\[
      \underline{\mathrm{Hom}}_{\Delta,\, i(\delta)}(\bar{C}_{\delta}, E)
      \;\xrightarrow{\ \approx\ }\;
      \underline{\mathrm{Hom}}_{\delta}(C, E)
    \]

\[
  C \;\xrightarrow{\ \alpha_i\ }\; C_i \;\xrightarrow{\ \tau_i\ }\; C_{i+1}
\]

\[
\begin{cases}
  C_0 = C, \quad \delta_0 = \delta = (\delta', \delta''), \\
  (C_{i+1}, \delta_{i+1}) = K(C_i, \delta_i), \\
  C_i = \varinjlim_{j<i,\ (\mathrm{Cat})} C_j \quad \text{si $i$ ordinal limite.}
\end{cases}
\]

\[
      \underline{\mathrm{Hom}}_{\delta_{i+1}}(C_{i+1}, E)
      \;\longrightarrow\;
      \underline{\mathrm{Hom}}_{\delta_i}(C_i, E).
    \]

\[
  (d^{*}) \qquad
  \boxed{\ \underline{\mathrm{Hom}}_{\delta_i}(C_i, E)
  \;\xrightarrow{\ \approx\ }\;
  \underline{\mathrm{Hom}}_{\delta}(C, E)\ }
  \qquad \varphi \mapsto \varphi \circ \alpha_i .
\]

\[
  (*) \qquad \forall\, j < \omega, \quad C_j \to C_\omega \ \text{est}\ \delta_j\text{-exact}.
\]

\[
  (**) \qquad \delta_\omega = \text{(cônes exacts de types $d'$ et $d''$)}
  \ \cup\ \alpha_\omega(\delta).
\]

\[
  \rho(C,\delta) = \rho_3 \rho_2 \rho_1 (C,\delta).
\]

\[
  K(C,\delta) = \sigma\rho(C,\delta)
  = \sigma_3 \sigma_2 \sigma_1 \rho_3 \rho_2 \rho_1 (C,\delta).
\]

\[
  K'(C,\delta) = \sigma_1 \rho_3 \sigma_2 \rho_1 \rho_2 \sigma_3 (C,\delta)\ !
\]

\[
  f^{(i)}_\lambda : R^{(i)}_\lambda \longrightarrow R'^{(i)}_\lambda
  \qquad (i = 0,1,2),
\]

\[
  \underline{\mathrm{Hom}}_{\delta}(C, E) = \underline{\mathrm{Hom}}_{\Delta,\delta}(C, E)
  \qquad \forall\, E \in \mathrm{Ob}\,\Delta .
\]

\[
\begin{cases}
  (C_0, \delta_0) = (C, \delta), \\
  (C_{i+1}, \delta_{i+1}) = K(C_i, \delta_i), \quad
    \tau_i : C_i \to C_{i+1}, \quad \delta_{i+1} = \tau_i(\delta_i), \\
  C_i = \varinjlim_{j < i,\ (\Delta)} C_j \ \text{ si $i$ ordinal limite,} \quad
    \delta_i = \textstyle\bigcup_{j<i} \mathrm{Im.\ de\ } \delta_j \ \text{dans } C_i .
\end{cases}
\]

\[
      \Bigl( R \ \overset{\tau_i \circ u' \varphi}
      {\underset{\tau_i \circ v' \varphi}{\rightrightarrows}}\ C_{i+1} \Bigr),
    \]

\[
      \underline{\mathrm{Hom}}_{\Delta,\delta_{i+1}}(C_{i+1}, E)
      \;\xrightarrow{\ \approx\ }\;
      \underline{\mathrm{Hom}}_{\Delta,\delta_i}(C_i, E).
    \]

\[
  (c^{*}) \qquad
  \underline{\mathrm{Hom}}_{\Delta,\delta_i}(C_i, E)
  \;\xrightarrow{\ \approx\ }\;
  \underline{\mathrm{Hom}}_{\Delta,\delta}(C, E),
\]

\[
      \mathrm{Im}\bigl(\varphi^{*}_C \mid \underline{\mathrm{Hom}}_{\bar\delta}(\bar{R}, C)\bigr)
      \ \subset\ \underline{\mathrm{Hom}}_{\delta}(R, C) ;
    \]

\[
  \underline{\mathrm{Hom}}_{\Delta'}(\bar{R}, C)
  \ \xrightarrow{\ \varphi^{*}_{\Delta',C} \,=\, \Psi\ }\
  \underline{\mathrm{Hom}}_{\Delta'}(R, C).
\]

\[
  \underline{\mathrm{Hom}}_{\Delta,\, (i(\delta'), i(\delta''))}(C', E)
  \;\xrightarrow{\ \approx\ }\;
  \underline{\mathrm{Hom}}_{\Delta,\, (\delta',\delta'')}(C, E)\ ].
\]

\[
  T_0(E) = \underline{\mathrm{Hom}}_{\Delta_0}(C, E)
  \;\xrightarrow{\ \approx\ }\;
  \underline{\mathrm{Hom}}_{\Delta}(\hat{C}, E)
  \quad
  \bigl[\ \xrightarrow{\ \approx\ } \underline{\mathrm{Hom}}_{\mathrm{top}}(E, \hat{C})^{\circ}
  \ \text{si $E$ un topos}\ \bigr].
\]

\[
  E \;\longmapsto\; \underline{\mathrm{Hom}}_{\Delta, \Sigma}(R, E),
  \qquad E \in \mathrm{Ob}\,\Delta .
\]

\[
  \underline{\mathrm{Hom}}_{\Delta, \Sigma}(R, E)
  \;\xrightarrow{\ \approx\ }\;
  \underline{\mathrm{Hom}}_{\Delta}(R', E).
\]

\[
      T_{R_2}(E) = \underline{\mathrm{Hom}}_{\Delta}(R_2, E)
      \;\xrightarrow{\ \approx\ }\; \underline{\mathrm{Hom}}(J_1, R)
      \;\longleftrightarrow\; \underline{\mathrm{Hom}}_{\Delta}(R, E) = T_R(E)
    \]

