Cote n° 150 · pages 2–98
· 247 displayed formulas · Espaces stratifiés et voisinages côniques (ou : déploiement des espaces stratifiés) : notes manuscrites (s.d.).
Inventory dating : 1981-1982
Édition de démonstration
\[x\le y\ \overset{\text{déf}}{\Longleftrightarrow}\ \overline{\lbrace x\rbrace}\subset\overline{\lbrace y\rbrace}\]
LaTeX source
\[
x\le y\ \overset{\text{déf}}{\Longleftrightarrow}\ \overline{\lbrace x\rbrace}\subset\overline{\lbrace y\rbrace}
\]\[\mathcal{E}=(\mathrm{Esp})\ \underset{\mathrm{Esp}}{\overset{\mathrm{Ord}}{\rightleftarrows}}\ (\mathrm{Préord})=\mathcal{O}\]
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\[
\mathcal{E}=(\mathrm{Esp})\ \underset{\mathrm{Esp}}{\overset{\mathrm{Ord}}{\rightleftarrows}}\ (\mathrm{Préord})=\mathcal{O}
\]\[\mathrm{Hom}_{\mathcal{E}}(\mathrm{Esp}(I),X)\ \xrightarrow{\ \sim\ }\ \mathrm{Hom}_{\mathcal{O}}(I,\mathrm{Ord}(X))\]
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\[
\mathrm{Hom}_{\mathcal{E}}(\mathrm{Esp}(I),X)\ \xrightarrow{\ \sim\ }\ \mathrm{Hom}_{\mathcal{O}}(I,\mathrm{Ord}(X))
\]\[X_i\overset{\text{déf}}{=}X_{\bar\imath}^{*}=\bigcup_{j\le i}X_j\]
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\[
X_i\overset{\text{déf}}{=}X_{\bar\imath}^{*}=\bigcup_{j\le i}X_j
\]\[\bigcup_{j\in\bar J}X_j^{*}\ \ \Big\Vert \qquad \overline{\bigcup_{j\in J}X_j^{*}}=\bigcup_{j\in J}\overline{X_j^{*}}\]
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\[
\bigcup_{j\in\bar J}X_j^{*}\ \ \Big\Vert \qquad \overline{\bigcup_{j\in J}X_j^{*}}=\bigcup_{j\in J}\overline{X_j^{*}}
\]\[\overline{X_j^{*}}=\bigcup_{i\in\bar j}X_i^{*}\ \Big(\overset{\text{déf}}{=}X_j\Big).\]
LaTeX source
\[
\overline{X_j^{*}}=\bigcup_{i\in\bar j}X_i^{*}\ \Big(\overset{\text{déf}}{=}X_j\Big).
\]\[X_i\cap X_j=\bigcup_{k\in\bar i\cap\bar j}X_k .\]
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\[
X_i\cap X_j=\bigcup_{k\in\bar i\cap\bar j}X_k .
\]\[\overline{X_i^{*}}=\bigcup_{j\le i}X_j^{*}\]
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\[
\overline{X_i^{*}}=\bigcup_{j\le i}X_j^{*}
\]\[X_i\cap X_j=\bigcup_{k\in\bar i\cap\bar j}X_k ,\]
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\[
X_i\cap X_j=\bigcup_{k\in\bar i\cap\bar j}X_k ,
\]\[J_0=J\setminus\lbrace j\rbrace=\bigcup_{j'\in J\setminus\lbrace j\rbrace}\overline{j'}\]
LaTeX source
\[
J_0=J\setminus\lbrace j\rbrace=\bigcup_{j'\in J\setminus\lbrace j\rbrace}\overline{j'}
\]\[X_j^{\bullet}\overset{\text{déf}}{=}\overline{X_j^{*}}\setminus X_j^{*}\subset X_{J_0}^{*}=X_{J_0}=\bigcup_{i\in J_0}X_i ,\]
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\[
X_j^{\bullet}\overset{\text{déf}}{=}\overline{X_j^{*}}\setminus X_j^{*}\subset X_{J_0}^{*}=X_{J_0}=\bigcup_{i\in J_0}X_i ,
\]\[\Sigma_i=X'_i=X_i\setminus\Big(X_i\cap\bigcup_{k<i}\overset{\circ}{\mathrm{Tub}}(X_k,X_i)\Big)\]
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\[
\Sigma_i=X'_i=X_i\setminus\Big(X_i\cap\bigcup_{k<i}\overset{\circ}{\mathrm{Tub}}(X_k,X_i)\Big)
\]\[\Sigma_{(i,j)}=\struck{\ill{}}\,\mathrm{Tub}^{\bullet}\Big(X'_i,\,X_j\setminus\bigcup\mathrm{Tub}(X_k,X_j)\Big)\hookrightarrow X'_j=\Sigma_j\]
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\[
\Sigma_{(i,j)}=\struck{\ill{}}\,\mathrm{Tub}^{\bullet}\Big(X'_i,\,X_j\setminus\bigcup\mathrm{Tub}(X_k,X_j)\Big)\hookrightarrow X'_j=\Sigma_j
\]\[\struck{\Sigma_{i,j}=T.}\]
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\[
\struck{\Sigma_{i,j}=T.}
\]\[V_{i,j}=V(X_i^{*},X_j)\setminus\bigcup_{i<k<j}V(X_i^{*},X_j)\cap X_k\hookrightarrow X_j^{*}\]
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\[
V_{i,j}=V(X_i^{*},X_j)\setminus\bigcup_{i<k<j}V(X_i^{*},X_j)\cap X_k\hookrightarrow X_j^{*}
\]\[\Sigma_{012}\subset\Sigma_0\times\Sigma_1\times\Sigma_2,\qquad
\Sigma_{012}=(\Sigma_{01}\times\Sigma_2)\cap(\Sigma_0\times\Sigma_{12})\]
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\[
\Sigma_{012}\subset\Sigma_0\times\Sigma_1\times\Sigma_2,\qquad
\Sigma_{012}=(\Sigma_{01}\times\Sigma_2)\cap(\Sigma_0\times\Sigma_{12})
\]\[\Sigma_{012}=\Sigma_{02}\cap\Sigma_{12}\]
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\[
\Sigma_{012}=\Sigma_{02}\cap\Sigma_{12}
\]\[(1)\qquad (\lambda,t)\longmapsto\lambda t\qquad \mathbf{I}\times T\longrightarrow T\]
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\[
(1)\qquad (\lambda,t)\longmapsto\lambda t\qquad \mathbf{I}\times T\longrightarrow T
\]\[f:\dot T\longrightarrow Y\qquad f(x)=0.x\]
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\[ f:\dot T\longrightarrow Y\qquad f(x)=0.x \]
\[(4)\qquad T\simeq\mathcal{C}(f)\]
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\[
(4)\qquad T\simeq\mathcal{C}(f)
\]\[(5)\qquad \overset{\circ}{T}=T\setminus\dot T\]
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\[
(5)\qquad \overset{\circ}{T}=T\setminus\dot T
\]\[\left\lbrace
\begin{array}{l}
T\setminus\overset{\circ}{T}_\lambda\overset{\text{déf}}{=}E_{1-\lambda=\varepsilon}(T)\\[4pt]
\dot E_\varepsilon(T)\overset{\text{déf}}{=}\dot T\cup(1-\varepsilon)\dot T
\end{array}
\right.\]
LaTeX source
\[
\left\lbrace
\begin{array}{l}
T\setminus\overset{\circ}{T}_\lambda\overset{\text{déf}}{=}E_{1-\lambda=\varepsilon}(T)\\[4pt]
\dot E_\varepsilon(T)\overset{\text{déf}}{=}\dot T\cup(1-\varepsilon)\dot T
\end{array}
\right.
\]\[(7)\qquad E_\varepsilon(T)\simeq[\underbrace{1-\varepsilon}_{\lambda},1]\times\dot T\]
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\[
(7)\qquad E_\varepsilon(T)\simeq[\underbrace{1-\varepsilon}_{\lambda},1]\times\dot T
\]\[(8)\qquad \dot T_\lambda=\lambda\dot T\]
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\[ (8)\qquad \dot T_\lambda=\lambda\dot T \]
\[(9)\qquad T\simeq Y\times\mathbf{I}\]
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\[
(9)\qquad T\simeq Y\times\mathbf{I}
\]\[(11)\qquad X^{*}=X\setminus\overset{\circ}{T}=(X\setminus T)\cup\dot T\]
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\[
(11)\qquad X^{*}=X\setminus\overset{\circ}{T}=(X\setminus T)\cup\dot T
\]\[T'=\mathrm{Im}(\mathbf{I}\times\dot T'\to T)\cup Y'.\]
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\[
T'=\mathrm{Im}(\mathbf{I}\times\dot T'\to T)\cup Y'.
\]\[(13)\qquad E\simeq\dot T\times[0,1]\]
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\[ (13)\qquad E\simeq\dot T\times[0,1] \]
\[E\simeq B\times[0,1]\]
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\[ E\simeq B\times[0,1] \]
\[(14)\qquad X_i\cap T=\emptyset\quad\text{ou}\quad X_i\supset Y .\]
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\[
(14)\qquad X_i\cap T=\emptyset\quad\text{ou}\quad X_i\supset Y .
\]\[\begin{array}{c}\dot X\longrightarrow Y,\\ \cup\\ (\dot X_i)\end{array}\]
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\[
\begin{array}{c}\dot X\longrightarrow Y,\\ \cup\\ (\dot X_i)\end{array}
\]\[(15)\qquad (X_i)_{i\in I}\]
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\[
(15)\qquad (X_i)_{i\in I}
\]\[X=\tilde X^{*}\amalg_{\dot T}Y\supset Y\]
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\[
X=\tilde X^{*}\amalg_{\dot T}Y\supset Y
\]\[X=\tilde X^{*}\amalg_{\dot X}Y .\]
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\[
X=\tilde X^{*}\amalg_{\dot X}Y .
\]\[\tilde X'_1\cap\dot X=\dot X'_1 .\]
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\[ \tilde X'_1\cap\dot X=\dot X'_1 . \]
\[X'=\underbrace{\mathrm{Im}\,(\tilde X'\ \text{par}\ \tilde X\to X)}_{X'\setminus Y\cap X'}\cup Y'\]
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\[
X'=\underbrace{\mathrm{Im}\,(\tilde X'\ \text{par}\ \tilde X\to X)}_{X'\setminus Y\cap X'}\cup Y'
\]\[\begin{array}{c}
\dot T\cap E\simeq\dot T_0\times\mathbf{I}\\
\cap\\
T\cap E\simeq T_0\times\mathbf{I}\\
\cup\\
Y\cap E\simeq Y_0\times\mathbf{I}
\end{array}
\qquad\text{où}\qquad
\begin{array}{l}
\dot T_0=\dot T\cap X_0\\
T_0=T\cap X_0\\
Y_0=Y\cap X_0
\end{array}\]
LaTeX source
\[
\begin{array}{c}
\dot T\cap E\simeq\dot T_0\times\mathbf{I}\\
\cap\\
T\cap E\simeq T_0\times\mathbf{I}\\
\cup\\
Y\cap E\simeq Y_0\times\mathbf{I}
\end{array}
\qquad\text{où}\qquad
\begin{array}{l}
\dot T_0=\dot T\cap X_0\\
T_0=T\cap X_0\\
Y_0=Y\cap X_0
\end{array}
\]\[T\cap E\simeq T_0\times\mathbf{I}\qquad(\text{compatible avec } E\simeq X_0\times\mathbf{I})\]
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\[
T\cap E\simeq T_0\times\mathbf{I}\qquad(\text{compatible avec } E\simeq X_0\times\mathbf{I})
\]\[\dot T\cap E\simeq\dot T_0\times\mathbf{I},\qquad Y\cap E\simeq Y_0\times\mathbf{I}\]
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\[
\dot T\cap E\simeq\dot T_0\times\mathbf{I},\qquad Y\cap E\simeq Y_0\times\mathbf{I}
\]\[(\tilde X_0,\ \mathcal E_{\tilde X_0},\ Y_0,\ \mathcal E_{Y_0})\]
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\[
(\tilde X_0,\ \mathcal E_{\tilde X_0},\ Y_0,\ \mathcal E_{Y_0})
\]\[(b,(t_\alpha)_{\alpha\in J})\longmapsto(b,(t'_\alpha)),\qquad
t'_\alpha=t_\alpha\ \text{si}\ \alpha\neq j,\quad t'_\alpha=\lambda t_\alpha\ \text{si}\ \alpha=j .\]
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\[
(b,(t_\alpha)_{\alpha\in J})\longmapsto(b,(t'_\alpha)),\qquad
t'_\alpha=t_\alpha\ \text{si}\ \alpha\neq j,\quad t'_\alpha=\lambda t_\alpha\ \text{si}\ \alpha=j .
