\[
S\,;\ (\mathrm{Cat}) \longrightarrow (\mathrm{Ssimpl}) \quad .
\]

\[
T : (\mathrm{Ssimpl}) \longrightarrow (\mathrm{Cat}) ,
\]

\[
\begin{array}{cccc}
X_{00} & X_{01} & \cdots & X_{0n} \\
X_{10} & X_{11} & \cdots & X_{1n} \\
\cdots & & & 
\end{array}
\]

\[
0 \leftarrow C_{0} \leftarrow C_{1} \leftarrow 0 \quad .
\]

\[
\mathrm{Hom}_{\underline{B}}(X, Y) = B(X \times Y) \quad .
\]

\[
(*) \qquad B(X \times Y) \to \mathrm{Hom}(X', Y') \quad .
\]

\[
X' \xrightarrow{\;F'^{\bullet}(p)\;} Z' \xrightarrow{\;F'_{\bullet}(q)\;} Y'
\quad .
\]

\[
A \xrightarrow{\ \Delta\ } A_{\mathrm{dr}} \otimes_{\mathcal{O}_{X_0}} {}_{\mathrm{g}}A
\]

\[
df = \mathrm{pr}_2^*(f) - \mathrm{pr}_1^*(f) \mod I^2 .
\]

\[
\mathfrak{g}^{\mathcal{C}} = \struck{\mathrm{d\acute{e}f}}\;
\underline{\mathrm{Hom}}_{\mathcal{O}_{X_0}}(\Omega_{\mathcal{C}}, \mathcal{O}_X)
= \varinjlim_{\alpha} \underline{\mathrm{Hom}}_{\mathcal{O}_{X_0}}(\Omega_{\mathcal{C}_\alpha}, \mathcal{O}_{X_0})
\]

\[
\delta_0 : \mathcal{O}_{X_0} \longrightarrow \Omega \qquad
\text{[i.e. } \Omega^1_{X_0/\mathbb{Z}} \xrightarrow{\ u\ } \Omega
\text{ par } \delta(f) = u(df)\text{ ]}
\]

\[
\alpha)\quad \delta_0(fg) = \delta_0(f)\,g + f\,\delta_0(g)
\]

\[
\delta_1 : \Omega \longrightarrow \overset{2}{\Lambda}\Omega
\]

\[
\begin{cases}
\beta)\quad \delta_1(f\omega) = \delta_0(f)\,\delta_1(\omega) + f\,\delta_1(\omega) \\
\gamma)\quad \delta_1\delta_0 = 0
\end{cases}
\]

\[
[X,Y]\ \struck{(U)} \in \Gamma(U, \mathfrak{g})
\]

\[
(*)\qquad \langle \omega, [X,Y] \rangle
= -\langle \delta\omega, X \wedge Y \rangle + \theta_X \langle \omega, Y \rangle
- \theta_Y \langle \omega, X \rangle ,
\qquad \omega \in \Gamma(U, \Omega)
\]

\[
\theta_X(f) = \langle \delta f, X \rangle ,
\qquad
\text{[NB } \boxed{\theta_{gX} = g\,\theta_X} \text{ ]}
\]

\[
(**)\qquad \boxed{\theta_{[X,Y]} = [\theta_X, \theta_Y]}
\;\overset{\mathrm{df}}{=}\; \theta_X\theta_Y - \theta_Y\theta_X
\]

\[
\boxed{[X,X] = 0}\ \text{(\uncertain{trivial})},
\qquad
\boxed{[[X,Y],Z] + [[Y,Z],X] + [[Z,X],Y] = 0}
\]

\[
\boxed{[X, fY] = \theta_X(f)\,Y + f\,[X,Y]}
\qquad \text{résulte de } \alpha)
\]

\begin{align*}
&\sum_{\substack{\text{perm. circ.}\\ \text{de } X,Y,Z}}
\Big[ -\langle \delta\omega, [X,Y] \wedge Z \rangle
+ \theta_{[X,Y]} \langle \omega, Z \rangle
- \theta_Z \langle \omega, [X,Y] \rangle \Big] \\
&= \sum \Big[ -\langle i_Z \delta\omega, [X,Y] \rangle
+ (\theta_X\theta_Y - \theta_Y\theta_X) \langle \omega, Z \rangle \\
&\qquad + \theta_Z \langle \delta\omega, X \wedge Y \rangle
- \theta_Z\theta_X \langle \omega, Y \rangle
+ \theta_Z\theta_Y \langle \omega, X \rangle \Big] \\
&= \sum -\langle i_Z \delta\omega, [X,Y] \rangle
+ \theta_Z \langle \delta\omega, X \wedge Y \rangle \\
&= \sum \langle \delta i_Z \delta\omega, X \wedge Y \rangle
- \theta_X i_Y i_Z \delta\omega + \theta_Y i_X i_Z \delta\omega
+ \theta_Z \langle \delta\omega, X \wedge Y \rangle \\
&= \sum \big( i_X i_Y \delta i_Z + \theta_Z i_X i_Y
- \theta_X i_Y i_Z + \theta_Y i_X i_Z \big) . (\delta\omega)
\end{align*}