\[
      T' \simeq T_R \times^{2}_{T_I} T_C
    \]

\[
  T \xrightarrow{\ f^{*}\ } R, \qquad T \xrightarrow{\ g^{*}\ } S .
\]

\[
  \underline{\mathrm{Hom}}(U_{0}, E)
  \;\xrightarrow{\ \approx\ }\;
  \underline{\mathrm{Hom}}(R_{0}, E)
    \times^{2}_{\underline{\mathrm{Hom}}(T_{0},E)}
  \underline{\mathrm{Hom}}(S_{0}, E)
\]

\[
  \underline{\mathrm{Hom}}_{\Delta}(R, E)
    \times^{2}_{\underline{\mathrm{Hom}}_{\Delta}(T,E)}
  \underline{\mathrm{Hom}}_{\Delta}(S, E)
\]

\[
  T = T_{R} \times^{(2)}_{T_{T}} T_{S}
\]

\[
  C = \varinjlim_{\alpha,\,(\mathrm{Cat})} C_{\alpha},
\]

\[
  T(E) \simeq \underline{\mathrm{Hom}}\,\mathrm{top}(E, R)^{\circ}
        \simeq \underline{\mathrm{Hom}}_{\Delta}(R, E)
\]

\[
  T_{R}(\mathrm{Ens}) \simeq \underline{\mathrm{Pt}}(R)^{\circ}
    \simeq \underline{\mathrm{Fib}}(R) .
\]

\[
  T_{V}(E) = \text{End. des sous-objets de } \struck{\ill{}}\, e_{E}
\]

\[
  T_{\mathrm{Unv}}(E) \subset E, \qquad \text{i.e. } x \xrightarrow{\ \sim\ } e_{E}
\]

\[
  E \longmapsto \mathcal{P}(E) = \mathcal{P}(e_{E}) .
\]

\[
  X \times Y \to Z, \qquad X \times Y \to \struck{X},
\]

\[
  X \times X \longrightarrow X \quad (!), \qquad
  X \times X \rightrightarrows X \quad (!!) .
\]

\[
        \underline{\mathrm{Hom}}_{\delta_{1}}(R_{1}, E)
        \;\xrightarrow{\ \approx\ }\;
        \underline{\mathrm{Hom}}_{\delta_{2}}(R_{2}, E) ;
      \]

\[
  \underline{\mathrm{Hom}}_{\Delta,\, i(\delta)}(C', E)
  \;\xrightarrow{\ \approx\ }\;
  \underline{\mathrm{Hom}}_{\mathrm{Cat},\, \delta}(C, E) .
\]

\begin{gather*}
  (C_{0}, \delta_{0}) = (C, \delta) \\
  (C_{i+1}, \delta_{i+1}) = K(C_{i}, \delta_{i}), \qquad
  \tau_{i} : C_{i} \to C_{i+1} \\
  \text{i.e. } K \text{ est une opération à expliciter plus bas} \\
  C_{i} = \varinjlim_{j < i,\,(\mathrm{Cat})} C_{j}, \qquad
  \delta_{i} = \bigcup_{j < i}
    \bigl(\mathrm{Im}\ \text{de } \delta_{j} \text{ par } C_{j} \to C_{i}\bigr)
  \quad \text{si } i \text{ ordinal limite.}
\end{gather*}

\[
      \underline{\mathrm{Hom}}_{\delta_{i+1}}(C_{i+1}, E)
      \longrightarrow
      \underline{\mathrm{Hom}}_{\delta_{i}}(C_{i}, E) ;
    \]

\[
  (1)\qquad
  \underline{\mathrm{Hom}}_{\delta_{i}}(C_{i}, E)
  \;\xrightarrow{\ \approx\ }\;
  \underline{\mathrm{Hom}}_{\delta}(C, E)
  \qquad (F \mapsto F \circ \alpha_{i}) .
\]

\[
  (2)\qquad
  \alpha_{i}(\delta) \subset \delta_{i} \subset
  \alpha_{i}(\delta) \cup
  \bigcup_{\substack{j < i \\ (S,\lambda) \in \sigma}} \;
  \bigcup_{u \in \underline{\mathrm{Hom}}(S, C_{j})} \tau_{ij}\, u(\lambda)
  \qquad \text{si } i \text{ ordinal limite.}
\]

\[
  \underline{\mathrm{Hom}}_{\Delta,\, \alpha_{\omega}(\delta)}(C_{\omega}, E)
  \ \struck{\ill{}} = 
  \underline{\mathrm{Hom}}_{\struck{\ill{}},\, \delta_{\omega}}(C_{\omega}, E)
  \;\xrightarrow[\ (1)\ ]{\ \approx\ }\;
  \underline{\mathrm{Hom}}_{\delta}(C, E) .]
\]

\[
  (3)\qquad \forall i < \omega, \quad
  \tau_{\omega,i} : C_{i} \to C_{\omega} \text{ est } \delta_{i}\text{-exact.}
\]

\[
  \alpha_{\omega}(\delta) \subset \delta_{\omega} \subset
  \alpha_{\omega}(\delta) \cup \bigcup_{(S,\lambda) \in \sigma} \;
  \bigcup_{u \in \underline{\mathrm{Hom}}_{\lambda}(S, C_{\omega})}
  \struck{\ill{}}
\]

\[
  H^{n}(K(G,1), \Pi) \xrightarrow{\ \sim\ } \mathrm{Simpl}(\mathrm{ENS}),
  \qquad \struck{Z_{i}(\Pi)} .
\]

\[
  S = \underline{\mathrm{Hom}}'(\mathcal{L}^{\circ}, \mathrm{ENS})
  \ \rightleftarrows\ \mathrm{Alg}(\mathfrak{T}),
  \qquad (\mathcal{L}^{\circ} \to \mathrm{ENS}), \qquad
  \mathrm{Mod}(\mathcal{L}^{\circ}) .
\]

\[
  G_{1} \rightrightarrows B_{2} \dashrightarrow S * S_{1} .
\]