\]\[\begin{array}{lll}
E_J\cap T\simeq T_J\times\mathbf{I}^J & & (\text{où } T_J=T\cap B_J)\\
\qquad\cup & & \\
\dot E_J\cap\dot T\simeq\dot T_J\times\mathbf{I}^J & \text{induisant} & (\text{où } \dot T_J=\dot T\cap B_J)\\
E_J\cap Y\simeq Y_J\times\mathbf{I}^J & \text{induisant} & (\text{où } Y_J=Y\cap B_J)
\end{array}\]
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\[
\begin{array}{lll}
E_J\cap T\simeq T_J\times\mathbf{I}^J & & (\text{où } T_J=T\cap B_J)\\
\qquad\cup & & \\
\dot E_J\cap\dot T\simeq\dot T_J\times\mathbf{I}^J & \text{induisant} & (\text{où } \dot T_J=\dot T\cap B_J)\\
E_J\cap Y\simeq Y_J\times\mathbf{I}^J & \text{induisant} & (\text{où } Y_J=Y\cap B_J)
\end{array}
\]\[(14)\qquad
\begin{array}{ll}
a) & \forall i,j\in I,\ \text{on a}\ X_i\cap X_j=\bigcup_{\alpha\le i,j}X_\alpha .\\[4pt]
b) & X=\bigcup_{i\in I}X_i
\end{array}\]
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\[
(14)\qquad
\begin{array}{ll}
a) & \forall i,j\in I,\ \text{on a}\ X_i\cap X_j=\bigcup_{\alpha\le i,j}X_\alpha .\\[4pt]
b) & X=\bigcup_{i\in I}X_i
\end{array}
\]\[E_J\cap X_i\simeq(X_{iJ}\times\mathbf{I}^J)\qquad\text{(induit par } E_J\simeq B_J\times\mathbf{I}^J\text{, où } X_{iJ}=X_i\cap B_J\text{)}.\]
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\[
E_J\cap X_i\simeq(X_{iJ}\times\mathbf{I}^J)\qquad\text{(induit par } E_J\simeq B_J\times\mathbf{I}^J\text{, où } X_{iJ}=X_i\cap B_J\text{)}.
\]\[X_{I_0}=\bigcup_{i\in I_0}X_i=\coprod_{i\in I_0}X_i .\]
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\[
X_{I_0}=\bigcup_{i\in I_0}X_i=\coprod_{i\in I_0}X_i .
\]\[E_{k;k}=p_{jk}^{-1}(E_{j';j}) .\]
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\[
E_{k;k}=p_{jk}^{-1}(E_{j';j}) .
\]\[\forall i,j\in I,\ \text{on a}\ X_i\cap X_j=\bigcup_{k\le i,j}X_k .\]
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\[
\forall i,j\in I,\ \text{on a}\ X_i\cap X_j=\bigcup_{k\le i,j}X_k .
\]\[(14')\qquad X_i\cap X_\beta=
\begin{cases}
\emptyset & \text{si } i\not\ge\beta\\
X_\beta & \text{si } i\ge\beta\ \text{(cas trivial)}
\end{cases}\]
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\[
(14')\qquad X_i\cap X_\beta=
\begin{cases}
\emptyset & \text{si } i\not\ge\beta\\
X_\beta & \text{si } i\ge\beta\ \text{(cas trivial)}
\end{cases}
\]\[(15)\qquad i\in I^{*}\setminus I_\beta\Longrightarrow X_i\cap T_\beta=\emptyset\]
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\[
(15)\qquad i\in I^{*}\setminus I_\beta\Longrightarrow X_i\cap T_\beta=\emptyset
\]\[(16)\qquad i\in I_\beta\Longrightarrow X_i\supset X_\beta\]
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\[ (16)\qquad i\in I_\beta\Longrightarrow X_i\supset X_\beta \]
\[(16')\qquad X_i\cap T_\beta=\mathbf{I}.\underbrace{X_i\cap\dot T_\beta}_{\dot X_i}\cup X_\beta\]
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\[
(16')\qquad X_i\cap T_\beta=\mathbf{I}.\underbrace{X_i\cap\dot T_\beta}_{\dot X_i}\cup X_\beta
\]\[(17)\qquad\bigl(X,\ (X_i)_{i\in I},\ (B_\alpha,\mathcal E_\alpha)_{\alpha\in R},\ (T_\beta)_{\beta\in I_0}\bigr)\]
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\[
(17)\qquad\bigl(X,\ (X_i)_{i\in I},\ (B_\alpha,\mathcal E_\alpha)_{\alpha\in R},\ (T_\beta)_{\beta\in I_0}\bigr)
\]\[(18)\quad\left\lbrace
\begin{array}{l}
\overbrace{X^{*},\ (\dot X_\beta)_{\beta\in I_0},\ (B^{*}_\alpha,\mathcal E^{*}_\alpha)_{\alpha\in\Lambda}}^{\text{syst. de cylindres-bords transversaux}},\ (X^{*}_i)_{i\in I^{*}}\ ;\ (X_\beta)_{\beta\in I},\ (B_{\beta,\alpha},\mathcal E_{\beta,\alpha})_{\alpha\in\Lambda,\ \beta\in I_0},\\[6pt]
(\dot X_\beta\xrightarrow{\ p_\beta\ }X_\beta)_{\beta\in I_0}
\end{array}\right\rbrace\]
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\[
(18)\quad\left\lbrace
\begin{array}{l}
\overbrace{X^{*},\ (\dot X_\beta)_{\beta\in I_0},\ (B^{*}_\alpha,\mathcal E^{*}_\alpha)_{\alpha\in\Lambda}}^{\text{syst. de cylindres-bords transversaux}},\ (X^{*}_i)_{i\in I^{*}}\ ;\ (X_\beta)_{\beta\in I},\ (B_{\beta,\alpha},\mathcal E_{\beta,\alpha})_{\alpha\in\Lambda,\ \beta\in I_0},\\[6pt]
(\dot X_\beta\xrightarrow{\ p_\beta\ }X_\beta)_{\beta\in I_0}
\end{array}\right\rbrace
\]\[(19)\quad\left\lbrace
\begin{array}{l}
\hat X=X^{*}\amalg_{\dot X=\coprod\dot X_\beta}\underbrace{\mathcal C\bigl(p=\textstyle\coprod p_\beta\bigr)}_{\coprod_\beta\mathcal C(p_\beta)},\\[10pt]
T_\beta=\mathcal C(p_\beta)\\[4pt]
X_\beta\ \text{pour}\ \beta\in I_0\ \text{clair}\\[4pt]
X_i\ \text{pour}\ i\in I^{*}\ \text{est donné par}\\[4pt]
\qquad\bar X_i=X^{*}_i\cup\bigcup_{\beta\in I_0,\ \beta\le i}\bigl(\mathbf{I}.\dot X_{i\beta}\cup X_\beta\bigr),\qquad \dot X_{i\beta}\overset{\text{déf}}{=}X^{*}_i\cap\dot X_\beta\\[10pt]
B_\alpha=B^{*}_\alpha\cup\bigcup_{\beta\in I_0}\mathbf{I}.\dot B_\alpha,\qquad \dot B_\alpha\overset{\text{déf}}{=}B^{*}_\alpha\cap\dot X_\beta\\[10pt]
\mathcal E_\alpha\ \text{défini par}\ \mathcal E^{*}_\alpha\ \text{et les}\ (B_{\beta,\alpha},\mathcal E_{\beta\alpha})
\end{array}\right.\]
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\[
(19)\quad\left\lbrace
\begin{array}{l}
\hat X=X^{*}\amalg_{\dot X=\coprod\dot X_\beta}\underbrace{\mathcal C\bigl(p=\textstyle\coprod p_\beta\bigr)}_{\coprod_\beta\mathcal C(p_\beta)},\\[10pt]
T_\beta=\mathcal C(p_\beta)\\[4pt]
X_\beta\ \text{pour}\ \beta\in I_0\ \text{clair}\\[4pt]
X_i\ \text{pour}\ i\in I^{*}\ \text{est donné par}\\[4pt]
\qquad\bar X_i=X^{*}_i\cup\bigcup_{\beta\in I_0,\ \beta\le i}\bigl(\mathbf{I}.\dot X_{i\beta}\cup X_\beta\bigr),\qquad \dot X_{i\beta}\overset{\text{déf}}{=}X^{*}_i\cap\dot X_\beta\\[10pt]
B_\alpha=B^{*}_\alpha\cup\bigcup_{\beta\in I_0}\mathbf{I}.\dot B_\alpha,\qquad \dot B_\alpha\overset{\text{déf}}{=}B^{*}_\alpha\cap\dot X_\beta\\[10pt]
\mathcal E_\alpha\ \text{défini par}\ \mathcal E^{*}_\alpha\ \text{et les}\ (B_{\beta,\alpha},\mathcal E_{\beta\alpha})
\end{array}\right.
\]\[(20)\quad
\begin{array}{ll}
(a) & \text{Les } X_\beta \text{ disjoints}\\
(b) & \text{Chaque } \dot X_\beta \text{ transverse : } (B^{*}_\alpha,\mathcal E^{*}_\alpha)\ \Longrightarrow\ \text{le syst. induit dans } (\dot X_\beta,\ldots)\ (\dot B^{\alpha},\dot{\mathcal E}^{\alpha})\\
(c) & (\dot X_\beta,\underbrace{\dot B_{\beta\alpha},\dot{\mathcal E}_{\beta\alpha}}_{\text{induit sur }\dot X_\beta\text{ par }(B_\alpha,\mathcal E_\alpha)})\longrightarrow(X_\beta,B_{\beta,\alpha},\mathcal E_{\beta,\alpha})\ \text{est transverse}\\
(d) & \text{Les } X^{*}_i \text{ transverses aux } \bigl((\dot X_\beta,\mathcal E_\beta)_{\beta\in I_0},(B^{*}_\alpha,\mathcal E^{*}_\alpha)_{\alpha\in\Lambda}\bigr)
\end{array}\]
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\[
(20)\quad
\begin{array}{ll}
(a) & \text{Les } X_\beta \text{ disjoints}\\
(b) & \text{Chaque } \dot X_\beta \text{ transverse : } (B^{*}_\alpha,\mathcal E^{*}_\alpha)\ \Longrightarrow\ \text{le syst. induit dans } (\dot X_\beta,\ldots)\ (\dot B^{\alpha},\dot{\mathcal E}^{\alpha})\\
(c) & (\dot X_\beta,\underbrace{\dot B_{\beta\alpha},\dot{\mathcal E}_{\beta\alpha}}_{\text{induit sur }\dot X_\beta\text{ par }(B_\alpha,\mathcal E_\alpha)})\longrightarrow(X_\beta,B_{\beta,\alpha},\mathcal E_{\beta,\alpha})\ \text{est transverse}\\
(d) & \text{Les } X^{*}_i \text{ transverses aux } \bigl((\dot X_\beta,\mathcal E_\beta)_{\beta\in I_0},(B^{*}_\alpha,\mathcal E^{*}_\alpha)_{\alpha\in\Lambda}\bigr)
\end{array}
\]\[X_i\cap X_j=\bigcup_{k\le i,j}X_k .\]
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\[
X_i\cap X_j=\bigcup_{k\le i,j}X_k .
