Remark

Let B be a topological space locally connected by paths and suppose that every point $b$ of B possesses a neighbourhood V such that the image of $\pi_1(V, b)$ in $\pi_1(B, b)$ reduces to the identity element (cf. IV, p. 340, def. 2). Then every operation of the groupoid $\varpi (B)$ is without local monodromy.

In fact, let us consider a set E, a mapping $p: E\rightarrow B$ and an operation $\varphi$ of the groupoid $\varpi (B)$ on E relative to $p$. Let $a$ be a point of B and let V be a neighbourhood of $a$ such that the image of $\pi_1(V, a)$ in $\pi_1(B, a)$ is reduced to the identity element. Let U be a path-connected neighbourhood of $a$ contained in V. Let $b$ be a point of U and let $\gamma \in \pi_1(U, b)$. Let $\delta$ be the class of a path joining $a$ to $b$ in U. Then $\delta \gamma \delta^{-1}$ is the class of a loop at $a$ in U; its image in $\pi_1(B, a)$ is therefore trivial. Thus $\varphi_{a,a}(\delta \gamma \delta^{-1}) =$ Id$_{E_a}$, whence $\varphi_{b,b}(\gamma ) =$ Id$_{E_b}$. This proves that the operation of $\varpi (U)$ on the U-space $E_U$ is without monodromy. Consequently, the operation $\varphi$ is without local monodromy.