\[
  A \xrightarrow[\ \sim\ ]{\ \Phi\ } \mathrm{Alg}(\mathbb{T}),
  \qquad F \dashv U : A \to B, \qquad B \xrightarrow{\ T\ } B,
  \qquad \mathbb{T} = (UF, -, -) .
\]

\[
  X_{1} \rightrightarrows X_{0} \longrightarrow Q' \ \text{ in } A,
  \qquad
  U(X_{1}) \rightrightarrows U(X_{0}) \rightleftarrows Q, \qquad U(Q') ,
\]

\[
  \Longrightarrow \quad X' \rightrightarrows X \rightrightarrows Y .
\]

\[
  T^{2} \ \substack{\xrightarrow{\ T(\xi)\ } \\ \xrightarrow{\ \mu\ }}\ TX
  \xrightarrow{\ \xi\ } X, \qquad \eta, \qquad T(X) \xrightarrow{\ \xi\ } X .
\]

\[
  X_{1} \to X, \qquad U X_{1} \to U X, \qquad C/S, \qquad C/S' .
\]

\[
  X_{1} \ \substack{\xrightarrow{\ d_{1}\ } \\ \xrightarrow{\ d_{0}\ }}\ X_{0}
  \longleftrightarrow u, \qquad X \longrightarrow X' .
\]

\[
  X_{1} \ \substack{\xrightarrow{\ d_{0}\ } \\ \xrightarrow{\ d_{1}\ }}\ X_{1}
  \xrightarrow{\ p\ } Q, \qquad Q \rightrightarrows Q ,
\]

\[
  d_{1} s_{1} = \mathrm{id}_{X_{1}}, \qquad d_{0} s_{1} = s_{0} p .
\]

\[
  \pi = f^{*} f_{*} : B \longrightarrow B
  \qquad \text{(exact à g., accessible)}, \qquad
  \begin{cases} \pi \to \mathrm{id} \\ \pi \to \pi^{2} \end{cases}
\]

\[
  \begin{cases}
    f^{*} \text{ conservatif} \\
    f^{*} \text{ commute aux \ill{} noyaux de couples}
  \end{cases}
\]

\[
  \mathrm{CoAlg}(B, \pi) \rightrightarrows B, \qquad
  \ill{} = B_{\pi}, \qquad
  \bigl(B \to B_{\pi}\bigr), \qquad
  B_{\pi} \xrightarrow{\ \simeq\ } B \quad \text{(th)} .
\]

\[
  \underline{\mathrm{Hom}}_{\mathrm{Ens}}(f^{*}X, Y)
  \simeq \underline{\mathrm{Hom}}_{G}(X, f_{*}(Y)),
  \qquad f_{*}(Y) = \underline{\mathrm{Hom}}(G, Y),
  \qquad Y \longmapsto \underline{\mathrm{Hom}}(G, Y) .
\]

\[
  \emptyset \xrightarrow{\ \sim\ } I(u) \subset A, \qquad
  \forall f(b)\ \exists\ \struck{e} \in \Pi(u) \text{ tel que } \ldots
\]

\[
  e\,(1-e), \qquad f^{n},\ f, \qquad V(f) \longrightarrow V(f^{n}) ,
\]

\[
  fg = e, \qquad f(1-e) = 0, \qquad f^{n}(1-e) = 0, \qquad f^{n} \overset{?}{=} e .
\]

\[
  X_{\infty} \subset {}^{e}\hat{R}, \qquad \text{crible de } R,
  \qquad \struck{X_{\infty}^{\mathrm{af}}} .
\]

\[
  X_{\infty}(\mathbb{Z}) =
  \begin{cases}
    \emptyset & \text{si } Z_{\infty} \struck{\ill{}}\ \emptyset \\
    \{e\} & \text{si } Z_{\infty} \neq \emptyset
  \end{cases}
  \qquad
  \underline{\mathrm{Hom}}_{\mathrm{Sch.}\,H \simeq \hat{R}}
    \bigl(\mathbb{Z}, \mathrm{Spec}(\mathbb{Q})\bigr) .
\]

\[
  B' = B/S \xrightarrow{\ \mathrm{can}\ } B, \qquad
  = \hat{C}_{R} \xrightarrow{\ i\ } B = \hat{R}, \qquad
  = \hat{C}_{B}\ \{\varepsilon \xrightarrow{\ \varphi\ } \ill{}\ \text{
    (avec hyp. can.)} \} \longrightarrow \ill{}
\]

\[
  B_{T'} \xrightarrow{\ \sim\ } \tilde{B} = \hat{B}\ \ill{} , \qquad
  S = i_{!}(\mathbb{1}_{B'}) \ \struck{\ill{}}
\]

\[
  C_{B} = \mathrm{Id}\ \text{ens.}\ i_{!}
  \;=\; B/S \xrightarrow{\ i\ } \hat{R}
  \;=\; \{\, x \in B \mid A \mid x \in T'(B_{/x}) \,\}
\]

\[
  T' = T_{B} \cdot (T_{i} \to T_{B}) = T_{B/S}, \qquad
  T'(\varepsilon) = \{\, \varepsilon \xrightarrow{\ f\ } B,\ f^{*}(S) \simeq
    e_{\varepsilon} \,\}
\]

\[
      \mathcal{O}_{A} \xrightarrow{\ \sim\ }
      \varinjlim_{n} \Bigl(\mathrm{Ker}\bigl(A \xrightarrow{\ x \mapsto x^{n}\ }
      A\bigr)\Bigr) \quad [\, I(u) \,]
    \]

\[
  F(X) \longrightarrow \prod_{i} F(X_{i}) \rightrightarrows
  \prod_{i,j} F(X_{i} \times_{X} X_{j})
\]

\[
  \begin{cases}
    \forall f, g \in \struck{A}(U), \quad
    \varinjlim_{n} V(f^{n}g^{n}) \Longleftarrow
    \varinjlim_{n} V(f^{n}) \vee \varinjlim_{n} V(g^{n}) \\
    V(1_{A}) \simeq \emptyset_{\varepsilon}
  \end{cases}
\]

