\[
  \delta_3(x,y) = (x,x,y), \qquad
  \delta_1(x,y) = (y,x,x), \qquad
  \delta_2(x,y) = (x,y,x),
\]

\[
  p_{12}(x,y,z) = (x,y), \qquad
  p_{23}(x,y,z) = (y,z), \qquad
  p_{31}(x,y,z) = (z,x),
\]

\[
  \rho(x,y,z) = (y,z,x).
\]

\[
  \begin{cases}
    p_{23} = p_{12}\rho \\
    p_{31} = p_{23}\rho = p_{12}\rho^2 \\
    p_{12} = p_{31}\rho = p_{23}\rho^2
  \end{cases}
  \qquad
  \begin{cases}
    \delta_1 = \rho\delta_2 = \rho^2\delta_3 \\
    \delta_2 = \rho\delta_3 = \rho^2\delta_1 \\
    \delta_3 = \rho\delta_1 = \rho^2\delta_2
  \end{cases}
\]

\[
  \sigma p_{12} = p_{12}\sigma_3, \qquad
  \sigma p_{23} = p_{23}\sigma_1, \qquad
  \sigma p_{31} = p_{31}\sigma_2.
\]

\[
  \boxed{p_0\delta_0 = \mathrm{id}_{E_0}}, \qquad \boxed{\sigma\delta_0 = \delta_0},
\]

\[
  \delta_0p_0 = p_{12}\,\underbrace{\rho\delta_1}_{\delta_3}
             = p_{23}\,\delta_1
             = p_{31}\,\underbrace{\rho^2\delta_1}_{\delta_2}.
\]

\[
  (p_{12}\rho)\delta_1 \neq p_{23}(\rho\delta_1) \qquad ??
\]

\[
  \begin{cases}
    \delta_0p_0 = p_{12}\,\underbrace{\rho\delta_1}_{\delta_3} \\
    \mathrm{id}_{E_1} = p_{12}\,\underbrace{\rho^2\delta_1}_{\delta_2} \\
    \sigma = p_{12}\,\delta_1
  \end{cases}
\]

\[
  \boxed{\sigma_3\delta_1 = \delta_1}, \qquad \boxed{p_{12}\sigma_3 = \sigma p_{12}},
\]

\[
  \boxed{\rho\,\delta_1\delta_0 = \sigma_1\,\delta_1\delta_0 = \delta_1\delta_0}, \qquad
  p_0\,p_{12} = p_0\,\underbrace{p_{12}\rho^2}_{p_{31}}.
\]

\[
  \boxed{p_0\delta_0 = \mathrm{id}_{E_0}}, \qquad \boxed{\sigma\delta_0 = \delta_0},
\]

\[
  \begin{cases}
    p_{12}\delta_1 = \sigma & \text{(involution)} \\
    p_{12}\,\underbrace{\rho\delta_1}_{\delta_3} = \delta_0p_0 & \text{(idempotent)} \\
    p_{12}\,\underbrace{\rho^2\delta_1}_{\delta_2} = \mathrm{id} &
  \end{cases}
\]

\[
  \boxed{\sigma_1\delta_1 = \delta_1}, \qquad
  \boxed{p_{12}\sigma_3 = \underbrace{\sigma\,p_{12}}_{p_{21}}},
\]

\[
  \boxed{\rho(\delta_1\delta_0) = \sigma_1(\delta_1\delta_0) = \delta_1\delta_0}
  \quad \text{i.e.} \quad g(\delta_1\delta_0) = \delta_1\delta_0 \ \text{ pour } g \in \mathfrak{S}_3,
\]

\[
  \boxed{p_0\,p_{12}\,\sigma_3 = \underbrace{p_0\,p_{12}}_{p_1}}.
\]

\[
  p_0p_{12} = p_0p_{13} = p_0\,\underbrace{p_{12}\rho^2}_{p_{31}}\,\sigma_2,
  \qquad \rho^2\sigma_2 = \sigma_3\rho^2,
\]

\[
  p_0\,p_{12}\,\sigma_1 = p_0\,p_{12}.
\]

\[
  \begin{cases}
    p_{12}\delta_1 = \sigma \\
    p_{12}\,\underbrace{\rho\delta_1}_{\delta_3} = \text{application constante de valeur } e_1 \\
    p_{12}\,\underbrace{\rho^2\delta_1}_{\delta_2} = \mathrm{id}_{E_1} \\
    \sigma_1\delta_1 = \delta_1 \\
    p_{12}\sigma_3 = \sigma\,p_{12}
  \end{cases}
\]

\[
  p_{12}\delta_1(x) = \sigma(x), \qquad
  \underbrace{p_{12}\,\rho\delta_1}_{p_{23}}(x) = \delta_0p_0(x), \qquad
  \underbrace{p_{12}\,\rho^2\delta_1}_{p_{31}}(x) = x.
\]

