0P9W Definition 7.3.9. We define 𝒮∙(Z,1){\mathcal{S}}^{\bullet}(Z,1) to be the pointed category with object set ZZ, with Hom𝒮∙(Z,1)(x,y)={0}⊔{admissible homotopy classes of paths x→y}\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z,1)}(x,y)=\{0\}\sqcup\{\text{admissible homotopy classes of paths }x\to y\} and αβ={α∘β if α∘β is admissible0 otherwise.\alpha\beta=\begin{cases}\alpha\circ\beta&\text{ if }\alpha\circ\beta\text{ is admissible}\\ 0&\text{ otherwise.}\end{cases}