\[
  X_{\mathrm{irr}} \simeq \tilde{R}_{\mathrm{topirr}}
\]

\[
  X_{\mathrm{cpct}} \simeq \tilde{R}_{\mathrm{top\,cpct}}
\]

\[
      A^{\ast} \amalg A \xrightarrow{\ \mathrm{incl},\ 0_{A}\ } A
      \quad \text{est épi}
    \]

\[
      X_{\mathrm{ps.corps}} \simeq \tilde{R}_{\mathrm{top\,\struck{\ill{}}}}
    \]

\[
      \simeq \tilde{R}_{\mathrm{cons}}
    \]

\[
      \forall f \in A(U),\ \exists (!)\ e \in A(U),\ e^{2} = e,\
      ef \text{ inversible dans } eA(U),\ \struck{e}\,(1-e)f \text{ nilpotent.}
    \]

\[
      i_{n} : D_{n} \hookrightarrow A
    \]

\[
      \forall f \in A(U),\quad A(U) \xrightarrow{\ \sim\ } A_{f} \times A/fA,
      \text{ i.e. } \exists (!)\ e \in A(U),\ e^{2} = e,\ ef \text{ inv.},\
      \uncertain{e(1-f) = 0}
    \]

\[
      X_{\mathrm{cpct}} \simeq \tilde{R}_{\mathrm{top\,ps\,corps}}
      \simeq \hat{R}_{\mathrm{cons}}
    \]

\[
      \hat{R}_{\mathrm{cons}} \subset \hat{R}_{\uncertain{\mathrm{cpct}}}
      \quad \text{formé des } R^{\circ} \to \mathrm{Ens} \text{ qui
      transforment les flèches précédentes \ill{}}
    \]

\[
      X_{\mathrm{corps\,hens}} \simeq \tilde{R}_{\mathrm{top\,corps\,sépclos}}
    \]

\[
      \simeq \widetilde{(R_{\mathrm{cons}})}
    \]

\[
      X_{\mathrm{corps\,alg\,clos}} \simeq
      \tilde{R}_{\mathrm{top\,corps\,alg\,clos}}
    \]

\[
      \simeq \tilde{R}_{\mathrm{cons},\,\mathrm{fppf}}
    \]

\[
      X_{\struck{n\text{-tors}}\ \mathbb{Z}/n\mathbb{Z}}
      \simeq \tilde{R}_{\mathrm{top}\ n\text{-tors}}
      \simeq \hat{R}_{\mathbb{Z}/n\mathbb{Z}}
    \]

\[
      X_{\mathbb{Z}[1/n]} \simeq \tilde{R}_{\mathrm{top}\,\mathbb{Z}[1/n]}
      \simeq \hat{R}_{\mathbb{Z}[1/n]}
    \]

\[
  X_{\Lambda\text{-alg}} \simeq \tilde{R}_{\Lambda\text{-top}}
  \simeq \hat{R}_{\Lambda}
\]

\[
  X_{\mathbb{Q}} = X_{\mathrm{car}\,0} = \struck{\ill{}}
  \bigcap_{n \geqslant 2} X_{\mathbb{Z}[1/n]}
\]

\[
  \mathcal{C} = (\mathrm{Pro}(R))^{\circ} = \mathrm{Ind}(\mathcal{S})
  \qquad (\struck{\ill{}} = R^{\circ})
\]

\[
  \hat{R}_{/S_{\alpha}} = \bigcap_{\alpha} \hat{R}_{/S_{\alpha}}
  = \widetilde{\varprojlim_{\alpha} R_{/S_{\alpha}}}
\]

\[
      p\,1_{A} = 0, \qquad A \xrightarrow{\ x \mapsto x^{p}\ } A \text{ isom}
    \]

\[
      X_{p\text{-parf}} = \tilde{R}_{\mathrm{top}\,\struck{\ill{}}}
      = \hat{R}_{p\text{-parf}}
    \]

\[
      X_{\struck{\ill{}}\ p\text{-parf}} \simeq
      \tilde{R}_{\struck{\ill{}}} \simeq \hat{R}_{\mathrm{cons}\ p\text{-parf}}
    \]

\[
      X_{\mathrm{corps}\ p\text{-parf}} \simeq
      \tilde{R}_{\mathrm{top\ corps}\ p\text{-parf}} \simeq
      \tilde{R}_{\mathrm{cons}\ \uncertain{p\text{-parf}}\ \mathrm{zar}}
    \]

\[
      e_{\varepsilon} \simeq \coprod_{c \in C} U_{c},
    \]

\[
      \Bigl\{\, V(p \cdot 1_{A}) \hookrightarrow e_{\varepsilon} \ ;\
      \bigcap_{n \geqslant 1} (e_{\varepsilon})_{n 1_{A}} \hookrightarrow
      e_{\varepsilon} \,\Bigr\}
    \]

\[
      X_{\mathrm{car}} = \tilde{R}_{\mathrm{top.car}}
    \]

\[
  B = \hat{R} = X_{\mathrm{ann}}, \qquad V = X_{(\mathrm{car} > 0)}
\]

\[
  X_{\mathrm{car}} \simeq \hat{R}_{\mathrm{car}},
\]

\[
      \Pi_{n} \subset A^{n} \ \hookleftarrow\ e_{1}, \ldots, e_{n} \text{ avec }
      \begin{cases}
        e_{i}^{2} = e_{i} \quad \forall i \\
        e_{i} e_{j} = 0 \quad \forall i, j,\ i \neq j \\
        \sum e_{i} = 1
      \end{cases}
    \]

\[
      A_{j} = A_{\Pi_{n}} \Big/ \sum_{j \neq i} e_{j} A_{\Pi_{n}}
      \qquad
      \Pi_{n} \amalg \Pi_{n} \xrightarrow{\ i_{n,j}\ } \Pi(A_{j})
      \quad \text{(proj. de } A_{j}\text{)}
    \]

\[
  N \mapsto M \otimes N \text{ exact}
\]

\[
  \Bigl(\sum f_{i} A\Bigr) \otimes M \longrightarrow M \quad \text{injectif}
\]

