Proposition 6

There exists a unique group homomorphism

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

$i\in$Orb(G)

such that

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

(8) $\Lambda (j) =\delta_{\varphi_0(j)}\tau_0(j)\delta^{-1}_{\psi_0(j)}$ for $j\in$ Orb(H).

The homomorphism Λ is surjective; its kernel is the smallest normal subgroup of $(*_iG_{a(i)})*$ F(Orb(H)) containing the elements $j$ of Fl(T) and the elements $\varphi_j(f)j\psi_j(f)^{-1}j^{-1}$, for $j\in$ Orb(H) and $f\in H_{b(j)}$.

The existence and uniqueness of the homomorphism Λ result from the universal property of free products and free groups (A, I, p. 85, prop. 8).

Let A denote the set of the $a(i)$ for $i\in$ Orb(G) and $G_A$ the full subgroupoid of G whose set of vertices is A. For every $x\in$ Som(G), let $\overline{x}$ denote the orbit of $x$ in G and choose an arrow $d_x$ of G joining $x$ to $a(\overline{x})$. The pair $v$ formed by the mapping $x\mapsto a(\overline{x})$ of Som(G) into A and the mapping which to $f\in$ Fl$_{x,y}(G)$ associates the element $d^{-1}_xf d_y$ of Fl$_{a(\overline{x}),a(\overline{y})}(G_A)$ is a groupoid morphism. It follows from Proposition 1 of II, p. 182 that $v$ is a homotopism. Let $\varphi '=v\circ \varphi$ and $\psi '=v\circ \psi$, and then let $w$ be the canonical morphism of Coh($\varphi , \psi$ ) into Coh($\varphi ', \psi '$) ; it is a homotopism (II, p. 187, Theorem 1).

The orbits of $G_A$ are the sets $\{a\}$, for $a\in A$, and the injection $G_A\rightarrow G$ induces a bijection of Orb(G$_A$) onto Orb(G), by means of which we shall identify these two sets. One defines a basic equipment $(a', b', \beta '_1, \beta '_2,T', i_0)$ of the pair $(\varphi ', \psi ')$ by setting $a'(i) =a(i)$ for $i\in$ Orb(G)$,b'(j) =b(j),\beta '_1(j) =v(c_1(j)),\beta '_2(j) =v(c_2(j))$ for $j\in$ Orb(H) and $T'= T$. The group homomorphisms $\varphi '_j$ and $\psi '_j$ (for $j\in$ Orb(H)), the quiver morphism $\tau '_0$, the arrows $\delta '_i$ (for $i\in$ Orb(G)), and therefore the group homomorphism $\Lambda '$, deduced from this basic equipment, are the composites with $w$ of the corresponding homomorphisms $\varphi_j,\psi_j$, of the quiver morphism $\tau_0$, of the corresponding arrows $\delta_i$ and of the homomorphism Λ.

Let B be the set of the $b(j)$ for $j\in$ Orb(H), let $H_B$ be the full subgroupoid of H with set of vertices B; let us denote by $u: H_B\rightarrow H$ the canonical injection; put $\varphi ''=\varphi '\circ u$ and $\psi ''=\psi '\circ u$. The morphism $u$ induces a bijection B $\rightarrow$ Orb(H) by which we shall identify these two sets. From theorem 1 of II, p. 187 one further deduces a canonical homotopism $w':$ Coh($\varphi '', \psi ''$)$\rightarrow$ Coh($\varphi ', \psi '$). Moreover, the pair $(\varphi '', \psi '')$ is endowed with a base equipment $(a'', b'', \beta ''_1, \beta ''_2,T'', i_0)$, so that $\Lambda ', \varphi '_j, \psi '_j, \tau '_0, \delta '_i(i, i'\in$ Orb(G), $j\in$ Orb(H)) are the composites with $w'$ of $\Lambda '', \varphi ''_j, \psi ''_j, \tau ''_0, \delta ''_i$.

The various morphisms of groupoids introduced are summarized by the following diagram:

$H_{B\psi''}^{\varphi''}G_A^{\alpha''}$ Coh($\varphi '', \psi ''$)

$w'$

(9) H $^{\varphi'}G_A^{\alpha'}$ Coh($\varphi '', \psi ''$)

$\psi '$

H $^{\varphi}_{\psi}$ G $^{\alpha}$ Coh($\varphi , \psi$ ) .

To prove the proposition, we may therefore suppose that A = Som(G) and B = Som(H), in other words that the canonical mappings Som(G) $\rightarrow$ Orb(G) and Som(H) $\rightarrow$ Orb(H) are bijective, hypotheses under which we shall place ourselves in the rest of the proof.