\]\[(21)\qquad I^{*}_\beta=\lbrace i\in I^{*}\mid i>\beta\rbrace\]
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\[
(21)\qquad I^{*}_\beta=\lbrace i\in I^{*}\mid i>\beta\rbrace
\]\[(e)\qquad i,j\in I^{*},\ i\le j\Longrightarrow X^{*}_i\subset X^{*}_j\]
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\[
(e)\qquad i,j\in I^{*},\ i\le j\Longrightarrow X^{*}_i\subset X^{*}_j
\]\[(f)\qquad i,j\in I^{*}\Longrightarrow X^{*}_i\cap X^{*}_j=\bigcup_{h\le i,j}X^{*}_h\]
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\[
(f)\qquad i,j\in I^{*}\Longrightarrow X^{*}_i\cap X^{*}_j=\bigcup_{h\le i,j}X^{*}_h
\]\[X_i\cap X_j=\bigcup_{h\le i,j}X_h\qquad(i,j\in I)\]
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\[
X_i\cap X_j=\bigcup_{h\le i,j}X_h\qquad(i,j\in I)
\]\[X_i\cap X_j\cap T_\beta=\bigcup_{h\le i,j}X_h\cap T_\beta\]
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\[
X_i\cap X_j\cap T_\beta=\bigcup_{h\le i,j}X_h\cap T_\beta
\]\[(X,(X_i)_{i\in I},(B_\alpha,\mathcal E_\alpha)_{\alpha\in\Lambda}).\]
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\[
(X,(X_i)_{i\in I},(B_\alpha,\mathcal E_\alpha)_{\alpha\in\Lambda}).
\]\[(22)\qquad \underbrace{X^{*},\ (X^{*}_i)_{i\in I^{*}=I\setminus I_0}}_{\text{stratification de } I^{*}},\
\underbrace{\bigl((B_\alpha,\mathcal E_\alpha)_{\alpha\in\Lambda},\ (\dot X_\beta,\mathcal E_\beta)_{\beta\in I_0}\bigr)}_{\text{famille transversale}}\]
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\[
(22)\qquad \underbrace{X^{*},\ (X^{*}_i)_{i\in I^{*}=I\setminus I_0}}_{\text{stratification de } I^{*}},\
\underbrace{\bigl((B_\alpha,\mathcal E_\alpha)_{\alpha\in\Lambda},\ (\dot X_\beta,\mathcal E_\beta)_{\beta\in I_0}\bigr)}_{\text{famille transversale}}
\]\[(23)\qquad \bigl((X_\beta)_{\beta\in I_0},\ (B_{\beta\alpha},\mathcal E_{\beta\alpha})_{\beta\in I_0,\ \alpha\in\Lambda}\bigr)\]
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\[
(23)\qquad \bigl((X_\beta)_{\beta\in I_0},\ (B_{\beta\alpha},\mathcal E_{\beta\alpha})_{\beta\in I_0,\ \alpha\in\Lambda}\bigr)
\]\[X_i^{\circ}\overset{\text{déf}}{=}X_i\setminus\bigcup_{j<i}X_j ,\]
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\[
X_i^{\circ}\overset{\text{déf}}{=}X_i\setminus\bigcup_{j<i}X_j ,
\]\[\Sigma:\mathrm{Drap}(I)\longrightarrow\mathcal M,\qquad (i_0,\ldots,i_n)\longmapsto\Sigma_{i_0\ldots i_n}\]
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\[
\Sigma:\mathrm{Drap}(I)\longrightarrow\mathcal M,\qquad (i_0,\ldots,i_n)\longmapsto\Sigma_{i_0\ldots i_n}
\]\[\Sigma_{i_0\ldots i_n}\longrightarrow\Sigma_{i_0,\ldots,i_p}\]
LaTeX source
\[
\Sigma_{i_0\ldots i_n}\longrightarrow\Sigma_{i_0,\ldots,i_p}
\]\[\Sigma_{i_0\ldots i_n}\longrightarrow\Sigma_{i_0\ldots\widehat{i_p}\ldots i_n},\qquad (p\neq n)\]
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\[
\Sigma_{i_0\ldots i_n}\longrightarrow\Sigma_{i_0\ldots\widehat{i_p}\ldots i_n},\qquad (p\neq n)
\]\[\underbrace{\Sigma_{ij}\times_{\Sigma_j}\Sigma_{jk}}_{\overset{\text{déf}}{=}\Sigma_{ijk}}\longrightarrow\Sigma_{i,k}\]
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\[
\underbrace{\Sigma_{ij}\times_{\Sigma_j}\Sigma_{jk}}_{\overset{\text{déf}}{=}\Sigma_{ijk}}\longrightarrow\Sigma_{i,k}
\]\[\underline i:\Sigma_{*}\longrightarrow\Sigma_{**}\]
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\[
\underline i:\Sigma_{*}\longrightarrow\Sigma_{**}
\]\[\Sigma_{\mathcal{M}}\longrightarrow I_{\mathcal{M}} .\]
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\[
\Sigma_{\mathcal{M}}\longrightarrow I_{\mathcal{M}} .
\]\[\Sigma_{i_0\cdots i_n}=\Sigma_{i_0i_1}\cap\cdots\cap\Sigma_{i_0i_n}\ \ill{}\]
LaTeX source
\[
\Sigma_{i_0\cdots i_n}=\Sigma_{i_0i_1}\cap\cdots\cap\Sigma_{i_0i_n}\ \ill{}
\]\[\Sigma_{ijk}=\Sigma_{ik}\cap\Sigma_{jk},\]
LaTeX source
\[
\Sigma_{ijk}=\Sigma_{ik}\cap\Sigma_{jk},
\]\[\Sigma_{*}(z)\hookrightarrow I(k),\qquad \Sigma_{*}(z)=\lbrace x\in\Sigma_{*}\mid x\le z\rbrace,\quad I(k)=\lbrace i\in I\mid i\le k\rbrace,\]
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\[
\Sigma_{*}(z)\hookrightarrow I(k),\qquad \Sigma_{*}(z)=\lbrace x\in\Sigma_{*}\mid x\le z\rbrace,\quad I(k)=\lbrace i\in I\mid i\le k\rbrace,
\]\[\Sigma_{i_0\cdots i_n}\longrightarrow\Sigma_{i_0\cdots\hat\imath_p\cdots i_n}\qquad(p\neq n)\]
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\[
\Sigma_{i_0\cdots i_n}\longrightarrow\Sigma_{i_0\cdots\hat\imath_p\cdots i_n}\qquad(p\neq n)
\]\[d_1\subset d_2\subset\cdots\subset d_p\subset d_{p+1}=d'\qquad(p=\mathrm{long.}\,d')\]
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\[
d_1\subset d_2\subset\cdots\subset d_p\subset d_{p+1}=d'\qquad(p=\mathrm{long.}\,d')
\]\[\Sigma_{d'}=\Sigma_{d_{p+1}}\longrightarrow\Sigma_{d_p}\longrightarrow\cdots\longrightarrow\Sigma_{d_2}\longrightarrow\Sigma_{d_1}\]
LaTeX source
\[
\Sigma_{d'}=\Sigma_{d_{p+1}}\longrightarrow\Sigma_{d_p}\longrightarrow\cdots\longrightarrow\Sigma_{d_2}\longrightarrow\Sigma_{d_1}
\]\[C_{\Sigma,d,d'} .\]
LaTeX source
\[
C_{\Sigma,d,d'} .
\]\[(24)\qquad \Sigma_{*},\]
LaTeX source
\[
(24)\qquad \Sigma_{*},
\]\[(25)\qquad \Sigma_{*}\longrightarrow I\]
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\[
(25)\qquad \Sigma_{*}\longrightarrow I
\]\[\Big(\text{d'où}\quad \Sigma_{*}=\coprod_{i\in I}\Sigma_i,\qquad \Sigma_{**}=\coprod_{i\le j}\Sigma_{ij}\Big)\]
LaTeX source
\[
\Big(\text{d'où}\quad \Sigma_{*}=\coprod_{i\in I}\Sigma_i,\qquad \Sigma_{**}=\coprod_{i\le j}\Sigma_{ij}\Big)
\]\[\Sigma_{ij}\longrightarrow\Sigma_j\qquad\text{« cylindrique »}\]
LaTeX source
\[
\Sigma_{ij}\longrightarrow\Sigma_j\qquad\text{« cylindrique »}
\]\[(26)\qquad \Sigma_{ii}=\Sigma_i .\]
LaTeX source
\[
(26)\qquad \Sigma_{ii}=\Sigma_i .
\]\[(27)\qquad
\begin{array}{ccc}
\Sigma(n) & \longrightarrow & \mathrm{Drap}_n(I)\quad(\text{drapeaux stricts})\\
\Vert & & \\
\displaystyle\coprod_{\substack{(i_0<\cdots<i_n)\ \text{drapeau strict}\\ \text{de } I \text{ de longueur } n}}\Sigma_{i_0\cdots i_n} & &
\end{array}
\qquad\text{OK}\]
LaTeX source
\[
(27)\qquad
\begin{array}{ccc}
\Sigma(n) & \longrightarrow & \mathrm{Drap}_n(I)\quad(\text{drapeaux stricts})\\
\Vert & & \\
\displaystyle\coprod_{\substack{(i_0<\cdots<i_n)\ \text{drapeau strict}\\ \text{de } I \text{ de longueur } n}}\Sigma_{i_0\cdots i_n} & &
\end{array}
\qquad\text{OK}
\]\[(28)\qquad \Sigma_{ijk}=\Sigma_{ik}\cap\Sigma_{jk}\qquad(\text{a priori } \Sigma_{ijk}\subset\Sigma_{ik}\cap\Sigma_{jk})\]
LaTeX source
\[
(28)\qquad \Sigma_{ijk}=\Sigma_{ik}\cap\Sigma_{jk}\qquad(\text{a priori } \Sigma_{ijk}\subset\Sigma_{ik}\cap\Sigma_{jk})
\]\[\Sigma_{*}\longrightarrow I\quad(\text{\uncertain{qui} \add{\uncertain{est} \ill{}} induit une application } \Sigma_{*\le z}\hookrightarrow I_{\le k})\ \text{induit}\]
LaTeX source
\[
\Sigma_{*}\longrightarrow I\quad(\text{\uncertain{qui} \add{\uncertain{est} \ill{}} induit une application } \Sigma_{*\le z}\hookrightarrow I_{\le k})\ \text{induit}
\]\[(29)\qquad (\Sigma_{*})_{\le z}\hookrightarrow I_{\le k}\]
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\[
(29)\qquad (\Sigma_{*})_{\le z}\hookrightarrow I_{\le k}
\]\[\Sigma_{ik}\cap\Sigma_{jk}=\emptyset\]
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\[
\Sigma_{ik}\cap\Sigma_{jk}=\emptyset
\]\[\Sigma_{ik}\cap\Sigma_{jk}\subset\ \ill{}\ )\]
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\[
\Sigma_{ik}\cap\Sigma_{jk}\subset\ \ill{}\ )
\]\[\Sigma_{ij}\longrightarrow\Sigma_i\]
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\[
\Sigma_{ij}\longrightarrow\Sigma_i
\]\[\Sigma_{ij}\longrightarrow\Sigma_i\]
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\[
\Sigma_{ij}\longrightarrow\Sigma_i
\]\[\mathrm{Drap}\big(I_{i_p<\cdot<i_{p+1}}\amalg I_{i_{p+1}<\cdot<i_{p+2}}\amalg\cdots\amalg I_{i_{n-1}<\cdot<i_n}\big)\]
LaTeX source
\[
\mathrm{Drap}\big(I_{i_p<\cdot<i_{p+1}}\amalg I_{i_{p+1}<\cdot<i_{p+2}}\amalg\cdots\amalg I_{i_{n-1}<\cdot<i_n}\big)
\]\[\Sigma^{*}(d')\longrightarrow\Sigma^{*}(d)\qquad d\subset d',\ d,d'\ \text{cofinaux (avec le plus grand élément)}\]
LaTeX source
\[
\Sigma^{*}(d')\longrightarrow\Sigma^{*}(d)\qquad d\subset d',\ d,d'\ \text{cofinaux (avec le plus grand élément)}
\]\[\Sigma^{*}(\lbrace\alpha\rbrace)\simeq\Sigma^{*}(0),\]
LaTeX source
\[
\Sigma^{*}(\lbrace\alpha\rbrace)\simeq\Sigma^{*}(0),
\]\[\Sigma^{*}(1)\longrightarrow\Sigma^{*}(0)\]
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\[
\Sigma^{*}(1)\longrightarrow\Sigma^{*}(0)
\]\[\Sigma^{*}(n)\ \overset{\varphi_{0,n}}{\underset{\varphi_{n-1,n}}{\rightrightarrows}}\ \Sigma^{*}(1)\]
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\[
\Sigma^{*}(n)\ \overset{\varphi_{0,n}}{\underset{\varphi_{n-1,n}}{\rightrightarrows}}\ \Sigma^{*}(1)
\]\[\Sigma^{*}(d)\longrightarrow\Sigma^{*}(0)\ni x\]
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\[
\Sigma^{*}(d)\longrightarrow\Sigma^{*}(0)\ni x
\]\[\underset{\tilde x}{\Sigma^{*}(n)}\ \xrightarrow{\ \text{imm.}\ }\ \underset{z}{\Sigma^{*}(d)}\ \Big(\overset{\text{imm.}}{\dashrightarrow}\ \underset{x}{\Sigma^{*}(0)}\Big)\]
LaTeX source
\[
\underset{\tilde x}{\Sigma^{*}(n)}\ \xrightarrow{\ \text{imm.}\ }\ \underset{z}{\Sigma^{*}(d)}\ \Big(\overset{\text{imm.}}{\dashrightarrow}\ \underset{x}{\Sigma^{*}(0)}\Big)
\]\[\Gamma : d\longmapsto\Sigma^{*}(d\amalg\lbrace\omega\rbrace)\]
LaTeX source
\[
\Gamma : d\longmapsto\Sigma^{*}(d\amalg\lbrace\omega\rbrace)
\]\[(35)\qquad \Gamma'(d)\simeq\mathfrak{S}_d\times\Gamma(d)\]
LaTeX source
\[
(35)\qquad \Gamma'(d)\simeq\mathfrak{S}_d\times\Gamma(d)
\]\[\Gamma_1\cup{}^{s}\Gamma_1=\Gamma_0\times\Gamma_0\setminus\mathrm{diag},\qquad \Gamma_1\cap{}^{s}\Gamma_1=\emptyset\]
LaTeX source
\[
\Gamma_1\cup{}^{s}\Gamma_1=\Gamma_0\times\Gamma_0\setminus\mathrm{diag},\qquad \Gamma_1\cap{}^{s}\Gamma_1=\emptyset
\]\[(36)\qquad (\Gamma_1\times_\Sigma\Gamma_0)\cap(\Gamma_0\times_\Sigma\Gamma_1)\xrightarrow{\ \mathrm{pr}_{13}\ }\Gamma_0\times\Gamma_0\quad\text{se factorise par }\Gamma_1,\]
LaTeX source
\[
(36)\qquad (\Gamma_1\times_\Sigma\Gamma_0)\cap(\Gamma_0\times_\Sigma\Gamma_1)\xrightarrow{\ \mathrm{pr}_{13}\ }\Gamma_0\times\Gamma_0\quad\text{se factorise par }\Gamma_1,
\]\[\Sigma^{*}_{f}(1)=\Sigma(1)\setminus\text{diagonale}.\]
LaTeX source
\[
\Sigma^{*}_{f}(1)=\Sigma(1)\setminus\text{diagonale}.