\[
  \begin{cases}
    \delta_0p_0(x) = 1 \\
    \sigma(x)\cdot x = 1 \\
    p_{12}(z)\,p_{23}(z)\,p_{31}(z) = 1
  \end{cases}
  \qquad
  \sigma(x)\,\underbrace{\delta_0p_0(x)}_{=\,1}\,x = 1
\]

\[
  E_2(\alpha) \ni \delta_1(\alpha) = \delta_1\sigma(\alpha),
\]

\[
  \Pi_0(F_\alpha \times_F F_\alpha) = \underbrace{p_{12}^{-1}(\alpha) \cap p_{13}^{-1}(\alpha)}_{E_2(\alpha)},
  \quad \text{stable par } \sigma_1.
\]

\[
  \varphi \colon A \longrightarrow X, \qquad
  \alpha \longmapsto U_\alpha\ (= \varphi(\alpha)),
\]

\[
  U_\gamma \longrightarrow U_{\alpha_1} \times_{U_{\alpha_0}} U_{\alpha_2}
\]

\[
  \varphi_! \colon \widehat{A} \longrightarrow X
\]

\[
  f \colon X \longrightarrow \widehat{A}
\]

\[
  \varphi_!\Bigl(\varprojlim_I F_i\Bigr) \longrightarrow \varprojlim_I \varphi_!(F_i)
\]

\[
  \partial_n \colon [1,n] \hookrightarrow [1,n+1] \quad \text{inclusion}, \qquad
  k_n \colon [1,n] \longleftarrow [1,n+1] \quad \text{rétraction}.
\]

\[
  \begin{cases}
    k_n\partial_n = \mathrm{id}_{E_n} \\
    \partial_n\sigma = i_n(\sigma)\,\partial_n & \text{si } \sigma \in \mathfrak{S}_n,\ \text{pour } n \geq 1 \\
    \sigma k_n = k_n\,i'_n(\sigma) & \text{si } \sigma \in \mathfrak{S}_{n-1},\ \text{pour } n \geq 2 \\
    k_n\sigma_{n+1} = k_n &
  \end{cases}
\]

\[
  (*) \quad
  \begin{cases}
    \partial^*_n k^*_n = \mathrm{id}_{E^*_n} \\
    \sigma\,\partial^*_n = \partial^*_n\,i_n(\sigma) & (\sigma \in \mathfrak{S}_n) \\
    k^*_n\,\sigma = i'_n(\sigma)\,k^*_n & \\
    \sigma_{n+1}k^*_n = k^*_n &
  \end{cases}
\]

\[
  (**) \quad
  E^*_{p+q} \xrightarrow{\ (\alpha^*_{p+q,p},\ \beta^*_{p+q,q})\ } E^*_p \times E^*_q
  \quad \text{surjectif pour } p, q \geq 1.
\]

\[
  \alpha^*_{n,p} \colon E^*_n \to E^*_p \quad (n > p \geq 1), \qquad
  \boxed{\alpha^*_{n,p} = \partial^*_p\,\partial^*_{p+1} \cdots \partial^*_{n-1}}
\]

\[
  E^*_n \xrightarrow{\ \partial^*_{n-1}\ } E^*_{n-1} \xrightarrow{\ \partial^*_{n-2}\ } E^*_{n-2} \cdots \xrightarrow{\ \partial^*_p\ } E^*_p
\]

\[
  \beta^*_{n,q} \colon E^*_n \to E^*_q \quad (n \geq q \geq 1), \qquad
  \boxed{\beta^*_{n,q} = \partial'^*_q\,\partial'^*_{q+1} \cdots \partial'^*_{n-1}}
\]

\[
  E^*_n \xrightarrow{\ \partial'^*_{n-1}\ } E^*_{n-1} \xrightarrow{\ \partial'^*_{n-2}\ } E^*_{n-2} \cdots \xrightarrow{\ \partial'^*_q\ } E^*_q
\]

\[
  \partial'^*_m \colon E^*_{m+1} \to E^*_m, \qquad \boxed{\partial'^*_m = \partial^*_m\,\rho^*_{m+1}}
\]

\[
  F = \varinjlim_{A/F}\alpha, \qquad G = \varinjlim_{A/G}\beta,
\]

\[
  F \times G = \varinjlim_{A/F}\bigl(\alpha \times G\bigr)
             = \varinjlim_{A/F \times A/G} \alpha \times \beta,
\]

\[
  \varphi_!(F \times G) = \varinjlim_{A/F \times A/G}
    \underbrace{\varphi_!(\alpha\times\beta)}_{\varphi_!(\alpha)\times\varphi_!(\beta)},
\]

