Proposition 3

Let B be a topological space locally path-connected and let E be a set endowed with an operation without local monodromy $\varphi$ of the groupoid $\varpi (B)$, relative to a mapping $p: E\rightarrow B$. Then there exists on E a unique topology for which the following conditions are satisfied:

(i) The set E endowed with this topology and the mapping $p$ is a covering of B;

(ii) The canonical operation of $\varpi (B)$ on this covering is identical with the operation $\varphi$.

The uniqueness of such a topology follows from lemma 1 of III, p. 312, where one takes for $g$ the identical mapping of E. To prove its existence, one may moreover suppose that B is connected and nonempty.

Let us suppose first of all that the operation $\varphi = (\varphi_{a,b})_{(a,b)\in B\times B}$ of the groupoid $\varpi (B)$ on the set E, relative to $p$, is without monodromy.

Let $a$ be a point of B. For every point $b\in B$, there exists a path $c$ in B joining $a$ to $b$. If $\gamma \in \varpi_{a,b}(B)$ denotes the class of the path $c$, the bijection $\varphi_{a,b}(\gamma ): E_a\rightarrow E_b$ is independent of the path $c$, since the operation is without monodromy; let $f_{a,b}$ denote this bijection. The mapping $\Phi_a: B\times E_a\rightarrow$ E defined by $(b, x)\mapsto f_{a,b}(x)$ is a bijection; the inverse bijection associates with $x\in E$ the pair $(p(x), f_{p(x),a}(x))$. Endow the set $E_a$ with the discrete topology, so that the B-space $B\times E_a$ is a trivial covering of B. By transport of structure, the bijection $\Phi_a$ endows E with a topology which makes it into a covering of B.

Let us prove that the canonical operation of $\varpi (B)$ on E is identical with the operation $\varphi$. Let $x$ be a point of $E_a$ and $b$ a point of B. Let $c$ be a path in B joining the point $a$ to the point $b$; the mapping $t\mapsto \Phi_a(c(t), x)$ is then a path in E joining the point $x$ to the point $\Phi_a(b, x) =f_{a,b}(x)$. If $\gamma \in \varpi_{a,b}(B)$ denotes the class of the path $c$, we thus have $x\cdot \gamma =f_{a,b}(x)$. This proves that the canonical operation of $\varpi (B)$ on the covering E and the operation $\varphi$ coincide on the classes of paths with origin $a$. Since these classes generate the groupoid $\varpi (B)$, the two operations are equal.

Let us now treat the general case. Let $\mathscr{B}$ be the set of open subsets V of E such that $\varpi (V)$ operates without monodromy on $\overset{-1}{p}(V)$. By hypothesis, the elements of $\mathscr{B}$ cover B. From the preceding, there exists for every $V\in \mathscr{B}$ a unique topology on the set $\overset{-1}{p}(V)$ such that $(\overset{-1}{p}(V), p_V)$ is a covering of V and that the canonical operation of $\varpi (V)$ on this covering coincides with the operation induced by $\varphi$.

Let V and $V'\in \mathscr{B}$. The topology on $\overset{-1}{p}(V\cap V')$ induced by the topology of $\overset{-1}{p}(V)$ (resp. of $\overset{-1}{p}(V')$) defined above makes it into a covering of $V\cap V'$ on which the canonical operation of $\varpi (V\cap V')$ is induced by the operation $\varphi$. These topologies therefore coincide with that of $\overset{-1}{p}(V\cap V')$. There then exists a unique topology on E inducing on each $\overset{-1}{p}(V)$ the topology previously defined (cf. I, § 2, p. 16).

When E is endowed with this topology, the mapping $p$ is continuous and the B-space E is a covering. The canonical operation of $\varpi (B)$ on this covering coincides with the operation $\varphi$ on the classes of paths whose image is contained in one of the open sets of $\mathscr{B}$. By lemma 4 of III, p. 272, these classes generate the groupoid $\varpi (B)$. It follows that these two operations are equal (II, p. 167).