\]\[\Sigma^{*}(d)\longrightarrow\Sigma=\Sigma^{*}(0),\qquad (x_0<\cdots<x_d)\longmapsto x_d\]
LaTeX source
\[
\Sigma^{*}(d)\longrightarrow\Sigma=\Sigma^{*}(0),\qquad (x_0<\cdots<x_d)\longmapsto x_d
\]\[(36)\qquad \Sigma^{*}(1)\xrightarrow{\ \underline s=\mathrm{pr}_1\ }\Sigma=\Sigma^{*}(0)\]
LaTeX source
\[
(36)\qquad \Sigma^{*}(1)\xrightarrow{\ \underline s=\mathrm{pr}_1\ }\Sigma=\Sigma^{*}(0)
\]\[(37)\qquad p_2 : \Sigma^{*}(2)\longrightarrow\Sigma^{*}(1)\]
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\[
(37)\qquad p_2 : \Sigma^{*}(2)\longrightarrow\Sigma^{*}(1)
\]\[(38)\qquad \underline b\circ p_2=\underline s\circ\underline b'\quad\text{i.e.\ le carré}\]
LaTeX source
\[
(38)\qquad \underline b\circ p_2=\underline s\circ\underline b'\quad\text{i.e.\ le carré}
\]\[p_1\circ p_2=p_1\circ\underline s'\quad\text{i.e.\ le carré}\]
LaTeX source
\[
p_1\circ p_2=p_1\circ\underline s'\quad\text{i.e.\ le carré}
\]\[(39)\qquad x\prec y\iff\exists\,u\in\Sigma^{*}(1)\ \text{tel que}\ x=p_1u,\ y=\underline b u\]
LaTeX source
\[
(39)\qquad x\prec y\iff\exists\,u\in\Sigma^{*}(1)\ \text{tel que}\ x=p_1u,\ y=\underline b u
\]\[x\prec y\ \text{et}\ y\prec z\ \Longrightarrow\ x\prec z\]
LaTeX source
\[
x\prec y\ \text{et}\ y\prec z\ \Longrightarrow\ x\prec z
\]\[p_1u=x,\quad \underline b u=y,\quad p_1v=y,\quad \underline b v=z\]
LaTeX source
\[ p_1u=x,\quad \underline b u=y,\quad p_1v=y,\quad \underline b v=z \]
\[\underline b u=p_1v\]
LaTeX source
\[ \underline b u=p_1v \]
\[\exists!\,d\in\Sigma^{*}(2),\ \text{avec}\quad \underline b'(d)=v,\quad p_2(d)=u\]
LaTeX source
\[
\exists!\,d\in\Sigma^{*}(2),\ \text{avec}\quad \underline b'(d)=v,\quad p_2(d)=u
\]\[w=\underline s'(d)\in\Sigma^{*}(1)\]
LaTeX source
\[
w=\underline s'(d)\in\Sigma^{*}(1)
\]\[\begin{aligned}
p_1(w)&=p_1(\underline s'(d))\overset{(38)}{=}p_1p_2(d)=p_1(u)=x\\
\underline b(w)&=\underline b(\underline s'(d))=\underline b(\underline b'(d))=\underline b(v)=z
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
p_1(w)&=p_1(\underline s'(d))\overset{(38)}{=}p_1p_2(d)=p_1(u)=x\\
\underline b(w)&=\underline b(\underline s'(d))=\underline b(\underline b'(d))=\underline b(v)=z
\end{aligned}
\]\[w : x\prec z .\]
LaTeX source
\[ w : x\prec z . \]
\[x\not\prec x\qquad\big(\text{car } \mathrm{Im}\big(\Sigma^{*}(1)\xrightarrow{(p_1,\underline b)}\Sigma\times\Sigma\big)\cap\mathrm{diag}=\emptyset\big)\]
LaTeX source
\[
x\not\prec x\qquad\big(\text{car } \mathrm{Im}\big(\Sigma^{*}(1)\xrightarrow{(p_1,\underline b)}\Sigma\times\Sigma\big)\cap\mathrm{diag}=\emptyset\big)
\]\[\Sigma^{*}(1)\ \overset{p_1=\underline s}{\underset{\underline b}{\rightrightarrows}}\ \Sigma\qquad\qquad \Sigma^{*}(2)\ \overset{p_2}{\underset{\underline b'}{\rightrightarrows}}\ \Sigma^{*}(1)\]
LaTeX source
\[
\Sigma^{*}(1)\ \overset{p_1=\underline s}{\underset{\underline b}{\rightrightarrows}}\ \Sigma\qquad\qquad \Sigma^{*}(2)\ \overset{p_2}{\underset{\underline b'}{\rightrightarrows}}\ \Sigma^{*}(1)
\]\[(39)\qquad\left\lbrace
\begin{array}{l}
(x\prec y)\xrightarrow{\ p_1=\underline s\ }x\\
(x\prec y)\xrightarrow{\ \underline b\ }y\\
(x\prec y\prec z)\xrightarrow{\ p_2\ }(x,y)\\
(x\prec y\prec z)\xrightarrow{\ \underline s'\ }(x,z)\\
(x\prec y\prec z)\xrightarrow{\ \underline b'\ }(y,z)
\end{array}\right.\]
LaTeX source
\[
(39)\qquad\left\lbrace
\begin{array}{l}
(x\prec y)\xrightarrow{\ p_1=\underline s\ }x\\
(x\prec y)\xrightarrow{\ \underline b\ }y\\
(x\prec y\prec z)\xrightarrow{\ p_2\ }(x,y)\\
(x\prec y\prec z)\xrightarrow{\ \underline s'\ }(x,z)\\
(x\prec y\prec z)\xrightarrow{\ \underline b'\ }(y,z)
\end{array}\right.
\]\[\Sigma^{*}(2)\ \overset{\underline s'}{\underset{\underline b'}{\rightrightarrows}}\ \Sigma^{*}(1)\xrightarrow{\ \underline b\ }\Sigma\]
LaTeX source
\[
\Sigma^{*}(2)\ \overset{\underline s'}{\underset{\underline b'}{\rightrightarrows}}\ \Sigma^{*}(1)\xrightarrow{\ \underline b\ }\Sigma
\]\[p_1=\underline s : \Sigma^{*}(1)\longrightarrow\Sigma\]
LaTeX source
\[
p_1=\underline s : \Sigma^{*}(1)\longrightarrow\Sigma
\]\[\Sigma^{*}(2)\xrightarrow{\ (p_1\circ\underline s',\ p_1\circ\underline b')\ }\Sigma\times\Sigma\]
LaTeX source
\[
\Sigma^{*}(2)\xrightarrow{\ (p_1\circ\underline s',\ p_1\circ\underline b')\ }\Sigma\times\Sigma
\]\[\Sigma^{*}(1)\xrightarrow{\ (p_1,\underline b)\ }\Sigma\times\Sigma\]
LaTeX source
\[
\Sigma^{*}(1)\xrightarrow{\ (p_1,\underline b)\ }\Sigma\times\Sigma
\]\[p_2 : \Sigma^{*}(2)\longrightarrow\Sigma^{*}(1)\]
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\[
p_2 : \Sigma^{*}(2)\longrightarrow\Sigma^{*}(1)
\]\[\Sigma_{ijk}=b_{ij}^{-1}(\Sigma_{ij})\]
LaTeX source
\[
\Sigma_{ijk}=b_{ij}^{-1}(\Sigma_{ij})
\]\[\Sigma=\coprod_{i\in I}\Sigma_i,\qquad \Sigma^{*}(1)=\coprod_{\substack{i,j\in I\\ i<j}}\Sigma_{ij}\]
LaTeX source
\[
\Sigma=\coprod_{i\in I}\Sigma_i,\qquad \Sigma^{*}(1)=\coprod_{\substack{i,j\in I\\ i<j}}\Sigma_{ij}
\]\[\begin{aligned}
\underline{\Sigma}=\Bigl(&\Sigma(0)=\mathrm{Ob}(\underline{\Sigma}),\ \Sigma(1)=\mathrm{Fl}(\underline{\Sigma}),\ \Sigma(1)\overset{\underline s}{\underset{\underline b}{\rightrightarrows}}\Sigma(0),\\
&(\Sigma(1),\underline b)\times_{\Sigma(0)}(\Sigma(1),\underline s)\overset{\text{déf}}{\Longrightarrow}\Sigma(2)\xrightarrow[\text{compos.\ des flèches}]{\text{compl.}}\Sigma(1)\Bigr),
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
\underline{\Sigma}=\Bigl(&\Sigma(0)=\mathrm{Ob}(\underline{\Sigma}),\ \Sigma(1)=\mathrm{Fl}(\underline{\Sigma}),\ \Sigma(1)\overset{\underline s}{\underset{\underline b}{\rightrightarrows}}\Sigma(0),\\
&(\Sigma(1),\underline b)\times_{\Sigma(0)}(\Sigma(1),\underline s)\overset{\text{déf}}{\Longrightarrow}\Sigma(2)\xrightarrow[\text{compos.\ des flèches}]{\text{compl.}}\Sigma(1)\Bigr),
\end{aligned}
\]\[\pi_0(\Sigma(1))\xrightarrow{\ \pi_0(\underline s,\underline b)\ }\pi_0(\Sigma(0)\times\Sigma(0))\simeq\pi_0(\Sigma(0))\times\pi_0(\Sigma(0))\]
LaTeX source
\[
\pi_0(\Sigma(1))\xrightarrow{\ \pi_0(\underline s,\underline b)\ }\pi_0(\Sigma(0)\times\Sigma(0))\simeq\pi_0(\Sigma(0))\times\pi_0(\Sigma(0))
\]\[\pi_0(\Sigma(2))\longrightarrow\bigl(\pi_0(\Sigma(1)),\pi_0(\underline b)\bigr)\times_{\pi_0(\Sigma(0))}\bigl(\pi_0(\Sigma(1)),\pi_0(\underline s)\bigr)\]
LaTeX source
\[
\pi_0(\Sigma(2))\longrightarrow\bigl(\pi_0(\Sigma(1)),\pi_0(\underline b)\bigr)\times_{\pi_0(\Sigma(0))}\bigl(\pi_0(\Sigma(1)),\pi_0(\underline s)\bigr)
\]\[\underline{\Sigma}\xrightarrow{\ f\ }\underline I\]
LaTeX source
\[
\underline{\Sigma}\xrightarrow{\ f\ }\underline I
\]\[\left\lbrace
\begin{array}{l}
\Sigma(0)\xrightarrow{\ \varphi_0\ }I_0=I\\
\Sigma(1)\xrightarrow{\ \varphi_1\ }I_1=\text{graphe de la relation d'ordre}\\
\qquad=\text{image de }\pi_0(\Sigma_1)\text{ dans }\pi_0(\Sigma_0)\times\pi_0(\Sigma_0)=I\times I
\end{array}\right.\]
LaTeX source
\[
\left\lbrace
\begin{array}{l}
\Sigma(0)\xrightarrow{\ \varphi_0\ }I_0=I\\
\Sigma(1)\xrightarrow{\ \varphi_1\ }I_1=\text{graphe de la relation d'ordre}\\
\qquad=\text{image de }\pi_0(\Sigma_1)\text{ dans }\pi_0(\Sigma_0)\times\pi_0(\Sigma_0)=I\times I
\end{array}\right.
