\[
  \Pi_1(X \times_S Y) \xrightarrow{\ \varphi\ } \Pi_1(X) \overset{2}{\times}_{\Pi_1(S)} \Pi_1(Y) .
\]

\[
  G = \pi_1(X, x_0), \quad H = \pi_1(Y, y_0), \quad \Sigma = \pi_1(S, s_0) ,
\]

\[
  F \subset Z \to Y
\]

\[
  \pi_0(Z) = \pi_0(F) / \pi_1(Y)
\]

\[
  F \subset X \to S
\]

\[
  \pi_0(Z) = \pi_1(Y) \backslash \pi_1(S) / \pi_1(X) ,
\]

\[
  \mathcal{C} = \mathrm{Cat}\, G , \qquad
  \mathcal{D} = \coprod_{i \in I} \mathrm{Cat}\, \mathbb{Z}
\]

\[
  \mathrm{Hom}((u_1, v_1, \alpha_1), (u_2, v_2, \alpha_2)) =
  \left\{ (\lambda, \mu) \;\middle|\;
  \begin{array}{l}
    \lambda \in \mathbb{Z}^I,\ \mu \in G' , \\
    \boxed{v_2 = \operatorname{int}(\mu) \circ v_1} , \\
    \alpha_2(i)\, {\ell'_{u(i)}}^{\lambda_i} = \mu\, \alpha_1(i)
  \end{array}
  \right\}
\]

\[
  \pi_0(\underline{\mathrm{End}}(\mathcal{C}, \mathcal{D}, \varphi))
  \overset{?}{=}
\]

\[
  \begin{array}{rcl}
    v_2(\ell_i) & = & \operatorname{int}(\mu)\, v_1(\ell_i) = \mu\, v_1(\ell_i)\, \mu^{-1} \\
    \| & & \| \\
    \alpha_2(i)\, \ell_{u(i)}\, \alpha_2(i)^{-1} & = & \mu\, \alpha_1(i)\, \ell_{u(i)}\, \alpha_1(i)^{-1} \mu^{-1}
  \end{array}
\]

\[
  (\operatorname{int} \alpha_2(i))\, \ell_{u(i)} =
  (\operatorname{int}(\mu\, \alpha_1(i))) . \ell_{u(i)}
\]

\[
  (u_1, v_1, \alpha_1) \simeq (u_2, v_2, \alpha_2) \quad \text{ssi} \quad
  u_1 = u_2 ,\ \overline{v}_2 = \overline{v}_1
\]

\[
  \pi_1(\underline{\mathrm{End}}) \overset{?}{=}
  \mathrm{Aut}(u = \mathrm{id},\ v = \mathrm{id},\ \alpha = 1 \in G^I) = ?
\]

\[
  \pi_1(\underline{\mathrm{End}}) = G \quad (\simeq \mathbb{Z})
\]

\[
  \pi_1(\underline{\mathrm{End}}) \simeq \mathbb{Z} \quad
  (\text{isom.\ canonique})
\]

\[
  \begin{array}{rcl}
    v_2(\ell_i) & = & \underline{\operatorname{int}(\mu_0)\, v_1(\ell_i)} \\
    \| & & \| \\
    \operatorname{int}(\alpha_2(i))\, \ell_i & & \operatorname{int}(\mu_0)\, \operatorname{int} \alpha_1(i)\, v_1(\ell_i)
  \end{array}
\]

\[
  0 \to \mathbb{Z}^I \dashrightarrow TS(\mathcal{C}, \mathcal{D}, \varphi)
  \xrightarrow{\ \Psi\ } T(\mathcal{C}, \mathcal{D}, \varphi) \to
  \mathfrak{S}_I \to 1
\]

\[
  \ell_i = \operatorname{int}(\alpha(i))\, \ell_i .
\]

\[
  \alpha(i) = \ell_i^{\lambda_i} \qquad \lambda_i \in \mathbb{Z} \text{ bien déterminé.}
\]

\[
  (\mathrm{id}_G, (\ell_i^{\lambda_i})) \simeq (\mathrm{id}_G, (\ell_i^{\lambda'_i}))
\]

\[
  \lambda'_1 = \mu + \lambda_1 , \qquad
  -\lambda'_2 = \mu - \lambda_2 \quad \text{i.e.} \quad \lambda'_2 = -\mu + \lambda_2
\]

\[
  \operatorname{Ker}(\mathbb{Z}^I \to \operatorname{Ker} \Psi) \simeq
  \operatorname{Ker}(\mathbb{Z}^I \to \mathbb{Z})
\]

\[
  \pi_1(\underline{\mathrm{End}}_{\mathcal{D}\ \mathrm{fixe}}(\mathcal{C},
  \mathcal{D}, \varphi)) = 1 .
\]

\[
    \begin{array}{c}
      \widetilde{\Pi}_b = \coprod_{i \in B} \widetilde{\Pi}_b(i) \\
      \big\downarrow \scriptstyle \varphi_b \\
      \Pi_b
    \end{array}
    \qquad \mathbb{Z}\text{-groupoïde 1-spécial}
  \]

\[
  T((\Sigma, \omega), \Pi_d, \Pi_b) = (\mathcal{C}, \mathcal{D}, \varphi)
\]

\[
  \begin{aligned}
    \mathcal{D} &= \Pi(\partial_t(\Sigma, \omega)) \amalg \widetilde{\Pi}_b \amalg \Pi_d \\
    \mathcal{C} &= \Pi(\Sigma) \amalg \Pi_b \amalg \mathrm{Cat}(\pi_0(\Pi_d)) \\
    \varphi &= \varphi_0 \amalg \varphi_b \amalg \varphi_d
  \end{aligned}
\]