\[
  \varphi_!(F) = \varinjlim_{A/F}\varphi_!(\alpha), \qquad
  \varphi_!(G) = \varinjlim_{A/G}\varphi_!(\beta).
\]

\[
  u(F) = \bigl(\alpha \longmapsto \mathrm{Hom}(\varphi(\alpha), F)\bigr), \qquad
  v(G) = \varinjlim_{\alpha \in A/G} \varphi(\alpha),
\]

\[
  X \longrightarrow \widehat{A}, \qquad Y \mapsto \bigl(a_i \to \varphi^*(Y)\bigr), \qquad \varphi(a_i) \to Y.
\]

\[
  \mathfrak{S}_2 \ \Bigl|\ X_2 \twoheadrightarrow X_1 \times X_1, \qquad
  \underbrace{p_0\sigma}_{\|} = p_1, \quad (p_0,p_1) \text{ épi},
\]

\[
  X_3 \xrightarrow{\ (p_{12},\,p_{13})\ \text{épi}\ } X_2 \times X_1
     \xrightarrow{\ (p_0,p_1)\times\mathrm{id}\ } X_1 \times X_1 \times X_1,
\]

\[
  X_3 \xrightarrow{\ \text{épi}\ (p_{12},\,p_{23})\ } X_2 \times_{X_1} X_2
     \xrightarrow{\ \text{épi}\ } X_2 \times_{X_1} (X_1 \times X_1),
\]

\[
  k_2\delta_2 = \mathrm{id}_{X_2}.
\]

\[
  \zeta = (x,y), \qquad
  d\sigma(\zeta) = \uncertain{(y,x,x)}, \qquad
  \rho^2 d(\zeta) = \uncertain{(y,x,y)}.
\]

\[
  \mathfrak{S}_2 = \{1,\sigma\}, \qquad
  \mathfrak{S}_3 = \{1,\rho,\rho^2,\sigma_1,\sigma_2,\sigma_3\},
\]

\[
  \sigma_i^2 = 1, \qquad
  \bigl[\ \rho^3 = 1, \quad
  \rho\,\sigma_i\,\rho^{-1} = \sigma_{i+1} \quad (\sigma_4 = \sigma_1)\ \bigr].
\]

\[
  d(x,y) = (x,y,y), \qquad
  \rho\,d(x,y) = (y,y,x), \qquad
  \rho^2 d(x,y) = (y,x,y),
\]

\[
  p(x,y,z) = (x,y),
\]

\[
  \rho(x,y,z) = (y,z,x), \qquad \rho^2(x,y,z) = (z,x,y),
\]

\[
  \sigma_1(x,y,z) = (x,z,y), \qquad
  \sigma_2(x,y,z) = (z,y,x), \qquad
  \sigma_3(x,y,z) = (y,x,z).
\]

\[
  \boxed{p\,d = \mathrm{id}_{X_2}}, \qquad
  \boxed{p\,\rho\,d = e}, \qquad
  \boxed{p\,\rho^2 d = \sigma},
\]

\[
  \boxed{\sigma_1 d = d}, \qquad \boxed{p\,\sigma_3 = \sigma\,p},
\]

\[
  \mathfrak{S}_3 = \{1,\sigma_1\}\cdot\{1,\rho,\rho^2\}
                 = \{1,\rho,\rho^2\}\cdot\{1,\sigma_1\}.
\]

\[
  \begin{cases}
    \text{si } j = 0 : & \xi' = \xi \\
    \text{si } j = 1 : & \xi' = p\,\rho\,d(\xi) = e_2 \\
    \text{si } j = 2 : & \xi' = p\,\rho^2 d(\xi) = \sigma\xi
  \end{cases}
\]

\[
  X'_3 \simeq \{e_3\} \amalg
    \bigl(\mathfrak{S}_3 \wedge_{\{1,\sigma_1\}} X^{*}_2\bigr),
\]

\[
  X^{*}_3 \overset{\text{déf}}{=} X_3 \setminus
    \underbrace{X'_3}_{\mathfrak{S}_3\cdot d(X_2)}
\]

\[
  X_3 \xrightarrow{\ (p,\,p\rho)\ } X_2 \times X_2 \quad \text{épi} :
\]

\[
  \underbrace{(X_2 \times \{e_2\})}_{\Sigma_1} \cup
  \underbrace{(\{e_2\} \times X_2)}_{\Sigma_2} \cup \Delta'
\]

\[
  X_3 = \{e_3\} \amalg \{d\xi,\ \rho\,d\xi,\ \rho^2 d\xi\}.
\]

\[
  \delta_3 = \rho^{\ill}\,d\sigma .
\]

\[
  p^{-1}(e_2) \overset{?}{=} \delta_3(X_2)
    \quad \bigl[\ = \rho\,d\sigma(X_2) = \rho\,d(X_2)\ \bigr],
\]

