\[
\mathcal{T}_n = (n\text{-}\underline{\mathrm{Hom}})(\mathcal{C}_n, ((n-1)\text{-}\mathrm{Cat}))
\]

\[
\underline{\mathrm{Hom}}_{(\mathrm{Top})}(T, \mathcal{B})
\xrightarrow{u \mapsto \varphi \circ \Pi_n(u)}
\tau_1 \underline{\mathrm{Hom}}(\Pi_n(T), \mathcal{C}_n)
\]

\[
\Gamma_X(\mathcal{F}) = \mathcal{F}(X) \simeq \underline{\mathrm{Hom}}(e^X_{n-1}, \mathcal{F})
\]

\[
\mathcal{C}_n \xrightarrow{\ \mathcal{F}\ } ((n-1)\text{-}\mathrm{Cat})
\]

\[
0 \to \mathcal{L}^0 \to \mathcal{L}^1 \to \mathcal{L}^2 \to \cdots \to \mathcal{L}^n \to 0
\]

\[
H^i(X, \mathcal{L}^\bullet) = \pi_{N-i}\Gamma_X(\mathcal{T}^{N-n}\mathcal{F}) \qquad (i \leqslant N).
\]

\[
\boxed{H^i(X, \mathcal{F}) = \pi_{N-i}\Gamma_X(\mathcal{T}^{N-n}\mathcal{F}) \quad \text{si } N \geqslant i, n}
\]

\[
\mathcal{T}(\Gamma_X\mathcal{F}) \to \Gamma_X(\mathcal{T}\mathcal{F}),
\]

\[
H^i(X, \mathcal{F}) = \pi_{n-i}(\Gamma_X(\mathcal{F})) \qquad 0 \leqslant i \leqslant n.
\]

\[
H^i(X, \mathcal{F}) = \pi_{n+1-i}(\Gamma_X(\mathcal{T}\mathcal{F}))
\]

\[
H^{n+1}(X, \mathcal{F}) = \pi_0(\Gamma_X(\mathcal{T}\mathcal{F})) = \text{sections de } \mathcal{T}\mathcal{F} \text{ à équivalences près}.
\]

\[
\mathcal{C}_{n+1} \xrightarrow{\ \mathcal{F}\ } (n\text{-}\mathrm{Cat.\ de\ Picard\ strictes}).
\]

\[
H^i(\mathcal{C}_{n+1}, \mathcal{F}) = \pi_{n-i}(\underline{\mathrm{Hom}}(e^{(\mathcal{C}_{n+1})}_n, \mathcal{F})),
\]

\[
H^i(\mathcal{C}_{n+1}, \mathcal{F}) \simeq H^i(X, \mathcal{F}),
\]

\[
\mathcal{C} \xrightarrow{\ \mathcal{F}\ } (n\text{-}\mathrm{cat.\ de\ Picard\ strictes})
\]

\[
H^i(\mathcal{C}, \mathcal{F}) = \pi_{N-i}\underline{\mathrm{Hom}}(e^{\mathcal{C}}_N, \mathcal{T}^{N-n}\mathcal{F})
\]

\[
H^i(\mathcal{C}, \mathcal{F}) = \pi_{n+1-i}\underline{\mathrm{Hom}}(e^{\mathcal{C}}_{n+1}, \mathcal{T}\mathcal{F}).
\]

\[
H^{n+1}(X, \mathcal{F}) \simeq H^{n+1}(\Pi_{n+1}X, \mathcal{F}) \ ?
\]

\[
f_{n+1}\colon \mathcal{C}_{n+1} \to \mathcal{D}_{n+1}
\]

\[
f_n\colon \mathcal{C}_n \to \mathcal{D}_n .
\]

\[
f_n^*\colon \underline{\mathrm{Hom}}(\mathcal{D}_n, ((n-1)\text{-}\mathrm{Cat})) \to \underline{\mathrm{Hom}}(\mathcal{C}_n, ((n-1)\text{-}\mathrm{Cat}))
\]

\[
\mathcal{F}\colon \mathcal{D}_{n+1} \to (n\text{-}\mathrm{Cat}),
\]

\[
\underbrace{\underline{\mathrm{Hom}}(e^{\mathcal{D}_{n+1}}_n, \mathcal{F})}_{=\, \Gamma_{\mathcal{D}_{n+1}}(\mathcal{F})\ \text{déf}}
\to
\underbrace{\underline{\mathrm{Hom}}(e^{\mathcal{C}_{n+1}}_n, f_{n+1}^*\mathcal{F})}_{=\, \Gamma_{\mathcal{C}_{n+1}}(\mathcal{F})\ \text{déf}}
\]

\[
(*) \qquad \text{simplicial sets} \longleftrightarrow \infty\text{-groupoids}
\]

\[
L_1 \xrightarrow{d} L_0
\]

\[
\theta\colon L_0 \to \mathrm{Aut}_{\mathrm{Gr}}(L_1)
\]

\[
d(\theta(x_0).x_1) = \mathrm{int}(x_0)\, d(x_1)
\]

\[
\theta(d(x_1) = \mathrm{int}(x_1) .
\]

\[
\alpha \in H^3(\pi_0, \pi_1) .
\]

\[
L_n \to L_{n-1} \to \cdots \to L_1 \to L_0 \to 1
\]

\[
(*) \qquad H^i(X, G_X) \simeq \mathbb{E}\mathrm{xt}^i(J_{*X/k}, G)
\]

\[
R^if_!(F) \times R^{2d-i}f_*(\mathbb{R}\,\mathrm{hom}(F, \mu_n^{\otimes d})) \longrightarrow \mathbb{Z}/n\mathbb{Z} \,.)
\]

\[
\Delta(Rf_!(F)) \simeq Rf_*(DF[2]) \qquad \text{« shift » of dimension}
\]

\[
\Delta_0 \mathbb{R}\Gamma_k(J^*) \simeq \mathbb{R}\Gamma_k(\Delta J^*[1]) \,,
\]

\[
\Delta_0(H_!(X, F)) \simeq H^*(X, D(F)[3])
\]

\[
F \longmapsto \mathbb{R}\underline{\Gamma}_K(F)
\]

\[
R\Gamma_K(F) \simeq R\Gamma_k(R\underline{\Gamma}_K(F)) \,.
\]

\[
\Delta R\underline{\Gamma}_K(F) \simeq R\underline{\Gamma}_K(DF[1])
\]

\[
\Delta_0 R\Gamma_k(F) \simeq R\Gamma_K(DF[2])
\]

\[
i^!(D_S(F)) = D_S(i^*(F))
\]

\[
H^1(K, F) \overset{?}{\simeq} \mathrm{Ext}^1_{k\text{-grp}}(G', \mathbb{Q}/\mathbb{Z})
\]

\[
\phi(F_1) \simeq p_*j^*(\tilde{F}_1)
\]

\[
F_0 \,,\ \tilde{F}_1 \,,\ \tilde{u} : p^*(F_0) \to j^*(\tilde{F}_1)
\]