\]\[\Sigma(0)=\coprod_{i\in I}\Sigma_i,\qquad \Sigma(1)=\coprod_{\substack{i,j\in I\\ i\le j}}\Sigma_{ij}\]
LaTeX source
\[
\Sigma(0)=\coprod_{i\in I}\Sigma_i,\qquad \Sigma(1)=\coprod_{\substack{i,j\in I\\ i\le j}}\Sigma_{ij}
\]\[\Sigma(0)\xrightarrow{\ \delta\ }\Sigma(1)\qquad(\delta(x)=\mathrm{id}_x^{\underline{\Sigma}})\]
LaTeX source
\[
\Sigma(0)\xrightarrow{\ \delta\ }\Sigma(1)\qquad(\delta(x)=\mathrm{id}_x^{\underline{\Sigma}})
\]\[\Sigma(1)\simeq\delta(\Sigma(0))\amalg\Sigma^{*}(1)\ )\]
LaTeX source
\[
\Sigma(1)\simeq\delta(\Sigma(0))\amalg\Sigma^{*}(1)\ )
\]\[\Sigma^{*}_{ii}\overset{\text{déf}}{=}\Sigma_{ii}\cap\Sigma^{*}(1)=\emptyset ,\]
LaTeX source
\[
\Sigma^{*}_{ii}\overset{\text{déf}}{=}\Sigma_{ii}\cap\Sigma^{*}(1)=\emptyset ,
\]\[\boxed{\Sigma_i\simeq\Sigma_{ii}}\]
LaTeX source
\[
\boxed{\Sigma_i\simeq\Sigma_{ii}}
\]\[\underline{\Sigma}=\bigl(\Sigma(0),\ \Sigma(1)=\delta(\Sigma(0))\amalg\Sigma^{*}(1)\bigr),\qquad \Sigma(1)\subset\Sigma\times\Sigma ,\]
LaTeX source
\[
\underline{\Sigma}=\bigl(\Sigma(0),\ \Sigma(1)=\delta(\Sigma(0))\amalg\Sigma^{*}(1)\bigr),\qquad \Sigma(1)\subset\Sigma\times\Sigma ,
\]\[\underline{\Sigma}\longrightarrow\underline I\qquad\text{avec } I=\pi_0(\Sigma),\]
LaTeX source
\[
\underline{\Sigma}\longrightarrow\underline I\qquad\text{avec } I=\pi_0(\Sigma),
\]\[\Sigma_{ij}\longrightarrow\Sigma_j\quad\text{soit \emph{mono} i.e.\ plongement fermé.}\]
LaTeX source
\[
\Sigma_{ij}\longrightarrow\Sigma_j\quad\text{soit \emph{mono} i.e.\ plongement fermé.}
\]\[\Sigma_{ik}\cap\Sigma_{jk}=\emptyset\quad\text{si } i,j<k,\ i,j\ \text{non comparables.}\]
LaTeX source
\[
\Sigma_{ik}\cap\Sigma_{jk}=\emptyset\quad\text{si } i,j<k,\ i,j\ \text{non comparables.}
\]\[\widetilde I\longrightarrow I\]
LaTeX source
\[ \widetilde I\longrightarrow I \]
\[\Sigma_1=\coprod_{i,j\ \text{tels que}\ X_i\subset X_j}X_i\ ),\]
LaTeX source
\[
\Sigma_1=\coprod_{i,j\ \text{tels que}\ X_i\subset X_j}X_i\ ),
\]\[X_i^{*}=X_i\setminus\bigcup_{j<i}X_j .\]
LaTeX source
\[
X_i^{*}=X_i\setminus\bigcup_{j<i}X_j .
\]\[(42)\qquad \Sigma_i=X_i\setminus X_i\cap\mathring T_{<i}\qquad \mathring T_{<i}\overset{\text{déf}}{=}T_{<i}\setminus\dot T_{<i} .\]
LaTeX source
\[
(42)\qquad \Sigma_i=X_i\setminus X_i\cap\mathring T_{<i}\qquad \mathring T_{<i}\overset{\text{déf}}{=}T_{<i}\setminus\dot T_{<i} .
\]\[(43)\qquad \Sigma_{ij}=\dot T'_{ij}\setminus\mathring T(S\dot T'_{ij})\]
LaTeX source
\[
(43)\qquad \Sigma_{ij}=\dot T'_{ij}\setminus\mathring T(S\dot T'_{ij})
\]\[\Sigma_{ij}\subset\Sigma_j\]
LaTeX source
\[
\Sigma_{ij}\subset\Sigma_j
\]\[I\xrightarrow{\ \delta\ }\mathbb N\]
LaTeX source
\[
I\xrightarrow{\ \delta\ }\mathbb N
\]\[(44)\qquad X_{(d)}=\bigcup_{\substack{i\in I\\ \delta(i)\le d}}X_i\]
LaTeX source
\[
(44)\qquad X_{(d)}=\bigcup_{\substack{i\in I\\ \delta(i)\le d}}X_i
\]\[(45)\qquad X_{-1}=\emptyset\subset X_{(0)}\subset X_{(1)}\subset\cdots\subset X_{(N)}=X\qquad\text{si } N=\sup_{i\in I}\delta(i)\]
LaTeX source
\[
(45)\qquad X_{-1}=\emptyset\subset X_{(0)}\subset X_{(1)}\subset\cdots\subset X_{(N)}=X\qquad\text{si } N=\sup_{i\in I}\delta(i)
\]\[(46)\qquad X^{(0)}=X\supset X^{(1)}\supset\cdots\supset X^{(N+1)}=\emptyset\]
LaTeX source
\[
(46)\qquad X^{(0)}=X\supset X^{(1)}\supset\cdots\supset X^{(N+1)}=\emptyset
\]\[(47)\qquad X^{(i)}\subset X\setminus X_{i-1}\]
LaTeX source
\[
(47)\qquad X^{(i)}\subset X\setminus X_{i-1}
\]\[(48)\quad\left\lbrace
\begin{array}{l}
X^{(1)}=X\setminus\mathring T_{(0)}\\[4pt]
X^{(1)}_{(d)}=X^{(1)}\cap X_{(d)}\qquad(=\emptyset\ \text{si}\ d\le0)\\[4pt]
X^{(1)}_i=X^{(1)}\cap X_i\qquad\text{de sorte que}\quad X^{(1)}_{(d)}=\bigcup_{\substack{i\in I\\ \delta(i)\le d}}X^{(1)}\cap X_i
\end{array}\right.\]
LaTeX source
\[
(48)\quad\left\lbrace
\begin{array}{l}
X^{(1)}=X\setminus\mathring T_{(0)}\\[4pt]
X^{(1)}_{(d)}=X^{(1)}\cap X_{(d)}\qquad(=\emptyset\ \text{si}\ d\le0)\\[4pt]
X^{(1)}_i=X^{(1)}\cap X_i\qquad\text{de sorte que}\quad X^{(1)}_{(d)}=\bigcup_{\substack{i\in I\\ \delta(i)\le d}}X^{(1)}\cap X_i
\end{array}\right.
\]\[(49)\qquad X^{(2)}=X^{(1)}\setminus\mathring T^{(1)}_{(1)}=X^{(1)}\cap(X\setminus\mathring T_1)=X\setminus(\mathring T_0\cup\mathring T_1)\]
LaTeX source
\[
(49)\qquad X^{(2)}=X^{(1)}\setminus\mathring T^{(1)}_{(1)}=X^{(1)}\cap(X\setminus\mathring T_1)=X\setminus(\mathring T_0\cup\mathring T_1)
\]\[(48)\qquad X^{(d)}=X-\bigcup_{\alpha<d}\mathring T_\alpha .\]
LaTeX source
\[
(48)\qquad X^{(d)}=X-\bigcup_{\alpha<d}\mathring T_\alpha .
