ScalingStacks

Proof. Let $\hat{Z}$ be the non-singular cover of $Z$ and $q:\hat{Z}\to Z$ be the quotient map.

Assume (i). Consider $z$, $U$, $K$ as in the lemma and let $\hat{\gamma}$ be a lift of $\gamma$. Consider $L_{i}\in\pi_{0}(U-\{z\})$ with $K\cap\overline{\gamma^{-1}(L_{i})}\neq\emptyset$ for $i\in\{1,2\}$. We have $\hat{\gamma}(K)\subset\overline{q^{-1}(L_{i})}$. Consequently, we have $\overline{q^{-1}(L_{1})}\cap\overline{q^{-1}(L_{2})}\neq\emptyset$. If $L_{1}$ and $L_{2}$ are not in the same $\iota$-orbit, then $\overline{q^{-1}(L_{1})}$ and $\overline{q^{-1}(L_{2})}$ are in distinct connected components of $\overline{q^{-1}(U)}$, a contradiction. So, (ii) holds.

Assume (ii). Since lifts of non-identity paths are unique if they exist (Lemma 7.1.20), it is enough to show the existence of lifts locally on $Z$. This is clear for a small open neighbourhood of a point of $Z-Z_{exc}$. Consider now $z\in Z_{exc}$ and a small open neighbourhood $U$ of $z$ in $Z_{o}$. Let $K$ be a connected component of $\gamma^{-1}(z)$ and let $W$ be the connected component of $\gamma^{-1}(U)$ containing $K$. There is $L\in\pi_{0}(U-\{z\})$ such that $\gamma(W)\subset L\cup\{z\}\cup\iota(L)$. Since $q$ splits over $L\cup\{z\}\cup\iota(L)$, it follows that the restriction of $\gamma$ to $W$ lifts to $\hat{Z}$. ∎