Theorem 1

Let X and Y be topological spaces, and let $f: X\rightarrow$ Y be a continuous surjective mapping. Suppose that the spaces X and Y are locally path-connected and semi-locally simply connected. Finally, suppose that one of the following properties is satisfied:

(i) The mapping $f$ is proper, separated, with locally connected fibres, and the space $X\times_YX$ is locally path-connected;

(ii) The mapping $f$ is proper, separated, with finite fibres, the diagonal $\Delta_X$ is open in $X\times_YX$, and its complement is locally connected;

(iii) The mapping $f$ is open and has the path-lifting property. Then the groupoid morphism $\varpi '(f)$ is an isomorphism of the groupoid Coeg($f$) onto the Poincaré groupoid $\varpi (Y)$.

First let us note that under these hypotheses, the mapping $f$ is surjective and strict (I, p. 18, example 2). Moreover, every descent datum relative to $f$ on a covering of X is effective, and the quotient space is a covering of Y; this follows indeed from IV, p. 393, corollary 2 of prop. 4 under hypotheses (i) and (ii), and from prop. 7 of IV, p. 394 under hypothesis (iii). By prop. 10, the groupoid morphism $\varpi '(f)$ is therefore injective and its image is a subgroupoid of $\varpi (Y)$.

By prop. 9 of IV, p. 395, this image is equal to $\varpi (Y)$ under hypotheses (i) and (ii), but also under hypothesis (iii), since the groupoid morphism $\varpi (f)$ is then surjective.

Consequently, $\varpi '(f)$ is an isomorphism.