\[
  \underline{\mathrm{Hom}}((K_1, \Pi_b), (K'_1, \Pi'_b)) \simeq
  \mathrm{Cat}(\mathrm{Isom}(K_1, K'_1)) \times
  \underline{\mathrm{Equ}}_{?}(\Pi_b, \Pi'_b)
\]

\[
  (E) \qquad 1 \to N \to G \to H \to 1
\]

\[
  H' \xrightarrow{\ u\ } H
\]

\[
  \begin{array}{c}
    \pi_0 G' \xrightarrow{\ c\ } \pi_0 G \times_{\pi_0 H} \pi_0 H' . \\
    \| \\
    \pi_0(G \times_H H')
  \end{array}
\]

\[
  \left\{ \begin{array}{l}
    \pi_0 G = \pi_0 G / N^0 \\
    \pi_0 G' = \pi_0 G' / N^0
  \end{array} \right.
\]

\[
  \left\{ \begin{array}{ll}
    \pi_i G \simeq \pi_i H & i \geqslant 2 \\
    \pi_1 G = \operatorname{Ker}(\pi_1 H \to \pi_0 N) & (\pi_0 N = N) \\
    \pi_0 G \text{ est ext.\ de } \pi_0 H \text{ par } \operatorname{Coker}(\pi_1 H \to N)
  \end{array} \right.
\]

\[
  \frac{\operatorname{Im}(\pi_1 H \xrightarrow{\varphi} N)}
       {\operatorname{Im}(\pi_1 H' \xrightarrow{\varphi'} N)}
  \simeq \pi_1 H / (\operatorname{Ker} \varphi + \operatorname{Im} \pi_1(u))
\]

\[
  1 \to \pi_1 H / (\operatorname{Ker} \varphi + \operatorname{Im} \pi_1(u))
  \to \pi_0 G' \xrightarrow{\ c\ } \pi_0 G \times_{\pi_0 H} \pi_0 H' \to 1 .
\]

\[
  \struck{\pi \to}\ \pi \overset{\mathrm{déf}}{=} \pi_1(H)
  \xrightarrow{\ \varphi\ } \mathfrak{z}(N)
\]

\[
  1 \to N \to \widetilde{G} \to \widetilde{H} \to 1
\]

\[
  1 \to N \to \widetilde{G}_0 \to \mathfrak{G} \to 1
\]

\[
  \varphi : \pi \to \mathfrak{z}(N) .
\]

\[
  \operatorname{Ext}(\mathfrak{G}, N) \times
  \operatorname{Hom}_{\mathfrak{G}}(\pi, \mathfrak{z}(N))
  \xrightarrow{\ \sim\ } \operatorname{Ext}(H \bmod \widetilde{H}_0, N)
  \to \operatorname{Ext}(H, N)
\]

\[
  1 \to N \to E \to \widetilde{H} \to 1
  \quad \text{équivaut à} \quad H^2(\mathfrak{G}, N)
\]

\[
  H^2(\mathfrak{G}, \mathfrak{z} N) \to H^2(H, \mathfrak{z} N)
\]

\[
  \operatorname{Ext}(\mathfrak{G}, N) \times \operatorname{Hom}(\pi, N)
  \to H^2(H, \mathfrak{z}(N)) .
\]

\[
  \mathbb{Z}^{n_i} \xrightarrow{\ \lambda\ } \operatorname{Centre}(L_{g_i, n_i})
\]

\[
  1 \to T_{g_i, n_i} \to L_{g_i, n_i} \to \mathfrak{G}_{n_i} \to 1
\]

\[
  \boxed{\operatorname{Aut}_{\mathrm{Circ}}(K) \to \operatorname{Aut}(T_{?}(K)) \simeq T_{g,n}}
\]

\[
  \boxed{H^0 = \widetilde{H}_0/\pi , \quad \mathfrak{G} = \pi_0(H)}
\]

\[
  \boxed{\left\{ \begin{array}{l}
    \widetilde{H^0} = \text{rev.\ universel de } H^0 \\
    \pi = \pi_1(H^0) = \pi_1(H)
  \end{array} \right.}
\]

\[
  \begin{array}{ll}
    \pi_i(G) \xrightarrow{\ \sim\ } \pi_i(H) & \text{si } i \geqslant 2 \\
    \pi_1(G) = \operatorname{Ker}(\pi_1(H) \to \pi_0(N)) &
  \end{array}
\]

\[
  \pi \to G_0 \to \pi_0(G) \to 1
\]

\[
  \pi \to \mathfrak{z}(N) \hookrightarrow N \hookrightarrow G_0 \qquad \dots
\]

\[
  \pi_i(G) \xrightarrow{\ \sim\ } \pi_i(H) \quad \text{si } i \geqslant 2
\]

\[
  \pi_1 G = \operatorname{Ker}\bigl(\pi_1(H) \longrightarrow \pi_0 N\bigr)
\]

\[
  \pi_0 G \simeq \operatorname{Coker}\bigl(\pi_1 H \longrightarrow G_1\bigr)
\]

\[
  H' \xrightarrow{\;u\;} H
\]

\[
  1 \longrightarrow N \longrightarrow \widetilde{G}''_0 \longrightarrow
  \mathfrak{G}' \longrightarrow 1
\]

\[
  \pi \xrightarrow{\;\varphi'\;} \mathfrak{z}(N) \qquad ]
\]