\[
  \Bigl(\sum t_{i} A_{U_{n}}\Bigr) \otimes M_{U_{n}} \longrightarrow
  M_{U_{n}} \quad \text{injectif.}
\]

\[
  Y \longleftarrow X
\]

\[
  M \to M_{i}, \qquad Y \leftarrow Y_{i}, \qquad Z \to Y, \quad N \text{ plat}
  \qquad\qquad M \to M_{i}, \quad A \to A_{i}, \quad N \text{ plat}, \quad B
\]

\[
  X_{\mathrm{mod}} = \widehat{\mathrm{Mod}^{\circ}_{\mathrm{pf}}}
\]

\[
  E_{\pi'} \xrightarrow{\ \approx\ } \mathrm{Ind}_{\pi}(E_{\pi})_{\pi'}
  \quad \add{= \mathrm{Ind}_{\pi}(\mathcal{C})_{\pi'}},
\]

\[
  \mathrm{Fil}^{\pi'}(E) = E_{\pi'} \quad \text{pour } \pi' \text{ grand.}
\]

\[
  \underline{\mathrm{Hom}}(E, F)_{\pi} \xrightarrow{\ \sim\ }
  \underline{\mathrm{Hom}}(E_{\pi}, F) \quad \add{\text{pl. fid.}}
\]

\[
  \mathrm{Hom}_{E}(\varphi(Y), X) \simeq \mathrm{Hom}_{\hat{\mathcal{C}}}
  (Y, \hat{\varphi}(X))
\]

\[
  \bar{\varphi} : \hat{\mathcal{C}} \longrightarrow E, \qquad
  \bar{\varphi}(F) = \varinjlim_{\mathcal{C}_{/F}} \varphi(X)
\]

\[
  \mathrm{Hom}_{E}(\bar{\varphi}(F), X) \simeq
  \mathrm{Hom}_{\hat{\mathcal{C}}}(F, \hat{\varphi}(X))
\]

\[
      \beta_{F} : \underline{\mathrm{Hom}}'(E, F) \longrightarrow
      \underline{\mathrm{Hom}}(\mathcal{C}, F)
    \]

\[
  \underline{\mathrm{Hom}}^{*}(\mathcal{C}^{\circ} \times \mathcal{C}'^{\circ},
  \mathrm{Ens})
\]

\[
  \underline{\mathrm{Hom}}^{*}(\mathcal{C}^{\circ} \times \mathcal{C}'^{\circ},
  \mathcal{D}) \simeq T_{\mathcal{C}^{\circ}} T_{\mathcal{C}'^{\circ}}(\mathcal{D})
  \simeq T_{\mathcal{C}'^{\circ}} T_{\mathcal{C}^{\circ}}(\mathcal{D})
\]

\[
  X = \varinjlim_{I} X_{i}, \qquad X_{i} \in \mathcal{C},\ I \text{ grand
  devant } \pi. \quad \struck{\ill{}}
\]

\[
  X = \varinjlim_{I' \in J} X_{I'} .
\]

\[
  X = \varinjlim_{J} X_{j}, \qquad X_{j} \in \mathrm{Fil}^{\pi'}(X) \cap
  E_{\pi'},\ J \text{ grand devant } \pi',
\]

\[
  X = \varinjlim_{j \in J} Y_{j}, \qquad
  Y_{j} = \varinjlim_{I_{j}} \struck{\ill{}} Y_{i} .
\]

\[
  \mathrm{Hom}_{\Delta}(R, C) \longrightarrow
  \mathrm{Hom}_{\mathrm{Cat}}(\tilde{I}, C) = T_{\tilde{I}}(C)
\]

\[
  \Delta' \to \Delta \qquad \rho_{\Delta', \Delta}
\]

\[
  \mathrm{Hom}_{\Delta}(R, \mathrm{Hom}_{\mathrm{Cat}}(J, C)) \simeq
  \mathrm{Hom}_{\mathrm{Cat}}(J, \mathrm{Hom}_{\Delta}(R, C))
\]

\[
  \boxed{\ T_{R}(\mathrm{Hom}_{\mathrm{Cat}}(J, C)) \simeq
  \mathrm{Hom}_{\mathrm{Cat}}(J, T_{R}(C))\ }
\]

\[
  T_{R}(\hat{J}) \simeq \mathrm{Hom}(J^{\circ}, T_{R}(\mathrm{Ens}))
\]

\[
          T_{R}(C) \longrightarrow T_{R}(\hat{C}) \simeq
          \mathrm{Hom}_{\mathrm{Cat}}(C^{\circ}, T_{R}(\mathrm{Ens}))
        \]

\[
      \boxed{\ R^{\circ} \xrightarrow{\ \partial\ } S\ }
    \]

\[
      R \xrightarrow{\ \partial^{\circ}\ } S^{\circ} \longrightarrow
      \widehat{S^{\circ}}
    \]

\[
      \mathrm{Hom}(I, C) \simeq \mathrm{Hom}(I^{\circ}, C^{\circ})^{\circ}
      \simeq \mathrm{Hom}_{\Delta^{\circ}}(\widehat{I^{\circ}},
      C^{\circ})^{\circ} \simeq \mathrm{Hom}_{\Delta}
      (\widehat{I^{\circ}}{}^{\circ}, C)
    \]

\[
  T(C) \simeq \mathrm{Hom}_{\Delta}(S^{\circ}, C)
\]

\[
  T(C) \longrightarrow \mathrm{Hom}(C^{\circ}, T(\mathrm{Ens})) =
  \mathrm{Hom}(C^{\circ}, S)
\]

\[
      \mathrm{Hom}(\psi(Y'), X) \simeq \mathrm{Hom}(Y', \varphi(X))
    \]

\[
  \mathrm{Hom}_{S}(x, i(y')) \simeq \mathrm{Hom}(jx, y')
\]

\[
      \mathrm{Hom}_{\struck{\mathrm{Cat}}}(I, C) \longrightarrow
      \mathrm{Hom}_{\Delta_{1}}(\bar{I}, \bar{C}),
    \]

