Definition 4

An underlying equipment of the pair $(\varphi , \psi )$ is said to be given by the data:

(i) For every $i\in$ Orb(G), of a vertex $a(i)$ in the orbit $i$ of G;

(ii) For every $j\in$ Orb(H), of a vertex $b(j)$ in the orbit $j$ of H;

(iii) For every $j\in$ Orb(H), of arrows $c_1(j)$ and $c_2(j)$ of G joining respectively $\varphi (b(j))$ to $a(\varphi_0(j))$ and $\psi (b(j))$ to $a(\psi_0(j))$;

(iv) Of a subquiver T of the framework $\Gamma_0$ whose associated graph is a maximal tree of the graph $\widetilde{\Gamma}_0$;

(v) Of an orbit $i_0$ of G.

Let us choose an underlying equipment $(a, b, c_1, c_2,T, i_0)$ of the pair $(\varphi , \psi )$. We define a quiver morphism $\tau_1$ of $\Gamma_0$ into Grp(G$_1$) by setting $\tau_1(i) =a(i)$ for $i\in$ Som(Γ$_0$) $=$ Orb(G) and

for $j\in$ Fl(Γ$_0$) $=$ Orb(H). We shall denote by $\tau_0$ the composite of $\tau_1$ and the canonical morphism $\theta_1$ of Grp(G$_1$) into Coh($\varphi , \psi$ ) ; it is a quiver morphism of $\Gamma_0$ into Coh($\varphi , \psi$ ).

For $i\in$ Orb(G), let us denote by $\alpha_i: G_{a(i)}\rightarrow$ Coh($\varphi , \psi$ )$_{a(i)}$ the group homomorphism deduced from the morphism $\alpha : G\rightarrow$ Coh($\varphi , \psi$ ) by restriction to the isotropy groups at $a(i)$.

For $j\in$ Orb(H), let us denote

$\varphi_j=$ Int($c_1(j)$)$^{-1}\circ \varphi_{b(j)}: H_{b(j)}\rightarrow G_{a(\varphi_0(j))}$

and

$\psi_j=$ Int($c_2(j)$)$^{-1}\circ \psi_{b(j)}: H_{b(j)}\rightarrow G_{a(\psi_0(j))}$,

so that one has, for every element $f$ of $H_{b(j)}$,

(6) $\varphi_j(f) =c_1(j)^{-1}\varphi (f)c_1(j)$ and $\psi_j(f) =c_2(j)^{-1}\psi (f)c_2(j)$.

For each vertex $i$ of $\Gamma_0$, let us again denote by $d_i$ the unique path class joining $i_0$ to $i$ in the tree $\widetilde{T}$; it is regarded as an arrow of Grp(Γ$_0$). Let us then denote by $\delta_i$ the arrow of Coh($\varphi , \psi$ ) which is the image of $d_i$ under the canonical morphism of Grp(Γ$_0$) into Coh($\varphi , \psi$ ) deduced from $\tau_0$; the origin of $\delta_i$ is $a(i_0)$, its end is $a(i)$.

The quiver morphism $\tau_0$, the group homomorphisms $\alpha_i$ (for $i\in$ Orb(G))$,\varphi_j$ and $\psi_j$ (for $j\in$ Orb(H)), and the arrows $\delta_i$ in Coh($\varphi , \psi$ ) (for $i\in$ Orb(G)) will be said to be deduced from the basic equipment.

If $(G_i)_{i\in I}$ is a family of groups, we denote by $_i*_{\in I}G_i$ their free product; the image of an element $g\in G_i$ under the canonical mapping of $G_i$ into $_i*_{\in I}G_i$ will be denoted by $[g]$, or even by $g$ if there is no possible confusion. If S is a set, we denote by F(S) the free group constructed on S (A, I, p. 84).