\[
  p\xi = \begin{cases}
    \eta \ (\neq e_2) & \text{si } i = 0 \\
    \sigma\eta \ (\neq e_2) & \text{si } i = 2
  \end{cases}
\]

\[
  (1) \qquad Y \hookrightarrow X
\]

\[
  (3) \qquad Y \hookrightarrow \mathcal{V}_{Y,X} \hookrightarrow X ,
\]

\[
  (4) \qquad Y \xrightarrow{\ \text{équiv. d'homotopie}\ } \mathcal{V}_{Y,X}.
\]

\[
  (5) \qquad \mathcal{V}^{*}_{Y,X} \overset{\text{déf}}{=}
    \mathcal{V}_{Y,X} \cap X^{*} = \mathcal{V}_{Y,X} \setminus Y
    \qquad (X^{*} = X \setminus Y),
\]

\[
  (9) \qquad \Pi_1(X) \simeq \Pi_1(X^{*})
    \wedge_{\Pi_1\mathcal{V}^{*}_{Y,X}}
    \underbrace{\Pi_1(\mathcal{V}_{Y,X})}_{\simeq\ \Pi_1(Y)} ,
\]

\[
  (11) \qquad \mathcal{V}^{*}_{Y,X} \longrightarrow \mathcal{V}_{Y,X}
\]

\[
  (12) \qquad \cdots \to \Pi_i(\mathcal{V}^{*}_{z,Z})
    \to \Pi_i(\mathcal{V}^{*}_{Y,X}) \to \Pi_i(Y)
    \to \Pi_{i-1}(\mathcal{V}^{*}_{z,Z}) \to \cdots ,
\]

\[
  (15) \qquad \Pi_i(Y),\ \Pi_i(X^{*}),\ \Pi_i(\mathcal{V}^{*}_{z,Z})
    \text{ nuls pour } i \geq 2 ,
\]

\[
  (16) \qquad F|X^{*} \text{ et } F|Y \text{ localement triviaux} ,
\]

\[
  (17) \qquad (E_{X^{*}},\ E_{Y,X}\,;\ \varphi)
\]

\[
  (18) \qquad \varphi \colon p^{*}(E_{Y,X}) \longrightarrow i^{*}(E_{X^{*}}) .
\]

\[
  (21) \qquad \begin{cases} X = \mathbb{C} \text{ plan complexe} \\ a = \{0\} \end{cases}
\]

\[
  (22) \qquad \mathfrak{F} \simeq \widehat{C}
\]

\[
  p^{*}(a) \longrightarrow b \ \text{ dans } B
  \qquad (\text{resp. } i^{*}(a) \longrightarrow c \ \text{ dans } C).
\]

\[
  \begin{cases}
    X = \mathbb{P}^1_{\mathbb{C}} \simeq S^2 \\
    Y = \{\infty\} \\
    X^{*} = \mathbb{C}
  \end{cases}
\]

\[
  \mathfrak{F} \simeq \widehat{C} ,
\]

\[
  (24) \qquad X \longrightarrow \mathfrak{F}
\]

\[
  (24) \qquad
  \mathcal{G}_{Y,X}\bigl|\,\mathcal{V}^{*}_{Y,X}
  \;\simeq\;
  \mathcal{G}_{X^{*}}\bigl|\,\mathcal{V}^{*}_{Y,X} .
\]

\[
  (25) \qquad H^{2}(S^{2}, A) \;\simeq\; A ,
\]

\[
  (1) \qquad (X_i)_{i \in I}, \qquad X_i \subset X
\]

\[
  (1') \qquad
  \begin{cases}
    \text{a) Les } X_i \text{ fermés, la famille } (X_i) \text{ localement finie.} \\
    \text{b) } i \leq j \implies X_i \subset X_j .
  \end{cases}
\]

\[
  (2) \qquad
  X_{\Delta_0} \;=\; \coprod_{i \in I} X_i
  \;=\; \bigl\{ (x,i) \in X \times I \ \big|\ x \in X_i \bigr\}
\]

\[
  (3) \qquad X_{\Delta_0} \longrightarrow X
\]

\[
  (4) \qquad
  X_{\Delta_1} \;\subset\; X_{\Delta_0} \times X_{\Delta_0}
  \;=\; \coprod_{(i,j) \in I \times I} X_i \times X_j ,
\]

\[
  \phantom{(4)} \qquad
  X_{\Delta_1} \underset{\text{déf}}{=} \coprod_{i \leq j} \Gamma_{i,j}
\]

\[
  (5) \qquad
  \Gamma_{ij} \subset X_i \times X_j \quad
  \text{« graphe de l'inclusion } X_i \hookrightarrow X_j \text{ ».}
\]