\[
      \mathrm{Hom}_{\mathrm{Cat}}(I, C) \longrightarrow
      \mathrm{Hom}_{\Delta}(\tilde{I}, C),
    \]

\[
      \rho(C) = \mathrm{Hom}_{\Delta}(C, \mathrm{Ens})^{\circ} =
      T_{C}(\mathrm{Ens})^{\circ},
    \]

\begin{gather*}
  \mathrm{Hom}_{\Delta_{1}}(C, D) \simeq \mathrm{Hom}_{\Delta_{1}}
  (\rho(C), D) \simeq \ill{}, \\
  \rho(C) \cong T_{\rho(C), \Delta_{1}}(\mathrm{Ens})^{\circ} \cong
  T_{C, \Delta}(\mathrm{Ens})^{\circ} = \mathrm{Hom}_{\Delta}(C,
  \mathrm{Ens})^{\circ} : \text{OK.}
\end{gather*}

\[
  \mathrm{Hom}_{\Delta, \Sigma}(C, D) \xrightarrow{\ \alpha\ }
  \mathrm{Hom}_{\Delta}(C', D),
\]

\[
  \tilde{I}'\Sigma^{-1} \longrightarrow R,
\]

\begin{align*}
      R :\quad & T = T_{R} : C \mapsto
        \underline{\mathrm{Hom}}_{\mathrm{lex}}(R, C) \\
      & S = \struck{\ill{}}\ \underline{\mathrm{Hom}}_{\mathrm{lex}}(R,
        \mathrm{Ens}) \\
      & \Sigma \cong R^{\circ} \subset S \text{ par foncteurs repr.} \\
      \Sigma :\quad & R = \Sigma^{\circ} \\
      & T : C \mapsto \underline{\mathrm{Hom}}_{\mathrm{lex}}
        (\Sigma^{\circ}, C) \\
      & S = \underline{\mathrm{Hom}}_{\mathrm{lex}}(S^{\circ}, C),
        \text{ contient } \Sigma \text{ comme sous-catégorie pleine.}
    \end{align*}

\[
      \mathrm{Ind}(\Sigma) \longrightarrow S
    \]

\begin{align*}
  S :\quad & \Sigma = S_{\mathrm{pf}} \subset S \\
  & R = \Sigma^{\circ} \\
  & T : C \mapsto \underline{\mathrm{Hom}}_{\mathrm{lex}}
    (S_{\mathrm{pf}}{}^{\circ}, C) \\
  T :\quad & S = T(\mathrm{Ens}) \\
  & \Sigma = S_{\mathrm{pf}} = T(\mathrm{Ens})_{\mathrm{pf}} \\
  & R = \Sigma^{\circ}
\end{align*}

\[
  \underline{\mathrm{Hom}}_{\mathrm{Th}_{\Delta_{0}}}(T, T')
  \longrightarrow \underline{\mathrm{Hom}}_{\mathrm{Th}_{\Delta_{1}}}
  (T_{1}, T'_{1}) \quad \text{pl. fidèle.}
\]

\[
  S = T(\mathrm{Ens}) \add{= T_{1}(\mathrm{Ens})} \longrightarrow S' =
  T'(\mathrm{Ens}) = T'_{1}(\mathrm{Ens})
\]

\[
  \Sigma = R^{\circ} \subset S = T_{R}(\mathrm{Ens}) =
  \underline{\mathrm{Hom}}_{\mathrm{prod}}(R, \mathrm{Ens}).
\]

\[
  \sigma(R) = (C \mapsto \mathrm{Hom}(R, C))
\]

\[
  \mathrm{Hom}(\sigma(R), T) \longrightarrow
  \mathrm{Hom}_{\Delta}(\rho(T), R) = (\sigma(\rho(T)))(R)
\]

\[
      \mathrm{Hom}_{\mathrm{Cat}}(I, C) \simeq \mathrm{Hom}_{\Delta}(\tilde{I}, C)
    \]

\[
  \mathrm{Hom}_{\mathrm{Cat}}(D, C) \xrightarrow{\ \approx\ }
  \varprojlim \mathrm{Hom}_{\mathrm{Cat}}(A_{i}, C)
\]

\[
  \mathrm{Hom}(T, T') \simeq T'(R) ;
\]

\[
      T'(C) = \prod^{(2)}_{\alpha, C^{I}} T_{\alpha}(C)
    \]

\[
      \mathrm{Hom}_{\Delta}(R, C) \xrightarrow{\ \text{fid}\ }
      \mathrm{Hom}_{\mathrm{Cat}}(\rho_{0}, C) \simeq
      \mathrm{Hom}_{\Delta}(\tilde{\rho}_{0}, R)
    \]

\[
  \Sigma = R^{\circ}, \qquad S = \mathrm{Hom}_{\lambda}(R, (\mathrm{Ens}))
  \subset \hat{\Sigma} \qquad\qquad
  \Sigma \subset S \subset \widehat{R^{\circ}} = \hat{\Sigma}
\]

\[
  \varepsilon(X) =
  \begin{cases}
    \emptyset & \text{si } X \neq \emptyset \\
    \{e\} & \text{si } X = \emptyset
  \end{cases}
\]

\begin{equation}
      \lambda = (\overleftarrow{\lambda}, \overrightarrow{\lambda})
      \tag{1.1}
    \end{equation}

\begin{equation}
      (\mathrm{Cat}_{\lambda}) = \text{2-catégorie}
      \tag{1.2}
    \end{equation}

\begin{equation}
      T_{\lambda} \text{ cat. cofibrée sur } \mathrm{Cat}_{\lambda}
      \tag{1.3}
    \end{equation}

\begin{equation}
      \mathrm{Hom}_{\lambda}(T, T') =
      \mathrm{Hom}_{\mathrm{fib\ sur\ }(\mathrm{Cat}_{\lambda})}
      (T_{\lambda}, T'_{\lambda})
      \tag{1.4}
    \end{equation}