The quiver $G_1$ then has for set of vertices A and for arrows the sum set of the sets $G_a,a\in A$, and of the set B. The arrows of $G_a$ are loops at $a$; if $b\in B$, the arrow $b$ joins $\varphi (b)$ to $\psi (b)$, the arrows $c_1(b)$ and $c_2(b)$ are loops respectively at $\varphi (b)$ and $\psi (b)$. The quiver T will be identified with an oriented tree of $G_1$; it is a maximal oriented tree since the set of its vertices is equal to the set of vertices of $G_1($II, p. 157, prop. 1). We put $a_0=a(i_0)$. The set of arrows of the framework $\Gamma_0$ of the pair $(\varphi , \psi )$ being identified with B, the morphism of quivers $\tau_1: \Gamma_0\rightarrow$ Grp(G$_1$) associates to the arrow $b$ the class of paths $c_1(b)^{-1}bc_2(b)$ in the graph $\widetilde{G}_1$.

Recall that $\theta_1$ denotes the canonical morphism of quivers of $G_1$ into Grp(G$_1$). Let

$\lambda :_a*_{\in A}F(G_a)*F(B)\rightarrow$ Grp(G$_1$)$_{a_0}$

be the unique group homomorphism such that one has

$\lambda (f) =\tau_1(d_a)\theta_1(f)\tau_1(d_a)^{-1}$ if $a\in A$ and $f\in G_a$;

$\lambda (b) =\tau_1(d_{\varphi(b)})\tau_1(b)\tau_1(d_{\psi(b)})^{-1}$ if $b\in B$.

We thus have $\lambda =\lambda '\circ \varepsilon$, where $\lambda '$ denotes the canonical group homomorphism from $_a*_{\in A}F(G_a)*F(B)$ into Grp(G$_1$)$_{a_0}$ defined by the maximal oriented tree T (II, p. 179, prop. 9), and where $\varepsilon$ is the unique automorphism of the group $_a*_{\in A}F(G_a)*F(B)$ such that $\varepsilon (f) =f$ for $a\in A$ and $f\in G_a$, and $\varepsilon (b) =c_1(b)^{-1}h_1(b)c_2(b)$ for $b\in B$. According to remark 2 of II, p. 179, the homomorphism $\lambda$ is surjective and its kernel is the smallest normal subgroup of $_a*_{\in A}F(G_a)*F(B)$ containing the arrows of T.

Let us denote by $\pi :$ Grp(G$_1$)$\rightarrow$ Coh($\varphi , \psi$ ) the canonical groupoid morphism. According to II, p. 177, cor. 1 of prop. 8, the group homomorphism $\pi_{a_0}$ from Grp(G$_1$)$_{a_0}$ into Coh($\varphi , \psi$ )$_{a_0}$ is surjective, and its kernel is the smallest normal subgroup of Grp(G$_1$)$_{a_0}$ containing the loops Int($\tau_1(d_a)$)$(\alpha_1(f)\alpha_1(g)\alpha_1(f g)^{-1})$, for $a\in A$ and $f,g\in G_a$, and the loops Int($\tau_1(d_{\varphi(b)})$)$(\varphi (f)b\psi (f)^{-1}b^{-1})$, for $b\in B$ and $f\in H_b$.

If $p: F(\bigcup G_a\cup B)\rightarrow (_a*_{\in A}G_a)*F(B)$ denotes the canonical surjective homomorphism, we thus have $\Lambda \circ p=\pi_{a_0}\circ \lambda$. This formula implies that the homomorphism Λ is surjective; it remains to determine its kernel.

For $a\in A$ and $f\in G_a$, let us denote by $[f]$ the image of $f\in F(G_a)$ in the group $_a*_{\in A}F(G_a)*F(B)$. For $a\in A,f, g\in G_a$, we then have

Int($\tau_1(d_a)$)$(\alpha_1(f)\alpha_1(g)\alpha_1(f g)^{-1}) =\lambda ([f][g][f g]^{-1})$.

Analogously, for $b\in B$ and $f\in H_b$, the definition of the homomorphisms $\varphi_b$ and $\psi_b$ (formula (6) of II, p. 193) implies that one has

Int($\tau_1(d_{\varphi(b)})$)$(\varphi (f)h_1(b)\psi (f)^{-1}h_1(b)^{-1})$

Consequently, the kernel of the homomorphism $\pi_{a_0}\circ \lambda$ is the smallest normal subgroup of $_a*_{\in A}F(G_a)*F(B)$ which contains the elements $[f][g][f g]^{-1}$ for $a\in A$ and $f,g\in G_a$, the elements $\varphi_b(f)[b]\psi_b(f)^{-1}[b]^{-1}$ for $b\in B$ and $f\in H_b$, and the elements $[b]$, for $b\in$ Fl(T).

Finally, the kernel of the homomorphism Λ is the smallest normal subgroup of $(_a*_{\in A}G_a)*F(B)$ which contains the images under $p$ of the preceding elements, in other words, the elements $[b]$, for $b\in$ Fl(T), and the elements $\varphi_b(f)[b]\psi_b(f)[b]^{-1}$, for $b\in B$ and $f\in H_b$. The proposition is thus proved.