Lemma 1

Let B be a topological space locally connected by paths, let $p: E\rightarrow B$ and $p': E'\rightarrow B$ be étale and separated mappings satisfying the path-lifting property (III, p. 302). Let $g: E\rightarrow E'$ be a mapping such that $p'\circ g=p$. Suppose that $g$ is compatible with the canonical operations of the groupoid $\varpi (B)$ on E and $E'$. Then $g$ is continuous.

Let $c$ be a path in E; put $c'=g\circ c$ and prove that the mapping $c'$ is continuous. Let $d$ denote the path $p\circ c$ in B and, for every $t\in \mathbf{I}$, let $d_t$ denote the path $s\mapsto d(st)$. For every $t\in \mathbf{I}$, we have $c(t) =c(0)\cdot d_t($III, p. 304, remark 3), whence $c'(t) =c'(0)\cdot d_t$ by the hypothesis made on $g$, which proves that $c'$ is a path lifting the path $d$, by the same remark. This proves that the mapping $g$ is pathwise continuous. Since the space E is locally connected by paths (III, p. 261, cor. 2), the mapping $g$ is continuous (III, p. 269, corollary to prop. 13).

Let B be a topological space. Consider an operation $\varphi$ = $(\varphi_{a,b})_{(a,b)\in B\times B}$ of the groupoid $\varpi (B)$ on a set E, relative to a mapping $p: E\rightarrow B$. One says that $\varpi (B)$ operates without monodromy on E (cf. II, p. 168) if for every $b\in$ B and every class of loops $\gamma \in \pi_1(B, b)$, the action of $\gamma$ on the fibre $E_b$ is trivial. If B is connected by paths, it is enough that this be so for one point of B (loc. cit.). We shall say that the operation $\varphi$ of the groupoid $\varpi (B)$ is without local monodromy if every point of B possesses a neighbourhood V such that $\varpi (V)$ operates without monodromy on the set $E_V$ = $\overset{-1}{p}(V)$ relative to the mapping $p_V=p|\overset{-1}{p}(V)$.