\begin{equation}
      (\lambda\text{-types}) \xrightarrow{\ \text{2-fidèle}\ }
      \text{2-Cofib}(\mathrm{Cat}_{\lambda})
      \tag{1.5}
    \end{equation}

\begin{equation}
      \tau^{I}(C) = C^{I}
      \tag{2.1}
    \end{equation}

\begin{equation}
      T_{\lambda}(C) \xrightarrow{\ b^{C,T}_{\lambda}\ } C^{I}
      \qquad [\text{foncteur 0-fidèle}]
      \tag{2.2}
    \end{equation}

\begin{equation}
      b^{T}_{\lambda} : T_{\lambda} \longrightarrow \tau^{I}
      \tag{2.3}
    \end{equation}

\begin{equation}
  \mathrm{Hom}_{\lambda}(I; T, T') = \text{catégorie des}
  \left(\text{morphismes } u : T_{\lambda} \to T'_{\lambda}
  \ \struck{\text{avec}}\ \text{isom. de } \struck{\text{commutation}}\ \alpha\right)
  \tag{2.4}
\end{equation}

\begin{equation}
  (\lambda\text{-}I\text{-types}) \hookrightarrow
  \text{Cat 2-cofib}(\mathrm{Cat}_{\lambda})/\tau^{I}
  \tag{2.5}
\end{equation}

\begin{equation}
  (\lambda\text{-}I\text{-types}) \longrightarrow (\lambda\text{-types})
  \tag{2.6}
\end{equation}

\begin{equation}
      \tau(C) = C
      \tag{3.1}
    \end{equation}

\begin{equation}
      \mathrm{Hom}_{\lambda}(T, \tau) \simeq R^{\lambda}_{T}
      \qquad \struck{(\text{ou } R_{T}(\ldots))}
      \tag{3.2}
    \end{equation}

\begin{equation}
      \xi^{\lambda} : R^{\lambda}_{T} \longrightarrow C
      \tag{3.3}
    \end{equation}

\begin{equation}
      R^{\lambda}_{T} \times T(C) \longrightarrow C
      \qquad \struck{\ill{}} \qquad
      \text{pour } C \in \mathrm{Ob}\,\mathrm{Cat}_{\lambda}
      \tag{3.4}
    \end{equation}

\begin{equation}
      R^{\lambda}_{T} \longrightarrow \mathrm{Hom}(T(C), C)
      \tag{3.5}
    \end{equation}

\begin{equation}
      I \longrightarrow \mathrm{Ob}\,R^{\lambda}_{T}
      \qquad (\text{fonctoriel en } T \in \mathrm{Ob}(\lambda\text{-}I\text{-types}))
      \tag{3.5}
    \end{equation}

\begin{equation}
      T(C) \cong \mathrm{Hom}_{\lambda}(R_{T}, C)
      \tag{4.1}
    \end{equation}

\begin{equation}
      \xi^{\lambda}_{T} \in T(R_{T})
      \tag{4.2}
    \end{equation}

\begin{equation}
  \mathrm{Hom}_{\mathrm{fib}/(\mathrm{Cat}_{\lambda})}(T_{\lambda}, \mathcal{F})
  \cong \mathcal{F}(R_{T})
  \tag{4.3}
\end{equation}

\begin{equation}
  \mathrm{Hom}_{\lambda}(T, T') \cong T'(R_{T})
  \qquad (\text{unique à isom. unique près})
  \tag{4.4}
\end{equation}

\begin{equation}
  \struck{(4.5)}\qquad
  \mathrm{Hom}_{\lambda}(T, \tau) \cong \tau(R_{T}) = R_{T}
\end{equation}

\begin{equation}
  R_{T} = R^{\lambda}_{T} \quad (\text{catégorie des } \lambda\text{-constructions})
  \tag{4.5}
\end{equation}

\begin{equation}
  T(\mathrm{Ens}) \cong \mathrm{Hom}_{\lambda}(R^{\lambda}_{T}, (\mathrm{Ens}))
  \quad \struck{\cong (R^{\lambda}_{T})^{\circ} \ldots}
  \tag{4.6}
\end{equation}

\[
      B \xrightarrow{\ \alpha\ } \mathrm{Ob}\,S^{\lambda}_{T,I} \times
      \mathrm{Ob}\,S^{\lambda}_{T,I}
    \]

\[
      \begin{cases}
        \text{couples } (\xi, (u_{\rho,\sigma})_{\rho,\sigma \in
        \mathrm{Ob}\,S^{\lambda}_{T,I}}) \\
        \text{avec } \xi \in \mathrm{Ob}\,T(C) \text{ et }
        u_{\rho,\sigma} \in \mathrm{Hom}_{C}(\rho(\xi), \sigma(\xi)) \\
        (\forall \rho, \sigma \in \mathrm{Ob}\,S^{\lambda}_{T,I})
      \end{cases}
    \]

\[
      \struck{\prod}\ \prod_{I}((T_{j})_{j \in J})(C) =
      \text{produit } \uncertain{\text{2-}}\text{fibré des } T_{j}(C)
      \text{ sur } C^{I}
    \]

\[
  \mathrm{Hom}(F, G) \longrightarrow \prod_{s \in S}
  \mathrm{Hom}(F(s), G(s))
\]

\begin{equation}
  (\lambda\text{-types}) \xrightarrow{\ \eta_{\lambda\lambda'}\ }
  (\lambda'\text{-types})
  \tag{6.1}
\end{equation}

\begin{equation}
  (\mathrm{Cat}_{\lambda}) \xrightarrow{\ \eta_{\lambda\lambda'}\ }
  (\mathrm{Cat}_{\lambda'})
  \tag{6.2}
\end{equation}

\begin{equation}
  T^{\circ}_{\lambda}(C) \cong T_{\lambda}(C^{\circ})^{\circ}
  \tag{7.1}
\end{equation}

\begin{equation}
  T^{\circ}_{\lambda} \cong (T_{\lambda})^{\circ} \circ (\ldots)
  \tag{7.1}
\end{equation}

\begin{equation}
  (\lambda\text{-types}) \cong (\lambda^{\circ}\text{-types})
  \tag{7.3}
\end{equation}