\]\[(49)\qquad
\begin{aligned}
\Sigma_i=X_i\cap X_{(d)}&=X_i\setminus\bigcup_{\alpha<d}X_i\cap\mathring T_\alpha\\
&=X_i\setminus\bigcup_{j<i}X_i\cap\mathring T_j\ \subset X_i^{*}
\end{aligned}\]
LaTeX source
\[
(49)\qquad
\begin{aligned}
\Sigma_i=X_i\cap X_{(d)}&=X_i\setminus\bigcup_{\alpha<d}X_i\cap\mathring T_\alpha\\
&=X_i\setminus\bigcup_{j<i}X_i\cap\mathring T_j\ \subset X_i^{*}
\end{aligned}
\]\[(49)\qquad \Sigma_{ij}=\dot T_i\cap\Sigma_j\subset\Sigma_j\]
LaTeX source
\[
(49)\qquad \Sigma_{ij}=\dot T_i\cap\Sigma_j\subset\Sigma_j
\]\[(50)\qquad
\begin{aligned}
\Sigma_{i_0\ldots i_n}&=\dot T_{i_0}\cap\dot T_{i_1}\cap\cdots\cap\dot T_{i_{n-1}}\cap\Sigma_{i_n}\\
&=\Sigma_{i_0i_n}\cap\Sigma_{i_1i_n}\cap\cdots\cap\Sigma_{i_{n-1}i_n}
\end{aligned}\]
LaTeX source
\[
(50)\qquad
\begin{aligned}
\Sigma_{i_0\ldots i_n}&=\dot T_{i_0}\cap\dot T_{i_1}\cap\cdots\cap\dot T_{i_{n-1}}\cap\Sigma_{i_n}\\
&=\Sigma_{i_0i_n}\cap\Sigma_{i_1i_n}\cap\cdots\cap\Sigma_{i_{n-1}i_n}
\end{aligned}
\]\[T_i^{(d)}=T_i\cap X^{(d)}\qquad(d=\delta(i))\]
LaTeX source
\[
T_i^{(d)}=T_i\cap X^{(d)}\qquad(d=\delta(i))
\]\[\Sigma_{ij}\subset\dot T_i^{(d)}\subset T_i^{(d)}\longrightarrow X_i^{(d)}=\Sigma_i\]
LaTeX source
\[
\Sigma_{ij}\subset\dot T_i^{(d)}\subset T_i^{(d)}\longrightarrow X_i^{(d)}=\Sigma_i
\]\[(51)\qquad X_i^{*}\longrightarrow \mathcal V_i\qquad\text{équivalence d'homotopie des provoisinages.}\]
LaTeX source
\[
(51)\qquad X_i^{*}\longrightarrow \mathcal V_i\qquad\text{équivalence d'homotopie des provoisinages.}
\]\[(52)\qquad \mathcal V_{ij}=\mathcal V_i\cap \mathcal V_j\]
LaTeX source
\[
(52)\qquad \mathcal V_{ij}=\mathcal V_i\cap \mathcal V_j
\]\[(53)\qquad X=\varinjlim\bigl(\mathcal V_i,\ \mathcal V_{ij},\ \rightrightarrows b_{ij},s_{ij}\bigr)\]
LaTeX source
\[
(53)\qquad X=\varinjlim\bigl(\mathcal V_i,\ \mathcal V_{ij},\ \rightrightarrows b_{ij},s_{ij}\bigr)
\]\[(54)\qquad \mathcal V_{i_0\ldots i_n}=\mathcal V_{i_0}\cap\cdots\cap \mathcal V_{i_n}\]
LaTeX source
\[
(54)\qquad \mathcal V_{i_0\ldots i_n}=\mathcal V_{i_0}\cap\cdots\cap \mathcal V_{i_n}
\]\[(55)\qquad \mathcal V_i^{Z}=\mathcal V_i\cap Z\qquad\text{provoisinage tubulaire de } X_i^{*} \text{ dans } Z\]
LaTeX source
\[
(55)\qquad \mathcal V_i^{Z}=\mathcal V_i\cap Z\qquad\text{provoisinage tubulaire de } X_i^{*} \text{ dans } Z
\]\[(56)\qquad X_i^{*}\hookrightarrow \mathcal V_i^{Z}\hookrightarrow \mathcal V_i\qquad\text{équivalences d'homotopie}\]
LaTeX source
\[
(56)\qquad X_i^{*}\hookrightarrow \mathcal V_i^{Z}\hookrightarrow \mathcal V_i\qquad\text{équivalences d'homotopie}
\]\[\mathcal V^{Z}_{i_0\ldots i_n}=\mathcal V_{i_0\ldots i_n}\cap Z=\mathcal V^{Z}_{i_0}\cap\cdots\cap \mathcal V^{Z}_{i_n}\]
LaTeX source
\[
\mathcal V^{Z}_{i_0\ldots i_n}=\mathcal V_{i_0\ldots i_n}\cap Z=\mathcal V^{Z}_{i_0}\cap\cdots\cap \mathcal V^{Z}_{i_n}
\]\[(57)\qquad \mathcal V^{Z}_{i_0\ldots i_n}\hookrightarrow \mathcal V_{i_0\ldots i_n}\qquad\text{équiv.\ d'homotopie}\]
LaTeX source
\[
(57)\qquad \mathcal V^{Z}_{i_0\ldots i_n}\hookrightarrow \mathcal V_{i_0\ldots i_n}\qquad\text{équiv.\ d'homotopie}
\]\[(58)\qquad X_J^{*}=\bigcup_{i\in J}X_i^{*},\qquad X_J=\bigcup_{i\in J}X_i\]
LaTeX source
\[
(58)\qquad X_J^{*}=\bigcup_{i\in J}X_i^{*},\qquad X_J=\bigcup_{i\in J}X_i
\]\[(59)\qquad X_J^{*}=X_{\widetilde J}\setminus X_{J'}\]
LaTeX source
\[
(59)\qquad X_J^{*}=X_{\widetilde J}\setminus X_{J'}
\]\[X_i^{J}\overset{\text{déf}}{=}X_i\cap X_J^{*}\qquad(i\in J)\]
LaTeX source
\[
X_i^{J}\overset{\text{déf}}{=}X_i\cap X_J^{*}\qquad(i\in J)
\]\[(60)\qquad \mathcal V^{J}_{j_0\ldots j_n}=\mathcal V_{j_0\ldots j_n}\cap X_J^{*}=\mathcal V^{Z}_{j_0\ldots j_n}\qquad\text{pour } Z=X_J^{*}\]
LaTeX source
\[
(60)\qquad \mathcal V^{J}_{j_0\ldots j_n}=\mathcal V_{j_0\ldots j_n}\cap X_J^{*}=\mathcal V^{Z}_{j_0\ldots j_n}\qquad\text{pour } Z=X_J^{*}
\]\[(61)\qquad
\begin{aligned}
X_J^{*}&\simeq\varinjlim_{i,j\in J}\bigl(\mathcal V_i^{J},\mathcal V_{ij}^{J}\bigr)\\
&\simeq\varinjlim_{(i_0<\cdots<i_n)\in\mathrm{Drap}(J)}\bigl(\mathcal V^{J}_{i_0\ldots i_n}\bigr)
\end{aligned}\]
LaTeX source
\[
(61)\qquad
\begin{aligned}
X_J^{*}&\simeq\varinjlim_{i,j\in J}\bigl(\mathcal V_i^{J},\mathcal V_{ij}^{J}\bigr)\\
&\simeq\varinjlim_{(i_0<\cdots<i_n)\in\mathrm{Drap}(J)}\bigl(\mathcal V^{J}_{i_0\ldots i_n}\bigr)
\end{aligned}
\]\[X_J^{*}\subset X_{J'}^{*}\]
LaTeX source
\[
X_J^{*}\subset X_{J'}^{*}
\]\[\mathcal V_{X_J^{*},X_{J'}^{*}}=\varinjlim_{i,j\in J}\bigl(\mathcal V_i^{J'},\mathcal V_{ij}^{J'}\bigr)\qquad\Bigl(X_J^{*}=\varinjlim_{i,j\in J}\bigl(\mathcal V_i^{J},\mathcal V_{ij}^{J}\bigr)\Bigr)\]
LaTeX source
\[
\mathcal V_{X_J^{*},X_{J'}^{*}}=\varinjlim_{i,j\in J}\bigl(\mathcal V_i^{J'},\mathcal V_{ij}^{J'}\bigr)\qquad\Bigl(X_J^{*}=\varinjlim_{i,j\in J}\bigl(\mathcal V_i^{J},\mathcal V_{ij}^{J}\bigr)\Bigr)
\]\[\mathcal V^{J'}_{J;K}=\mathcal V(X_J^{*},X_{J'}^{*})\setminus X_K\cap \mathcal V(X_J^{*},X_{J'}^{*})\]
LaTeX source
\[
\mathcal V^{J'}_{J;K}=\mathcal V(X_J^{*},X_{J'}^{*})\setminus X_K\cap \mathcal V(X_J^{*},X_{J'}^{*})
\]\[(63)\qquad \mathcal V^{J'}_{J;k}\overset{\text{déf}}{=}\mathcal V^{J'}_{J}\cap X_k=\emptyset\ \Longleftrightarrow\ X_J^{*}\cap X_k=\emptyset\]
LaTeX source
\[
(63)\qquad \mathcal V^{J'}_{J;k}\overset{\text{déf}}{=}\mathcal V^{J'}_{J}\cap X_k=\emptyset\ \Longleftrightarrow\ X_J^{*}\cap X_k=\emptyset
\]\[(62)\qquad
\begin{aligned}
\mathcal V_J&=\text{provoisinage tubulaire de } X_J^{*} \text{ dans } X\\
&=\bigcup_{i\in J}\mathcal V_i=\varinjlim_{i,j\in J}\bigl(\mathcal V_i,\mathcal V_{ij}\bigr)
\end{aligned}\]
LaTeX source
\[
(62)\qquad
\begin{aligned}
\mathcal V_J&=\text{provoisinage tubulaire de } X_J^{*} \text{ dans } X\\
&=\bigcup_{i\in J}\mathcal V_i=\varinjlim_{i,j\in J}\bigl(\mathcal V_i,\mathcal V_{ij}\bigr)
\end{aligned}
\]\[(63)\qquad \mathcal V_J^{L}\overset{\text{déf}}{=}\mathcal V_J\cap X_L^{*},\]
LaTeX source
\[
(63)\qquad \mathcal V_J^{L}\overset{\text{déf}}{=}\mathcal V_J\cap X_L^{*},
\]\[(64)\qquad \mathcal V_J^{J}=X_J^{*}\]
LaTeX source
\[
(64)\qquad \mathcal V_J^{J}=X_J^{*}
\]\[(65)\qquad \mathcal V_J^{L}=\mathcal V_J^{L_0}=\mathcal V_{J_0}^{L}=\mathcal V_{J_0}^{L_0}\]
LaTeX source
\[
(65)\qquad \mathcal V_J^{L}=\mathcal V_J^{L_0}=\mathcal V_{J_0}^{L}=\mathcal V_{J_0}^{L_0}
\]\[(66)\qquad J\prec L\]
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\[ (66)\qquad J\prec L \]
\[(67)\qquad I'=\lbrace i\in I \mid \exists j\in J,\ l\in L \text{ avec } j\le i\le l\rbrace\]
LaTeX source
\[
(67)\qquad I'=\lbrace i\in I \mid \exists j\in J,\ l\in L \text{ avec } j\le i\le l\rbrace
\]\[\mathcal V_J^L=\mathcal V_J\cap X^{*}_L=\Big(\bigcup_{j\in J}\mathcal V_j^{X'}\Big)\cap\underbrace{\Big(\bigcup_{l\in L}\mathcal V_l^{X'}\Big)}_{=X^{*}_L}=\bigcup_{j\in J,\ l\in L}\mathcal V^{X'}_{j,l}\]
LaTeX source
\[
\mathcal V_J^L=\mathcal V_J\cap X^{*}_L=\Big(\bigcup_{j\in J}\mathcal V_j^{X'}\Big)\cap\underbrace{\Big(\bigcup_{l\in L}\mathcal V_l^{X'}\Big)}_{=X^{*}_L}=\bigcup_{j\in J,\ l\in L}\mathcal V^{X'}_{j,l}
\]\[\mathcal V^{X'}_{j,l}=
\begin{cases}
\emptyset & \text{si } j,l \text{ non comparables}^{*}\\
\mathcal V^{X'}_j & \text{si } j=l, \text{ cas dans } j,l\in J\cap L\\
\mathcal V^{X'}_{j,l} \text{ standard} & \text{si } j<l\\
\mathcal V^{X'}_{l,j} \text{ standard} & \text{si } l<j \text{ dans } l,j\in J\cap L
\end{cases}\]
LaTeX source
\[
\mathcal V^{X'}_{j,l}=
\begin{cases}
\emptyset & \text{si } j,l \text{ non comparables}^{*}\\
\mathcal V^{X'}_j & \text{si } j=l, \text{ cas dans } j,l\in J\cap L\\
\mathcal V^{X'}_{j,l} \text{ standard} & \text{si } j<l\\
\mathcal V^{X'}_{l,j} \text{ standard} & \text{si } l<j \text{ dans } l,j\in J\cap L
\end{cases}
\]\[\mathcal V_J^L=\bigcup_{\substack{j\in J\\ l\in L\\ j\le l}}\mathcal V^{X'}_{j,l}\]
LaTeX source
\[
\mathcal V_J^L=\bigcup_{\substack{j\in J\\ l\in L\\ j\le l}}\mathcal V^{X'}_{j,l}
\]\[\mathcal V^{X'}_{j,l}\cap\mathcal V^{X'}_{j',l'}\]
LaTeX source
\[
\mathcal V^{X'}_{j,l}\cap\mathcal V^{X'}_{j',l'}
\]\[(70)\quad
\begin{cases}
\mathcal V^{X'}_i & i\in J\cap L\\
\mathcal V^{X'}_{ij} & i\in J\smallsetminus J\cap L,\ j\in L\smallsetminus J\cap L \qquad (i<j)\\
\mathcal V^{X'}_{ijk} & i\in J,\ k\in L,\ j\in J\cup L \qquad (i<j<k)\\
\mathcal V^{X'}_{ijkl} & i,j\in J,\ k,l\in L \qquad (i<j<k<l)
\end{cases}\]
LaTeX source
\[
(70)\quad
\begin{cases}
\mathcal V^{X'}_i & i\in J\cap L\\
\mathcal V^{X'}_{ij} & i\in J\smallsetminus J\cap L,\ j\in L\smallsetminus J\cap L \qquad (i<j)\\
\mathcal V^{X'}_{ijk} & i\in J,\ k\in L,\ j\in J\cup L \qquad (i<j<k)\\