\begin{align*}
&\sum i_Y i_X \delta i_Z + i_Z \delta i_Y i_X + \struck{\ill{}}\, \delta i_Z i_Y i_X
= i_X \delta i_Y i_Z - \delta i_X i_Y i_Z + i_Y \delta i_X i_Z + \delta i_Y i_X i_Z \\
&\sum i_Y i_X \delta i_Z + \big( i_Z \delta i_Y i_X + i_Y \delta i_X i_Z - i_X \delta i_Y i_Z \big)
= 3\, \delta i_X i_Y i_Z \\
&= \struck{9\, \delta i_X i_Y i_Z +}
\end{align*}

\[
\sum -\theta_X i_Y i_Z + \theta_Y i_X i_Z = -2 \sum \theta_Z i_X i_Y
\]

\[
(\text{Jacobi})(\omega) = T(\delta\omega), \quad \text{avec} \quad
T = \sum i_X i_Y \delta i_Z - \theta_Z i_X i_Y \;\struck{= \ill{}}
\]

\[
\boxed{\theta_X = i_X \delta + \delta i_X}
\]

\[
T = \sum i_X i_Y \delta i_Z - i_Z \delta i_X i_Y \;\struck{\ill{}}
\]

\[
T(\delta\omega) = \sum [\, i_{X \wedge Y}, \theta_Z \,](\delta\omega)
\]

\[
C^p = \underline{\mathrm{Hom}}(\overset{p}{\Lambda}\mathfrak{g}, \mathcal{O}_{X_0})
\]

\[
\overset{*}{\Lambda}\Omega \otimes_{\mathcal{O}_{X_0}} M
\]

\[
(*)\qquad \delta_M(fx) = \delta(f)\,x \;\struck{\ill{}}\; + (-1)^{\deg f} f\,\delta_M(x) .
\]

\[
M \xrightarrow{\ \delta^1_M\ } \Omega \otimes M
\]

\[
M \xrightarrow{\ \delta_M\ } \Omega \otimes M
\]

\[
\delta_M(fx) = \delta(f) \otimes x + f\,\delta_M(x) .
\]

\[
\delta_M(\omega x) = \delta(\omega)\, x + \omega\,(\delta_M x)
\qquad \big(\omega \in \Gamma\overset{*}{\Lambda}\Omega,;
x \in \Gamma\,\overset{*}{\Lambda}\Omega \otimes M\big)
\]

\[
M \xrightarrow{\ \delta^0_M\ } \Omega \otimes M
\xrightarrow{\ \delta^1_M\ } \overset{2}{\Lambda}\Omega \otimes M
\]

\[
\theta^M_X(x) = (X \otimes \mathrm{id}_M)(\delta^0_M(x))
\]

\[
\begin{cases}
a)\quad \theta^M_X(fx) = \theta_X(f)\,x + f\,\theta^M_X(x) \\
b)\quad \theta^M_{fX}(x) = f\,\theta^M_X(x)
\end{cases}
\]

\[
c)\quad \theta^M_{[X,Y]} = [\theta^M_X, \theta^M_Y] .
\]

\[
K \in C^2(\mathfrak{g}, \underline{\mathrm{End}}_{\mathcal{O}_{X_0}}(M))
\]

\[
K(X,Y) = \theta^M_{[X,Y]} - [\theta^M_X, \theta^M_Y]
\]

\[
\theta^M_{[X,Y]} = [\theta^M_X, \theta^M_Y] ,
\]

\[
M \longrightarrow \overset{2}{\Lambda}\Omega \otimes M \longrightarrow C^2(\mathfrak{g}, M)
\]

\[
K(X,Y)(u) = K'(u)(X,X) .
\]

\[
\overset{*}{\Lambda}\Omega \otimes M \quad \text{et} \quad C^{*}(\mathfrak{g}, M) .
\]

\[
\overset{*}{\Lambda}\Omega \otimes M \simeq C^{*}(\mathcal{C}, M)
\]

\[
S : (\mathrm{Cat}) \longrightarrow (\mathrm{Ssimpl}) .
\]

\[
T : (\mathrm{Ssimpl}) \longrightarrow (\mathrm{Cat}) ,
\]

\[
\begin{array}{cccc}
X_{00} & X_{01} & \cdots & X_{0n} \\
X_{10} & X_{11} & \cdots & X_{1n} \\
\cdots & & &
\end{array}
\]

\[
0 \longleftarrow C_0 \longleftarrow C_1 \longleftarrow 0 .
\]

\[
\begin{array}{ccccc}
C_{00} & \longleftarrow & 0 & \longleftarrow & 0 \\
\uparrow & & \uparrow & & \\
C_{10} & \longleftarrow & C_{11} & \longleftarrow & 0
\end{array}
\]

\[
\mathrm{Hom}_{\mathbf{B}}(X, Y) = B(X \times Y) .
\]

\[
B(X \times Y) \longrightarrow \mathrm{Hom}(X', Y') . \tag{$*$}
\]

\[
X' \xrightarrow{F'^{\bullet}(p)} Z' \xrightarrow{F'_{\cdot}(q)} Y' .
\]

\[
\overline{T}' \xrightarrow{\alpha_{i}} Z'_{i} \xrightarrow{\alpha_{i*}}
\overline{T}'
\]