\[
  (7) \qquad
  \begin{cases}
    \text{a) } X_{\Delta_1} \xrightarrow{\ \sigma\ } X_{\Delta_0}
      \text{ est un revêtement étale} \\
    \qquad (\text{sur chaque pièce } X_i)
      \text{ — en fait constant,} \\
    \qquad \text{de fibre égale à } I_i = \{ j \in I \mid j \geq i \}, \\
    \qquad \text{donc rev. fini si les } I_i \text{ sont finis} \\
    \text{b) } X_{\Delta_1} \xrightarrow{\ b\ } X_{\Delta_0}
      \text{ est une immersion locale.}
  \end{cases}
\]

\[
  (8) \qquad (x,i) \leq (y,j) \iff x = y, \quad i \leq j
\]

\[
  (10) \qquad
  \begin{cases}
    X_{\Delta_r} = \bigl\{ (x_0, \ldots, x_r) \in X_{\Delta_0}^{\,r+1}
      \ \big|\ x_0 \leq \cdots x_r \bigr\} \\[2pt]
    \phantom{X_{\Delta_r}} \simeq \bigl\{ (x, i_0, \ldots, i_r)
      \in X \times I^{\,r+1} \ \big| \\
    \qquad\qquad x \in X_{i_0},\ i_0 \leq \cdots \leq i_r \bigr\} \\[2pt]
    \phantom{X_{\Delta_r}} \simeq \coprod_{i_{*} \in I(\Delta_r)} X_{i_{*}} \\[2pt]
    \phantom{X_{\Delta_r}} \simeq \coprod_{i_0 \in I} X_{i_0} \times I_i(\Delta_{r-1})
  \end{cases}
\]

\[
  (11) \qquad
  \begin{cases}
    I(\Delta_r) = \mathrm{Hom}_{\text{ens. ord.}}(\Delta_r, I) \\[2pt]
    X_{i_{*}} = X_{i_0, i_1, \ldots, i_r} \underset{\text{déf}}{=} X_{i_0}
      \quad \text{si } i_{*} \in I(\Delta_r) .
  \end{cases}
\]

\[
  (12) \qquad X_{\Delta_{*}} = (X_{\Delta_r})_{r \geq 0}
\]

\[
  (13) \qquad \alpha^{*} : X_{\Delta_r} \longrightarrow X_{\Delta_{r'}}
\]

\[
  (14) \qquad \alpha : \Delta_{r'} \longrightarrow \Delta_r
\]

\[
  \alpha^{*}(x, i_{*}) = \bigl( x,\ \alpha^{*}(i_{*}) \bigr),
  \qquad \alpha^{*}(i_{*}) \underset{\text{déf}}{=} i_{*} \circ \alpha
\]

\[
  (15) \qquad X_{\Delta_{*}} \subset X \times I_{\Delta_{*}} .
\]

\[
  i \leq j \iff X_i \subseteq X_j .
\]

\[
  X_i \subseteq X_j \implies i \leq j
\]

\[
  (16) \qquad
  \begin{cases}
    \text{a) si } \alpha : \Delta_{r'} \to \Delta_r
      \text{ est tel que } \alpha(0) = 0, \text{ alors} \\
    \qquad X_{\Delta_r} \xrightarrow{\ \alpha^{*}\ } X_{\Delta_{r'}}
      \text{ est un morphisme de rev. étale} \\
    \qquad \text{(en particulier } X_{\Delta_r} \to X_{\Delta_0}, \\
    \qquad\quad (x_0, \ldots, x_r) \mapsto x_0
      \text{ ou } (x, i_0, \ldots, i_r) \mapsto x, \\
    \qquad\quad \text{est un morphisme de rev. étale).}
  \end{cases}
\]

\[
  (17) \qquad
  \alpha^{*-1}\bigl(
    \underbrace{X_{i'_{*}}}_{\textstyle = X_{i'_0} \times \{i'_{*}\}}
  \bigr)
  \;\simeq\;
  X_{i'_{*}} \times
  \bigl\{ i_{*} \in I_{\Delta_r} \ \big|\ \alpha^{*}(i_{*}) = i'_{*} \bigr\} .
\]

\[
  (18) \qquad X_{\Delta_r} \longrightarrow X \quad \text{imm. locale.}
\]

\[
  (20) \qquad X \;\simeq\; X_{\Delta_0} \amalg_{X_{\Delta_1}} X_{\Delta_0} .
\]

\[
  (21) \qquad
  \begin{cases}
    \text{c) } X = \bigcup X_i \\[2pt]
    \text{d) } \forall (i,j) \in I \times I, \ \text{on a}\ \
      X_i \cap X_j = \bigcup_{\substack{k \in I \\ k \leq i,\ k \leq j}} X_k ,
  \end{cases}
\]

