Proposition 5

Let G and H be groupoids; let $\varphi$ and $\psi$ be morphisms of groupoids from H into G. Suppose that the groupoid Coeg($\varphi , \psi$ ) is transitive. Let $\gamma : G\rightarrow$ Coeg($\varphi , \psi$ ) denote the canonical morphism of groupoids.

Let us endow the pair $(\varphi , \psi )$ with a complete equipment

For $j\in$ Orb(H), let $\varphi_j: H_{b(j)}\rightarrow G_{a(\varphi_0(j))}$ and $\psi_j: H_{b(j)}\rightarrow G_{a(\psi_0(j))}$ be the group homomorphisms Int($c_1(j)$)$^{-1}\circ \varphi_{b(j)}$ and Int($c_2(j)$)$^{-1}\circ \psi_{b(j)}$ respectively, where $\varphi_0$ and $\psi_0$ denote the canonical mappings deduced from $\varphi$ and $\psi$ by passing to orbits. Let $\tau$ be the morphism of the framework $\Gamma_0$ of the pair $(\varphi , \psi )$ into Coeg($\varphi , \psi$ ) defined by Som($\tau$ )$(i) =\gamma (a(i))$ if $i\in$ Orb(G) and such that Fl($\tau$ )$(j)$ is the path $\gamma (c_1(j))^{-1}\gamma (c_2(j))$ in Coeg($\varphi , \psi$ ). If $i$ is an orbit of G, let $c_i$ be the unique class of paths in $\widetilde{T}$ joining $i_0$ to $i$ and put $\delta_i=\widetilde{\tau}(c_i)$, where $\widetilde{\tau}:$ Grp(G) $\rightarrow$ Coeg($\varphi , \psi$ ) is the canonical groupoid morphism deduced from $\tau$.

There then exists a unique group homomorphism

$\lambda :(*G_{a(i)})*$ F(Orb(H)) $\rightarrow$ Coeg($\varphi , \psi$ )$_{\gamma(a(i_0))}$

$i\in$Orb(G)

such that

$\lambda (f) =\delta_i\gamma_{a(i)}(f)\delta_i^{-1}$ for $i\in$ Orb(G) and $f\in G_{a(i)}$,

$\lambda (j) =\delta_{\varphi_0(j)}\tau (j)\delta_{\psi_0(j)}^{-1}$ for $j\in$ Orb(H).

The homomorphism $\lambda$ is surjective; its kernel is the smallest normal subgroup containing the following elements:

($R_1$)$r_1(j) =j$ for $j$ in Fl(T) ;

($R_2$)$r_2(j, f) =\varphi_j(f)j\psi_j(f)^{-1}j^{-1}$

for $j\in$ Orb(H) and $f\in H_{b(j)}$;

($R_3$)$r_3(z) =\lambda_1(z)j_1^{\varepsilon_1}\lambda_2(z)j_2^{\varepsilon_2}. . . \lambda_n(z)j_n^{\varepsilon_n}$

for $z= ((y_1, \varepsilon_1), . . . ,(y_n, \varepsilon_n))\in Z$,

where the loops $\lambda_i(z)$ are defined by formula (1), p. 208.

The existence and uniqueness of such a morphism result from the universal property of the free product of a family of groups (A, I, p. 85, prop. 8). Let us denote by $\alpha : G\rightarrow$ Coh($\varphi , \psi$ ) and $\beta :$ Coh($\varphi , \psi$ )$\rightarrow$ Coeg($\varphi , \psi$ ) the canonical morphisms, so that $\gamma$ = $\beta \circ \alpha$. Let also $h:$ Som(H) $\rightarrow$ Fl(Coh($\varphi , \psi$ )) be the canonical homotopy joining $\alpha \circ \varphi$ to $\alpha \circ \psi$. Since $\beta$ is the groupoid morphism obtained by contraction of the arrows in the image of $h$, we have

Fl($\tau$ )$(j) =\gamma (c_1(j))^{-1}\gamma (c_2(j)) =\gamma (c_1(j)^{-1})\beta (h(j))\gamma (c_2(j)) =\beta (\tau_0(j))$ in Coeg($\varphi , \psi$ ), where $\tau_0: \Gamma_0\rightarrow$ Coh($\varphi , \psi$ ) denotes the quiver morphism deduced from the basic equipment $(a, b, c_1, c_2,T, i_0)$. Consequently, the homomorphism $\lambda$ is the composite of the homomorphism Λ defined in prop. 6 (II, p. 193) and the surjective homomorphism $\beta_{a(i_0)}$. In particular it is surjective.

Let $z\in Z$; by construction, $\Lambda (r_3(z))$ belongs to the conjugacy class $c(z)$. By definition of a complementary equipment, the kernel of the homomorphism $\beta_{a(i_0)}$ is therefore the smallest invariant subgroup of Coh($\varphi , \psi$ )$_{a(i_0)}$ containing the elements $\Lambda (r_3(z))$ for $z\in Z$. Since the homomorphism Λ is surjective, the kernel of the homomorphism $\lambda =\beta_{a(i_0)}\circ \Lambda$ is therefore the smallest invariant subgroup of the group $(*G_{a(i)})*F(\pi_0(H))$ containing the generators of the kernel

$i\in$Orb(G)

of Λ given by formulae ($R_1$)$, (R_2)$, together with the elements defined by formulae ($R_3$). The proposition is thus proved.