks

7.3.2. Pointed category of admissible paths

We now define a category associated with admissible paths.

Definition 7.3.9. We define ${\mathcal{S}}^{\bullet}(Z,1)$ to be the pointed category with object set $Z$ , with

\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z,1)}(x,y)=\{0\}\sqcup\{\text{admissible homotopy classes of paths }x\to y\}

and

\alpha\beta=\begin{cases}\alpha\circ\beta&\text{ if }\alpha\circ\beta\text{ is admissible}\\ 0&\text{ otherwise.}\end{cases}

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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 $\Pi_{o}(Z)$ .

We put ${\mathcal{S}}(Z,1)={\mathbf{F}}_{2}[{\mathcal{S}}^{\bullet}(Z,1)]$ .

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Example 7.3.11. We describe below some examples of products in ${\mathcal{S}}^{\bullet}(Z,1)$ . Here $Z$ is the third singular curve of example 7.2.11 and the paths are drawn in the smooth cover.