\[
  (22) \qquad
  \varinjlim X_{\Delta_{*}}
  \ \bigl( X_{\Delta_0} \leftleftarrows X_{\Delta_1} \Lleftarrow X_{\Delta_2} \cdots \bigr)
  \ \xrightarrow{\ \sim\ } \ X
\]

\[
  (23) \qquad
  \begin{cases}
    \text{a) } X_{\Delta_1} \text{ est un sous-espace top. fermé} \\
    \qquad \text{de } X_{\Delta_0} \times X_{\Delta_0}
      \text{ (avec la top. induite)} \\[2pt]
    \text{b) } \forall (i,j) \in I \times I,\
      \Gamma_{ij} \underset{\text{déf}}{=}
      X_{\Delta_1} \cap (X_i \times X_j) \\
    \qquad \text{est ou bien vide, ou bien le graphe} \\
    \qquad \text{d'une immersion fermée } X_i \hookrightarrow X_j .
  \end{cases}
\]

\[
  X = \varinjlim_{i \in I} X_i
  \ \simeq\ X_{\Delta_0} / (X_{\Delta_1} \rightrightarrows)
\]

\[
  (24) \qquad
  \begin{cases}
    \forall\, i,j \in I, \quad i \wedge j \underset{\text{déf}}{=}
      \mathrm{Inf}(i,j) \text{ existe, et} \\[2pt]
    X_{i \wedge j} = X_i \cap X_j
  \end{cases}
  \qquad ?
\]

\[
  (25) \qquad i(x) = \text{plus petit des } i \in I
  \text{ tels que } x \in X_i .
\]

\[
  (26) \qquad
  \begin{cases}
    \dot{X}_i \underset{\text{déf}}{=} \bigcup_{j < i} X_j
      \quad (\text{fermé de } X_i) \\[2pt]
    X_i^{*} \underset{\text{déf}}{=} X_i \smallsetminus \dot{X}_i
      \quad (\text{ouvert de } X_i) .
  \end{cases}
\]

\[
  (27) \qquad
  \begin{cases}
    \dot{X}_{\Delta_0} = \coprod_{i \in I} \dot{X}_i \\[2pt]
    X^{*}_{\Delta_0} = X_{\Delta_0} \smallsetminus \dot{X}_{\Delta_0}
      = \coprod_{i \in I} X_i^{*}
  \end{cases}
\]

\[
  (28) \qquad
  \begin{cases}
    \dot{X}_{\Delta_r} = \text{image inverse de } \dot{X}_{\Delta_0}
      \text{ par } \sigma_r : X_{\Delta_r} \to X_{\Delta_0} \\
    \qquad (\text{fermé de } X_{\Delta_r}) \\[2pt]
    X^{*}_{\Delta_r} = X_{\Delta_r} \smallsetminus \dot{X}_{\Delta_r}
      = \text{image inverse de } X^{*}_{\Delta_0} \text{ par } \sigma_r \\
    \qquad (\text{ouvert de } X_{\Delta_r}) .
  \end{cases}
\]

\[
  (29) \qquad
  \alpha^{*} : X_{\Delta_r} \longrightarrow X_{\Delta_{r'}} \quad (\text{étale})
\]

\[
  \alpha^{*}\bigl( X^{*}_{\Delta_{r'}} \bigr) = X^{*}_{\Delta_r},
  \qquad \text{i.e.} \qquad
  \alpha^{*}\bigl( \dot{X}_{\Delta_{r'}} \bigr) = \dot{X}_{\Delta_r} .
\]

\[
  (1) \qquad
  \overline{\mathcal{V}}_{i,j} = \mathcal{V}_{X_i, X_j}
  \qquad (\text{voisinage tubulaire de } X_i \text{ dans } X_j)
\]

\[
  (2) \qquad
  \dot{\mathcal{V}}_{i,j}
  = \overline{\mathcal{V}}_{ij} \cap \bigl( R^{*}_{i,j} \bigr)
  = \overline{\mathcal{V}}_{ij} \cap R_i
\]

\[
  (3) \qquad
  R^{*}_{i,j} = \Biggl(\ \bigcup_{\substack{k \in I \text{ tels que} \\ X_k \not\supset X_i}} X_k \Biggr) \cap X_j .
\]

\[
  (4) \qquad
  R^{*}_{i,j} \cap X_i = R_{ij} \cap X_i = \dot{X}_i .
\]

\[
  (5) \qquad
  \mathcal{V}_{ij}
  = \overline{\mathcal{V}}_{ij} \smallsetminus \dot{\mathcal{V}}_{ij}
  = \overline{\mathcal{V}}_{ij} - \overline{\mathcal{V}}_{ij} \cap R_{ij}
\]

\[
  (6) \qquad \mathcal{V}_{ij} \cap X_i = X_i^{*} .
\]