\begin{equation}
  (\lambda\text{-}I\text{-types}) \cong
  (\lambda^{\circ}\text{-}I\text{-types})\ )
  \tag{7.4}
\end{equation}

\begin{equation}
  (\mathrm{Cat}_{\lambda}) \cong (\mathrm{Cat}_{\lambda^{\circ}}),
  \qquad C \longmapsto C^{\circ}
  \tag{7.6}
\end{equation}

\[
  \rho : T(C) \longrightarrow C
\]

\begin{equation}
  i : C \longrightarrow C'
  \tag{8.2.1}
\end{equation}

\begin{equation}
  T(i) : T(C) \longrightarrow T(C')
  \tag{8.2.2}
\end{equation}

\begin{equation}
  i : C \hookrightarrow \hat{C}
  \tag{8.2.4}
\end{equation}

\begin{equation}
  C^{\circ} \longrightarrow \mathrm{Hom}_{\lambda}(\hat{C},
  (\mathrm{Ens})), \qquad X \longmapsto (F \mapsto F(X))
  \tag{8.2.6}
\end{equation}

\[
  \mathrm{Hom}_{\lambda}(\hat{C}, (\mathrm{Ens})) \longrightarrow
  \mathrm{Hom}(T(\hat{C}), T((\mathrm{Ens})))
\]

\begin{equation}
  C^{\circ} \longrightarrow \mathrm{Hom}(T(\hat{C}), T(\mathrm{Ens}))
  \tag{8.2.7}
\end{equation}

\begin{equation}
  T(\hat{C}) \xrightarrow{\ \alpha\ \approx\ } \mathrm{Hom}(C^{\circ},
  S_{T}) \qquad \text{où } S_{T} = T(\mathrm{Ens})
  \tag{8.2.8}
\end{equation}

\begin{equation}
  \begin{cases}
    S = S_{T} = T(\mathrm{Ens}), \text{ avec les foncteurs} \\
    \beta_{i} : S_{T} \longrightarrow (\mathrm{Ens})
  \end{cases}
  \tag{8.2.11}
\end{equation}

\begin{equation}
  \beta \longmapsto \beta_{\mathrm{Ens}} : R = R^{\lambda}_{T}
  \longrightarrow \mathrm{Hom}(S_{T}, (\mathrm{Ens})) =
  \widehat{S^{\circ}_{T}}
  \tag{8.2.13}
\end{equation}

\begin{equation}
  R \longrightarrow \mathrm{Hom}(\mathrm{Hom}_{\lambda}(R, \mathrm{Ens}),
  (\mathrm{Ens}))
  \tag{8.2.14}
\end{equation}

\begin{equation}
  R^{\circ} \subset S \subset \widehat{R^{\circ}}
  \tag{8.2.15}
\end{equation}

\[
  \pi : R \times S \longrightarrow (\mathrm{Ens})
\]

\begin{equation}
  \pi(c, \xi) = \xi(c) =
  \mathrm{Hom}_{\widehat{R^{\circ}}}(c, \xi) = \mathrm{Hom}_{S}(c, \xi)
  \tag{8.2.16}
\end{equation}

\begin{equation}
  R \hookrightarrow S^{\circ}
  \hookrightarrow \widehat{S^{\circ}} =
  \mathrm{Hom}(S, (\mathrm{Ens}))
  \tag{8.2.17}
\end{equation}

\[
  \widetilde{\xi} : C^{\circ} \longrightarrow S, \qquad x \longmapsto
  \mathrm{Hom}(x, \xi)
\]

\[
  x \longmapsto \mathrm{Hom}_{S}(\varphi, \mathrm{Hom}(x, \xi))
\]

\[
  \Sigma^{\circ} \longrightarrow \underline{\mathrm{Hom}}(T(C), C),
  \qquad T(C) \longrightarrow \mathrm{Hom}_{\lambda}(\Sigma^{\circ}, C)
\]

\[
  O_{E} \in \mathrm{Ob}\,T(E) \simeq \mathrm{Ob}\,\mathrm{Hom}_{sex}(C, E)
  \ni f_{0}
\]

\[
  f : E \longrightarrow B_{T}
\]

\[
  O_{E} \simeq f^{*}_{T}(O_{B}) \qquad f_{0} = f^{*} \mid C
\]

\begin{equation}
  \mathrm{Hom}_{E}(U, f^{*}X) \longrightarrow
  \mathrm{Hom}(O_{B}(X), O_{E}(U))
  \tag{1}
\end{equation}

\[
  \mathrm{Hom}_{E}(U, f^{*}(X)) \longrightarrow
  \mathrm{Hom}_{T(\mathrm{Ens})}(O_{E}(f^{*}(X)), O_{E}(U))
  \longrightarrow \mathrm{Hom}_{T(\mathrm{Ens})}(O_{B}(X), O_{E}(U))
\]

\[
  O_{B}(X) \longrightarrow O_{E}(f^{*}(X))
\]

\begin{gather*}
  C \longrightarrow \hat{C} \xrightarrow{\ h_{X}\ } (\mathrm{Ens}),
  \qquad Y \longmapsto \mathrm{Hom}_{C}(X, Y) =
  \mathrm{Hom}_{C^{\circ}}(Y, X) \\
  \to \mathrm{Hom}_{\hat{C}}(X, Y)
  \to \mathrm{Hom}_{E}(f^{*}(X), f^{*}(Y))
\end{gather*}

\[
  O_{B}(X) : C \longrightarrow (\mathrm{Ens}), \qquad
  Y \longmapsto \mathrm{Hom}_{C}(X, Y) = \mathrm{Hom}_{C^{\circ}}(Y, X)
\]

\[
  O_{E}(U) : \qquad\qquad
  Y \longmapsto \mathrm{Hom}_{E}(U, f_{0}(Y))
\]

\[
  \mathrm{Hom}(O_{B}(X), O_{E}(U)) = O_{E}(U)(X) =
  \mathrm{Hom}_{E}(U, f_{0}(X))
\]