\mathcal V^{X'}_{ijkl} & i,j\in J,\ k,l\in L \qquad (i<j<k<l)
\end{cases}
\]\[(71)\qquad \mathcal V^{X'}_{i_0\ldots i_n}\quad\text{avec } i_0<\cdots<i_n,\quad
\begin{cases} i_0,\ldots,i_n\in J\cup L\\ i_0\in J,\ i_n\in L\end{cases}\]
LaTeX source
\[
(71)\qquad \mathcal V^{X'}_{i_0\ldots i_n}\quad\text{avec } i_0<\cdots<i_n,\quad
\begin{cases} i_0,\ldots,i_n\in J\cup L\\ i_0\in J,\ i_n\in L\end{cases}
\]\[(72)\qquad \mathrm{Drap}^L_J(I)\subset\mathrm{Drap}(J\cup L)\subset\mathrm{Drap}(I)\]
LaTeX source
\[
(72)\qquad \mathrm{Drap}^L_J(I)\subset\mathrm{Drap}(J\cup L)\subset\mathrm{Drap}(I)
\]\[(73)\qquad H^{*}(\mathcal V^L_J,F)\Longleftarrow E_2^{pq}=H^p\big(\mathrm{Drap}^L_J(I),\ d\mapsto H^q(\mathcal V^{X'}_d,F)\big)\]
LaTeX source
\[
(73)\qquad H^{*}(\mathcal V^L_J,F)\Longleftarrow E_2^{pq}=H^p\big(\mathrm{Drap}^L_J(I),\ d\mapsto H^q(\mathcal V^{X'}_d,F)\big)
\]\[\Sigma_J\subset X^{*}_J,\qquad \Sigma_J=X^{*}_J\smallsetminus X^{*}_J\cap\bigcup_{i\in\overline J\smallsetminus J}\overset{\circ}{T}_i\]
LaTeX source
\[
\Sigma_J\subset X^{*}_J,\qquad \Sigma_J=X^{*}_J\smallsetminus X^{*}_J\cap\bigcup_{i\in\overline J\smallsetminus J}\overset{\circ}{T}_i
\]\[(71)\quad
\begin{cases}
\Sigma_\emptyset=\emptyset,\quad \Sigma_{\lbrace j\rbrace}=\Sigma_j,\quad \Sigma_I=X\\
\Sigma_J=X_J\quad\text{si } J \text{ un idéal i.e. } J=\overline J
\end{cases}\]
LaTeX source
\[
(71)\quad
\begin{cases}
\Sigma_\emptyset=\emptyset,\quad \Sigma_{\lbrace j\rbrace}=\Sigma_j,\quad \Sigma_I=X\\
\Sigma_J=X_J\quad\text{si } J \text{ un idéal i.e. } J=\overline J
\end{cases}
\]\[(72)\qquad \Sigma_J\simeq\varinjlim_{i,j\in J}\big(\Sigma_j,\ \Sigma_{ij},\ (s_{ij},b_{ij}:\Sigma_{ij}\rightrightarrows\Sigma_i,\Sigma_j)\big)\simeq\varinjlim_{d\in\mathrm{Drap}(J)}\Sigma_d\]
LaTeX source
\[
(72)\qquad \Sigma_J\simeq\varinjlim_{i,j\in J}\big(\Sigma_j,\ \Sigma_{ij},\ (s_{ij},b_{ij}:\Sigma_{ij}\rightrightarrows\Sigma_i,\Sigma_j)\big)\simeq\varinjlim_{d\in\mathrm{Drap}(J)}\Sigma_d
\]\[(73)\qquad \Sigma_J\overset{h}{\underset{\sim}{\hookrightarrow}}X^{*}_J\]
LaTeX source
\[
(73)\qquad \Sigma_J\overset{h}{\underset{\sim}{\hookrightarrow}}X^{*}_J
\]\[(74)\qquad \dot\Sigma_J=X_J\cap\big(\text{bord de }\underbrace{T_{\overline J\smallsetminus J}}_{\bigcup_{i\in\overline J\smallsetminus J}T_i}\big)
=\bigcup_{\substack{i<j\\ i,j\in J}}\mathrm{Im}(\Sigma_{ij}\to\Sigma_j)
=\varinjlim_{\substack{d\in\mathrm{Drap}\\ d \text{ ayant au moins 2 termes}}}\Sigma_d\]
LaTeX source
\[
(74)\qquad \dot\Sigma_J=X_J\cap\big(\text{bord de }\underbrace{T_{\overline J\smallsetminus J}}_{\bigcup_{i\in\overline J\smallsetminus J}T_i}\big)
=\bigcup_{\substack{i<j\\ i,j\in J}}\mathrm{Im}(\Sigma_{ij}\to\Sigma_j)
=\varinjlim_{\substack{d\in\mathrm{Drap}\\ d \text{ ayant au moins 2 termes}}}\Sigma_d
\]\[J\subset J'\subset I'\]
LaTeX source
\[ J\subset J'\subset I' \]
\[J_0=\overline J\smallsetminus J,\quad J'_0=\overline{J'}\smallsetminus J',\quad I'_0=\overline{I'}\smallsetminus I'.\]
LaTeX source
\[
J_0=\overline J\smallsetminus J,\quad J'_0=\overline{J'}\smallsetminus J',\quad I'_0=\overline{I'}\smallsetminus I'.
\]\[\begin{array}{ccccc}
\Sigma_J & \subset & \Sigma_{J'} & \subset & \Sigma_{I'}\\
\| & & \| & & \|{}^{?}\\
X_J\smallsetminus X_J\cap\overset{\circ}{T}_{I'_0} & & X_{J'}\smallsetminus X_{J'}\cap\overset{\circ}{T}_{I'_0} & & X_{I'}\smallsetminus X_{I'}\cap\overset{\circ}{T}_{I'_0}
\end{array}\]
LaTeX source
\[
\begin{array}{ccccc}
\Sigma_J & \subset & \Sigma_{J'} & \subset & \Sigma_{I'}\\
\| & & \| & & \|{}^{?}\\
X_J\smallsetminus X_J\cap\overset{\circ}{T}_{I'_0} & & X_{J'}\smallsetminus X_{J'}\cap\overset{\circ}{T}_{I'_0} & & X_{I'}\smallsetminus X_{I'}\cap\overset{\circ}{T}_{I'_0}
\end{array}
\]\[(75)\qquad T_{\Sigma_J,\Sigma_{I'}}=\Sigma_{I'}\cap\bigcup_{j\in J}T_j\quad\Big(=\Sigma_{I'}\cap\bigcup_{j\in\overline J}T_j\Big)\]
LaTeX source
\[
(75)\qquad T_{\Sigma_J,\Sigma_{I'}}=\Sigma_{I'}\cap\bigcup_{j\in J}T_j\quad\Big(=\Sigma_{I'}\cap\bigcup_{j\in\overline J}T_j\Big)
\]\[(76)\qquad \dot T_{\Sigma_J,\Sigma_{I'}}=\varinjlim_{d\in\mathrm{Drap}(I')}(\Sigma_d)\qquad\big(\simeq\Sigma_{J,I'\smallsetminus J}\big)\]
LaTeX source
\[
(76)\qquad \dot T_{\Sigma_J,\Sigma_{I'}}=\varinjlim_{d\in\mathrm{Drap}(I')}(\Sigma_d)\qquad\big(\simeq\Sigma_{J,I'\smallsetminus J}\big)
\]\[(77)\quad
\begin{cases}
\Sigma_{J,L}=\dot T_{\Sigma_J,\Sigma_{I'}}\smallsetminus\underbrace{\dot T_{\Sigma_J,\Sigma_{I'}}\cap\underbrace{\overset{\circ}{T}_{J'}}_{\bigcup_{j\in J'}\overset{\circ}{T}_j}}_{\dot T_{\Sigma_J,\Sigma_{I'}}\cap\bigcup_{j\in J'}\overset{\circ}{T}_j}\\[1ex]
\qquad=\dot T_{\Sigma_J,\Sigma_{I'}}\cap\Sigma_L
\end{cases}\]
LaTeX source
\[
(77)\quad
\begin{cases}
\Sigma_{J,L}=\dot T_{\Sigma_J,\Sigma_{I'}}\smallsetminus\underbrace{\dot T_{\Sigma_J,\Sigma_{I'}}\cap\underbrace{\overset{\circ}{T}_{J'}}_{\bigcup_{j\in J'}\overset{\circ}{T}_j}}_{\dot T_{\Sigma_J,\Sigma_{I'}}\cap\bigcup_{j\in J'}\overset{\circ}{T}_j}\\[1ex]
\qquad=\dot T_{\Sigma_J,\Sigma_{I'}}\cap\Sigma_L
\end{cases}
\]\[(78)\qquad \Sigma_{J,L}\simeq\varinjlim_{d\in\mathrm{Drap}(J\cup L)}\Sigma_d\]
LaTeX source
\[
(78)\qquad \Sigma_{J,L}\simeq\varinjlim_{d\in\mathrm{Drap}(J\cup L)}\Sigma_d
\]\[(79)\qquad
\begin{array}{ccc}
\Sigma_{J,L} & \hookrightarrow & \Sigma_L\\
\downarrow & & \\
\Sigma_J & &
\end{array}\]
LaTeX source
\[
(79)\qquad
\begin{array}{ccc}
\Sigma_{J,L} & \hookrightarrow & \Sigma_L\\
\downarrow & & \\
\Sigma_J & &
\end{array}
\]\[(80)\qquad (j_1\ldots j_p\,l_1\ldots l_q)\qquad j_1\ldots j_p\in J,\quad l_1\ldots l_q\in L,\quad p,q\ge 1\]
LaTeX source
\[ (80)\qquad (j_1\ldots j_p\,l_1\ldots l_q)\qquad j_1\ldots j_p\in J,\quad l_1\ldots l_q\in L,\quad p,q\ge 1 \]
\[(81)\qquad
\begin{array}{ccc}
\Sigma_{j_1\ldots j_p l_1\ldots l_q} & \hookrightarrow & \Sigma_{l_1\ldots l_q}\\
\downarrow & & \\
\Sigma_{j_1\ldots j_p} & &
\end{array}\]
LaTeX source
\[
(81)\qquad
\begin{array}{ccc}
\Sigma_{j_1\ldots j_p l_1\ldots l_q} & \hookrightarrow & \Sigma_{l_1\ldots l_q}\\
\downarrow & & \\
\Sigma_{j_1\ldots j_p} & &
\end{array}
\]\[(82)\qquad \overline A=\lbrace i\in I\mid \exists j\in A \text{ tel que } i\le j\rbrace\]
LaTeX source
\[
(82)\qquad \overline A=\lbrace i\in I\mid \exists j\in A \text{ tel que } i\le j\rbrace
\]\[(83)\qquad \widetilde B=\lbrace i\in I\mid \exists j\in B \text{ tel que } i\ge j\rbrace\]
LaTeX source
\[
(83)\qquad \widetilde B=\lbrace i\in I\mid \exists j\in B \text{ tel que } i\ge j\rbrace
\]\[(85)\qquad J\prec K\ \overset{\text{déf}}{\Longleftrightarrow}\ J\subset\overline K,\ K\subset\widetilde J\]
LaTeX source
\[
(85)\qquad J\prec K\ \overset{\text{déf}}{\Longleftrightarrow}\ J\subset\overline K,\ K\subset\widetilde J
\]\[(86)\qquad J\sim K\iff J\subset\underbrace{\overline K\cap\widetilde K}_{B(K)},\ K\subset\underbrace{\overline J\cap\widetilde J}_{B(J)}\iff B(J)=B(K)\]
LaTeX source
\[
(86)\qquad J\sim K\iff J\subset\underbrace{\overline K\cap\widetilde K}_{B(K)},\ K\subset\underbrace{\overline J\cap\widetilde J}_{B(J)}\iff B(J)=B(K)
\]\[(87)\qquad I'=\lbrace i\in I\mid \exists j\in J,\ k\in K \text{ t.q. } j\le i\le k\rbrace=\widetilde J\cap\overline K\]
LaTeX source
\[
(87)\qquad I'=\lbrace i\in I\mid \exists j\in J,\ k\in K \text{ t.q. } j\le i\le k\rbrace=\widetilde J\cap\overline K
\]\[I\longrightarrow\mathfrak P(X)\qquad i\longmapsto X_i\]
LaTeX source
\[ I\longrightarrow\mathfrak P(X)\qquad i\longmapsto X_i \]
\[d=(i_0<\cdots<i_n)\]
LaTeX source
\[ d=(i_0<\cdots<i_n) \]
\[(88)\qquad X_d=X_{i_0},\]
LaTeX source
\[
(88)\qquad X_d=X_{i_0},
\]\[(89)\qquad H^{*}(X,F)\Longleftarrow E_2^{pq}=H^p\big(I,\ i\mapsto H^q(X_i,F)\big)=H^p_{ss}\big(\mathrm{Drap}(I),\ d\mapsto H^q(X_d,F)\big)\]
LaTeX source
\[
(89)\qquad H^{*}(X,F)\Longleftarrow E_2^{pq}=H^p\big(I,\ i\mapsto H^q(X_i,F)\big)=H^p_{ss}\big(\mathrm{Drap}(I),\ d\mapsto H^q(X_d,F)\big)
\]\[(90)\qquad X_i^{*}=X_i\smallsetminus\bigcup_{j<i}X_j\]
LaTeX source
\[
(90)\qquad X_i^{*}=X_i\smallsetminus\bigcup_{j<i}X_j
\]\[(91)\qquad X_i=\bigcup_{j\in\bar i}X_j^{*}\qquad(\text{i.e. } j\le i)\]
LaTeX source
\[
(91)\qquad X_i=\bigcup_{j\in\bar i}X_j^{*}\qquad(\text{i.e. } j\le i)
\]\[(94)\qquad \bigcup_\alpha X^{*}_{J_\alpha}=X^{*}_{\bigcup_\alpha J_\alpha},\qquad \bigcap_\alpha X^{*}_{J_\alpha}=X^{*}_{\bigcap_\alpha J_\alpha}\]
LaTeX source
\[
(94)\qquad \bigcup_\alpha X^{*}_{J_\alpha}=X^{*}_{\bigcup_\alpha J_\alpha},\qquad \bigcap_\alpha X^{*}_{J_\alpha}=X^{*}_{\bigcap_\alpha J_\alpha}
\]\[X^{*}_\emptyset=\emptyset,\qquad X^{*}_I=X.\]
LaTeX source
\[
X^{*}_\emptyset=\emptyset,\qquad X^{*}_I=X.