\[
  (7) \qquad
  \mathcal{V}^{*}_{ij}
  = \mathcal{V}_{ij} \smallsetminus \bigcup_{k < j} X_k \cap \overline{\mathcal{V}}_{ij}
  = \overline{\mathcal{V}}_{ij} \smallsetminus \dot{X}_j
  = \mathcal{V}_{ij} \cap X_j^{*} \subset \mathcal{V}_{i,j} .
\]

\[
  (8) \qquad
  \overline{\mathcal{V}}_{ii} = X_i , \qquad
  \dot{\mathcal{V}}_{i,i} = \dot{X}_i , \qquad
  \mathcal{V}_{i,i} = \mathcal{V}^{*}_{i,i} = X_i^{*} .
\]

\[
  \mathcal{V}^{\,d'_{*}}_{d_{*},\, d''_{*}}
  \;\simeq\;
  \widetilde{N}_{d_{*};\,(d'_0,\, d''_0)} \bigl|\ D^{*}_{d_{*}}
\]

\[
  i \leq j \leq k \qquad \text{d'où} \qquad X_i \subset X_j \subset X_k
\]

\[
  (10) \qquad
  \mathcal{V}^{\,j}_{i,k} \underset{\text{déf}}{=}
  \overline{\mathcal{V}}_{X_i, X_k} \smallsetminus
  \bigl( R_{j,k} \cup S_{j,k} \bigr)
\]

\[
  (11) \qquad
  S_{j,k} \underset{\text{déf}}{=}
  \bigcup_{\substack{\ell \in I \\ \text{avec } j \leq \ell \leq k}} X_\ell
\]

\[
  I^{(k)'} = \bigl\{ \ell \in I^{(k)} \ \big|\ X_\ell \not\supset X_j \bigr\}
  \quad \text{et}
\]

\[
  I^{(k)''} = \bigl\{ \ell \in I^{(k)} \ \big|\ X_\ell \supset X_j \bigr\},
\]

\[
  (12) \qquad
  \begin{cases}
    R_{j,k} = \bigcup_{\ell \in I^{(k)'}} X_\ell \\[2pt]
    S_{j,k} = \bigcup_{\ell \in I^{(k)''}} X_\ell
  \end{cases}
  \qquad
  \begin{aligned}
    &\text{Donc } R_{j,k} \cup S_{j,k} = \dot{X}_k \\
    &(\text{avec } R_{j,k} \cap S_{j,k} = \dot{X}_i) .
  \end{aligned}
\]

\[
  (13) \qquad
  \mathcal{V}^{\,i}_{i,k} = \mathcal{V}^{*}_{i,k} , \qquad
  \mathcal{V}^{\,k}_{i,k} = \mathcal{V}_{i,k} ,
\]

\[
  \phantom{(13)} \qquad
  \mathcal{V}^{\,i}_{i,i} = \mathcal{V}_{i,i} = \mathcal{V}^{*}_{i,i}
  \ \ \text{d'où} \ = X_i^{*} .
\]

\[
  (14) \qquad
  \begin{cases}
    \mathcal{V}_{\Delta_1} = \coprod\limits_{\substack{(i,j) \in I^2 \\ i \leq j}}
      \mathcal{V}_{i,j} \\[10pt]
    \mathcal{V}^{*}_{\Delta_1}
      = \coprod\limits_{\substack{(i,j) \in I^2 \\ i \leq j}} \mathcal{V}^{*}_{i,j}
      \qquad \subset \mathcal{V}_{\Delta_1}
  \end{cases}
\]

\[
  (15)
\]

\[
  (17)
\]

\[
  (18) \qquad \mathcal{V}^{\,j}_{i,k} \subset \mathcal{V}^{\,j'}_{i',k'}
\]

\[
  (19) \qquad i \leq j \leq k , \qquad i' \leq j' \leq k'
\]

\[
  X_i^{*} , \quad \mathcal{V}^{*}_{i,j} , \quad \mathcal{V}_{i,j}
  \qquad (j \text{ successeur de } i) \ ?
\]

\[
  \begin{cases}
    X_i^{*} \neq \emptyset \\
    X_i \subset X_j \Rightarrow i \leq j
  \end{cases}
\]

\[
  X_i^{*} \longrightarrow \mathcal{V}_{i,j}
  \qquad (\text{dans les cas « raisonnables »})
\]

\[
  \mathcal{V}^{*}_{i,j} \longrightarrow X_j^{*}
\]

\[
  \mathcal{V}^{*}_{i,j} \longrightarrow \mathcal{V}_{i,j}
\]

\[
  (19)
\]

\[
  (20) \qquad \widetilde{\Delta}_n
\]

\[
  (21)
\]

\[
  (22)
\]

\[
  (23) \qquad I \xrightarrow{\ d\ } \mathbb{Z}
  \qquad \Bigl( \text{i.e.} \quad I = \coprod_m I_m \Bigr)
\]

