ScalingStacks

Definition 7.3.6. Given two smooth non-identity homotopy classes of paths $\zeta_{1}$ and $\zeta_{2}$ contained in the same component of $Z$, there is a unique $\varepsilon\in\{\pm 1\}$ such that there is a minimal smooth path $\gamma$ in $Z$ with the property that $\zeta_{1}$ and $\zeta_{2}^{\varepsilon}$ are equal to the classes of restrictions of $\gamma$. We say that $\zeta_{1}$ and $\zeta_{2}$ have the same orientation (resp. opposite orientation) if $\varepsilon=1$ (resp. $\varepsilon=-1$).