Theorem 2

Suppose that every point of Y possesses a neighbourhood above which there exists a continuous section of the mapping $f$. The groupoid morphism $\varpi '(f)$ from Coeg($f$) into $\varpi (Y)$ is an isomorphism.

By hypothesis, there exists a covering $(U_j)_{j\in J}$ of Y by open sets and, for every $j\in J$, a continuous section $s_j$ of $f_{U_j}$.

If $c$ is a path in X, we denote by $[c]$ its strict homotopy class in $\varpi (X)$ and by $\{c\}$ the image of $[c]$ in Coeg($f$) by the groupoid morphism $\varpi '(f)$. If $c$ and $c'$ are two paths in X$,\{c\}$ and $\{c'\}$ are composable in Coeg($f$) if and only if the paths $f\circ c$ and $f\circ c'$ in Y are juxtaposed.

Let $c'$ be a path in Y. By Lemma 4 of III, p. 272, applied

to the compact space $\mathbf{I}$ and to the covering ($(^-{c'}^1)(U_j)$)$_{j\in J}$ of $\mathbf{I}$, there exists an integer $n$ such that, for every integer $k$ satisfying $1\leqslant k\leqslant n$, the image of the interval $[\frac{k-1}{n},\frac{k}{n}]$ by $c'$ is contained in an open set $U_{j(k)}$. For every integer $k,1\leqslant k\leqslant n$, let $c'_k$ be the path in Y defined by $s\mapsto c'(\frac{k+s-1}{n})$; one has $[c'] = [c'_1][c'_2]. . .[c'_n]$ (cf. III, p. 291, remark 1), and $c'$ is the path denoted by $c'_1*c'_2* \cdots *c'_n$. For every $k\in \{1, . . . , n\}$, let us denote by $c_k$ the path $s_{j(k)}\circ c'_k$ in X and put $\{c_k\}=\gamma ([c_k])$. Since for every $k$, the paths $c'_{k-1}=f\circ c_{k-1}$ and $c'_k=f\circ c_k$ can be juxtaposed, the sequence $(\{c_1\}, . . . ,\{c_n\})$ is composable in Coeg($f$). By construction,

which proves that the groupoid morphism $\varpi '(f)$ is surjective.

Let $u$ and $v$ be arrows of Coeg($f$). Since the groupoid Coeg($f$) is generated by the image of $\varpi (X)$ (II, p. 200, corollary), there exist finite sequences $(c_1, . . . , c_n)$ and $(d_1, . . . , d_n)$ of paths in X such that one has $u=\{c_1\}. . .\{c_n\}$ and $v=\{d_1\}. . .\{d_n\}$. The paths $(f\circ c_1, . . . , f\circ c_n)$ in Y can then be juxtaposed and one has $\varpi '(f)(u) = [(f\circ c_1)]. . .[(f\circ$ $c_n)]$; analogously, $\varpi '(f)(v) = [(f\circ d_1)]. . .[(f\circ d_n)]$.

Suppose that one has $\varpi '(f)(u) =\varpi '(f)(v)$. There then exists a strict homotopy $\sigma$ joining $(f\circ c_1)* \cdots *(f\circ c_n)$ to $(f\circ d_1)* \cdots *(f\circ d_n)$. By lemma 4 of III, p. 272, applied to the compact space $\mathbf{I}\times \mathbf{I}$ and to the covering $(\overset{-1}{\sigma}(U_i))_{i\in I})$ of $\mathbf{I}\times \mathbf{I}$, there exists an integer $m\geqslant 1$ such that, for every pair of integers $(j, k)$ satisfying $1\leqslant j\leqslant m$ and $1\leqslant k\leqslant m$, the image of $[\frac{j-1}{m},\frac{j}{m}]\times [\frac{k-1}{m},\frac{k}{m}]$ by $\sigma$ is contained in an open set $U_{i(j,k)}$ of the covering $(U_i)_{i\in I}$.

Every path $c$ in X is of the form $c_1* \cdots *c_m$, where $c_k$ is the path $t\mapsto c(\frac{k-1+t}{m})$. Replacing the integers $m$ and $n$ by their product $mn$ if necessary, one may therefore suppose that $m=n$.

For every pair $(j, k)$ of integers of $\{1, . . . , n\}$ and every pair $(s, t)\in$ $\mathbf{I}\times \mathbf{I}$, set

For $t\in \mathbf{I}$, also set $h^0_{j,k}(t) =\sigma_{j,k}(t,0)$, $h^1_{j,k}(t) =\sigma_{j,k}(t,1)$, $v^0_{j,k}(t) =\sigma_{j,k}(0, t)$ and $v_{j,k}^1(t) =\sigma_{j,k}(1, t)$. By lemma 1 of III, p. 295, the paths $h^0_{j,k}*v_{j,k}^1$ and $v_{j,k}^0*h^1_{j,k}$ are strictly homotopic, whence the relation

in $\varpi (X)$, for every pair $(j, k)\in \{1, . . . , n\}^2$. On the other hand, for every pair of integers $(j, k)$, with 2 $\leqslant j\leqslant n$ and 1 $\leqslant k\leqslant n$, one has $f\circ v_{j,k}^0=f\circ v_{j-1,k}^1$, whence the relation

in Coeg($f$). Analogously, for every pair of integers $(j, k)$ such that $1\leqslant j\leqslant n$ and $2\leqslant k\leqslant n$, one has

For $j\in \{1, . . . , n\}$, the paths $f\circ c_j$ and $f\circ h^0_{j,1}$ coincide. By definition of the coequalizer Coeg($f$), one therefore has $\{c_j\}=\{h^0_{j,1}\}$. Analogously, for every $j\in \{1, . . . , n\},\{d_j\}=\{h^1_{j,n}\}$. Hence one has the relations

By lemma 3 below, one has

Since the paths $t\mapsto \sigma (0, t)$ and $t\mapsto \sigma (1, t)$ are constant, one has, for $1\leqslant k\leqslant n,\{v^0_{1,k}\}=e_a$ and $\{v^1_{n,k}\}=e_b$ where $a$ and $b\in Y$ are the origin and the end of the arrows $u$ and $v$ of Coeg($f$). It follows that

that is, $u=v$.