\[
  (24) \qquad \text{si } i,j \in I ,\ i \leq j ,\
  \text{ alors } d(j) = d(i)+1 \iff j \text{ succes. de } i
\]

\[
  (25) \qquad
  \begin{cases}
    X_m^{*} = \coprod\limits_{i \in I_m} X_i^{*} \\[10pt]
    \mathcal{V}_{m,m+1} = \coprod\limits_{\substack{(i,j) \in I \times I \\
      i \leq j ,\ d(i)=m ,\ d(j)=m+1}} \mathcal{V}_{i,j} \\[16pt]
    \mathcal{V}^{*}_{m,m+1} = \coprod\limits_{i,j \text{ comme dessus}}
      \mathcal{V}^{*}_{i,j}
  \end{cases}
\]

\[
  (26) \qquad (\mathrm{Diag})_m
\]

\[
  (27)
\]

\[
  (1) \qquad
  \begin{cases}
    \text{a)} & X_i \text{ fermés},\ (X_i)_{i \in I} \text{ loc. fini} \\
    \text{b)} & i \leq j \Rightarrow X_i \subset X_j \\
    \text{c)} & X = \bigcup X_i \\
    \text{d)} & \forall i,j \in I ,\ \ldots \quad
      X_i \cap X_j = \bigcup\limits_{k \leq i,j} X_k
  \end{cases}
\]

\[
  (2) \qquad X_i' \overset{\text{déf}}{=} X_i \times_X X'
\]

\[
  (3) \qquad
  \begin{cases}
    X'_{\Delta_r} \simeq X_{\Delta_r} \times_X X' \\
    X'^{\,*}_{\Delta_r} \simeq X^{*}_{\Delta_r} \times_X X' \\
    \dot{X}'_{\Delta_r} = \dot{X}_{\Delta_r} \times_X X'
  \end{cases}
\]

\[
  (4) \qquad
  \begin{cases}
    R'_{ij} = R_{ij} \times_X X' \\
    S'_{ij} = S_{ij} \times_X X'
  \end{cases}
\]

\[
  (5) \qquad
  \begin{cases}
    \mathcal{V}'_{i,j} \simeq \mathcal{V}_{i,j} \times_X X' \\
    \mathcal{V}'^{\,*}_{i,j} \simeq \mathcal{V}^{*}_{i,j} \times_X X'
  \end{cases}
  \qquad i \leq j
\]

\[
  (6) \qquad \mathcal{V}'^{\,j}_{i,k} \simeq \mathcal{V}^{\,j}_{i,k} \times_X X' ,
\]

\[
  (7) \qquad i \leq j \in I' \implies i \in I'
\]

\[
  (8) \qquad X_{I'} = \bigcup_{i \in I'} X_i \qquad (\text{partie fermée de } X)
\]

\[
  (9) \qquad \text{On a bien sûr } I' \subset I'' \implies X_{I'} \subset X_{I''} ,
  \text{ plus gén.!}
\]

\[
  (10) \qquad X_{I' \cup I''} = X_{I'} \cup X_{I''}
  \qquad \text{plus gén.}
  \begin{cases}
    X_{\cup I'_\alpha} = \bigcup\limits_\alpha X_{I'_\alpha} \\
    \text{avec } X_\emptyset = \emptyset
  \end{cases}
\]

\[
  (11) \qquad X_{I' \cap I''} = X_{I'} \cap X_{I''}
  \qquad (\text{donc } X_I = X)
\]

\[
  (12) \qquad X_i^{*} ,\ \mathcal{V}^{*}_{ij} ,\ \mathcal{V}_{ij}
  \quad \text{pour } i,j \in I' .
\]

\[
  (13) \qquad \mathcal{U}' \cap X_i^{*} ,\
  \mathcal{U}' \cap \mathcal{V}^{*}_{ij} ,\
  \mathcal{U} \cap \mathcal{V}_{ij} \qquad (i,j \in I')
\]

\[
  (14) \qquad I'' \subset I' \subset I
\]

\[
  (15) \qquad X_{I''} \subset X_{I'} \quad \text{et} \quad
  \mathcal{U}_{I',I''} \overset{\text{déf}}{=} X_{I'} \setminus X_{I''} ,
\]

\[
  (16) \qquad
  X_{i',\, \mathcal{U}_{I',I''}}
  = X_{i'} \setminus X_{i'} \cap X_{I''} \subset \mathcal{U}_{I',I''}
\]

\[
  (17)
\]

\[
  (18) \qquad
  \begin{array}{c} I' \\ I''' \end{array} \subset I'' \subset I ,
\]

\[
  (19) \qquad
  \begin{array}{c} X_{I'} \\ X_{I'''} \end{array}
  \subset X_{I''} \hookrightarrow X_I = X
\]