ScalingStacks

Given $z\in Z_{o}$, we denote by $C(z)^{+}$ the set of $c\in C(z)$ such that there is an oriented path $\gamma$ in $Z$ with $m_{c}^{+}(\gamma)=1$. Note that $C(z)=C(z)^{+}\coprod\iota(C(z)^{+})$. Note also that given $\zeta$ an oriented homotopy class of paths in $Z$, we have

(7.3.2)

$$m_{c}(\zeta)e_{c}+m_{\iota(c)}(\zeta)e_{\iota(c)}=(m_{c}^{+}(\zeta)+m_{\iota(c)}^{-}(\zeta))e_{c}\text{ for }z\in Z_{o}\text{ and }c\in C(z)^{+}.$$