\]\[(95)\qquad X^{*}_J\subset X_J=X^{*}_{\overline J}\]
LaTeX source
\[
(95)\qquad X^{*}_J\subset X_J=X^{*}_{\overline J}
\]\[(96)\qquad X^{*}_J=X^{*}_{\overline J}\smallsetminus X^{*}_{\underbrace{\scriptstyle\overline J\smallsetminus J}_{J_0}}\]
LaTeX source
\[
(96)\qquad X^{*}_J=X^{*}_{\overline J}\smallsetminus X^{*}_{\underbrace{\scriptstyle\overline J\smallsetminus J}_{J_0}}
\]\[(97)\qquad \overline{X^{*}_J}=X_J\ (=X^{*}_{\overline J})=X^{*}_J\amalg X^{*}_{\overline J\smallsetminus J}\]
LaTeX source
\[
(97)\qquad \overline{X^{*}_J}=X_J\ (=X^{*}_{\overline J})=X^{*}_J\amalg X^{*}_{\overline J\smallsetminus J}
\]\[(98)\qquad X^{*}_L=\complement_X X^{*}_J\]
LaTeX source
\[
(98)\qquad X^{*}_L=\complement_X X^{*}_J
\]\[X_i\subset X_j\iff\forall i'\in\bar i,\ X^{*}_{i'}\subset X_j=\bigcup_{i''\in\bar j}X^{*}_{i''}\]
LaTeX source
\[
X_i\subset X_j\iff\forall i'\in\bar i,\ X^{*}_{i'}\subset X_j=\bigcup_{i''\in\bar j}X^{*}_{i''}
\]\[(92)\qquad X_i\subset X_j\iff X_i^{*}\subset X_j\iff i\le j\]
LaTeX source
\[
(92)\qquad X_i\subset X_j\iff X_i^{*}\subset X_j\iff i\le j
\]\[(93)\qquad
\begin{cases}
X^{*}_J=\bigcup_{j\in J}X^{*}_j & (\text{pas nécessairement fermé})\\
X_J=\bigcup_{j\in J}X_j & \text{fermé dans } X
\end{cases}\]
LaTeX source
\[
(93)\qquad
\begin{cases}
X^{*}_J=\bigcup_{j\in J}X^{*}_j & (\text{pas nécessairement fermé})\\
X_J=\bigcup_{j\in J}X_j & \text{fermé dans } X
\end{cases}
\]\[(94)\qquad \mathcal V_i=\mathcal V(X_i^{*},X)\]
LaTeX source
\[
(94)\qquad \mathcal V_i=\mathcal V(X_i^{*},X)
\]\[(95)\qquad \mathcal V_i\cap X_j^{*}\ne\emptyset\iff\mathcal V_i\cap\overline{X_j^{*}}\ne\emptyset\iff X_i^{*}\cap\overline{X_j^{*}}\ne\emptyset
\ \overset{(a)}{\Longrightarrow}\ X_i^{*}\cap X_j\ne\emptyset\ \overset{(b)}{\Longrightarrow}\ i\le j\]
LaTeX source
\[
(95)\qquad \mathcal V_i\cap X_j^{*}\ne\emptyset\iff\mathcal V_i\cap\overline{X_j^{*}}\ne\emptyset\iff X_i^{*}\cap\overline{X_j^{*}}\ne\emptyset
\ \overset{(a)}{\Longrightarrow}\ X_i^{*}\cap X_j\ne\emptyset\ \overset{(b)}{\Longrightarrow}\ i\le j
\]\[\mathcal V_i\cap X_j^{*}\ne\emptyset\iff i\le j.\]
LaTeX source
\[
\mathcal V_i\cap X_j^{*}\ne\emptyset\iff i\le j.
\]\[(96)\qquad \mathcal V_i\cap\mathcal V_j\ne\emptyset\ \overset{(c)}{\Longrightarrow}\ i \text{ et } j \text{ comparables i.e. } i\le j \text{ ou } j\le i\]
LaTeX source
\[
(96)\qquad \mathcal V_i\cap\mathcal V_j\ne\emptyset\ \overset{(c)}{\Longrightarrow}\ i \text{ et } j \text{ comparables i.e. } i\le j \text{ ou } j\le i
\]\[(97)\qquad \mathcal V_{ij}\overset{\text{déf}}{=}\mathcal V_i\cap\mathcal V_j,\]
LaTeX source
\[
(97)\qquad \mathcal V_{ij}\overset{\text{déf}}{=}\mathcal V_i\cap\mathcal V_j,
\]\[(98)\qquad \mathcal V_{i_0\ldots i_n}=\mathcal V_{i_0}\cap\cdots\cap\mathcal V_{i_n},\]
LaTeX source
\[
(98)\qquad \mathcal V_{i_0\ldots i_n}=\mathcal V_{i_0}\cap\cdots\cap\mathcal V_{i_n},
\]\[i_0<i_1<\cdots<i_n\]
LaTeX source
\[ i_0<i_1<\cdots<i_n \]
\[(99)\qquad \mathcal V^Z_{i_0,\ldots,i_n}=\mathcal V_{i_0\ldots i_n}\cap Z\qquad(\hookrightarrow\mathcal V_{i_0\ldots i_n})\]
LaTeX source
\[
(99)\qquad \mathcal V^Z_{i_0,\ldots,i_n}=\mathcal V_{i_0\ldots i_n}\cap Z\qquad(\hookrightarrow\mathcal V_{i_0\ldots i_n})
\]\[(100)\qquad \mathcal V^Z_{i_0\ldots i_n}\overset{(h)}{\hookrightarrow}\mathcal V_{i_0\ldots i_n}\qquad(\text{équiv.\ d'homotopie})\]
LaTeX source
\[
(100)\qquad \mathcal V^Z_{i_0\ldots i_n}\overset{(h)}{\hookrightarrow}\mathcal V_{i_0\ldots i_n}\qquad(\text{équiv.\ d'homotopie})
\]\[(101)\qquad \mathcal V_J=\mathcal V(X^{*}_J,X)=\bigcup_{j\in J}\mathcal V_j=\varinjlim_{d\in\mathrm{Drap}(J)}\mathcal V_d\]
LaTeX source
\[
(101)\qquad \mathcal V_J=\mathcal V(X^{*}_J,X)=\bigcup_{j\in J}\mathcal V_j=\varinjlim_{d\in\mathrm{Drap}(J)}\mathcal V_d
\]\[(102)\qquad \mathcal V^Z_J=\mathcal V_J\cap Z\]
LaTeX source
\[ (102)\qquad \mathcal V^Z_J=\mathcal V_J\cap Z \]
\[(103)\qquad X^{*}_J\overset{(h)}{\hookrightarrow}\mathcal V_J\qquad(\text{équiv.\ d'hom.})\]
LaTeX source
\[
(103)\qquad X^{*}_J\overset{(h)}{\hookrightarrow}\mathcal V_J\qquad(\text{équiv.\ d'hom.})
\]\[(104)\qquad X^{*}_J\overset{(h)}{\hookrightarrow}\underbrace{\mathcal V^Z_J}_{=\mathcal V(X^{*}_J,Z)}\overset{(h)}{\hookrightarrow}\mathcal V_J\qquad(\text{équiv.\ d'hom.})\ \text{si } Z\supset X_J\]
LaTeX source
\[
(104)\qquad X^{*}_J\overset{(h)}{\hookrightarrow}\underbrace{\mathcal V^Z_J}_{=\mathcal V(X^{*}_J,Z)}\overset{(h)}{\hookrightarrow}\mathcal V_J\qquad(\text{équiv.\ d'hom.})\ \text{si } Z\supset X_J
\]\[(105)\qquad \mathcal V^Z_J=\bigcup_{j\in J}\mathcal V^Z_j=\varinjlim_{d\in\mathrm{Drap}(J)}\mathcal V^Z_d\]
LaTeX source
\[
(105)\qquad \mathcal V^Z_J=\bigcup_{j\in J}\mathcal V^Z_j=\varinjlim_{d\in\mathrm{Drap}(J)}\mathcal V^Z_d
\]\[(106)\quad
\begin{cases}
\mathcal V_{J,L}=\mathcal V_J\cap\mathcal V_L, & \mathcal V^Z_{J,L}=\mathcal V_{J,L}\cap Z\\
\mathcal V^{L}_{J}=\mathcal V_J\cap X^{*}_L=\mathcal V_{J,L}\cap X^{*}_L=\mathcal V^{X^{*}_L}_{J,L}
\end{cases}\]
LaTeX source
\[
(106)\quad
\begin{cases}
\mathcal V_{J,L}=\mathcal V_J\cap\mathcal V_L, & \mathcal V^Z_{J,L}=\mathcal V_{J,L}\cap Z\\
\mathcal V^{L}_{J}=\mathcal V_J\cap X^{*}_L=\mathcal V_{J,L}\cap X^{*}_L=\mathcal V^{X^{*}_L}_{J,L}
\end{cases}
\]\[(107)\qquad \mathcal V_{J,L}=\bigcup_{\substack{j\in J\\ l\in L}}\underbrace{\mathcal V_j\cap\mathcal V_l}_{\mathcal V_{j,l}}
=\bigcup_{i\in J\cap L}\mathcal V_i\ \cup\bigcup_{\substack{j<l\\ j\in J,\ l\in L}}\mathcal V_{jl}\ \cup\bigcup_{\substack{l<j\\ j\in J,\ l\in L}}\mathcal V_{lj}\]
LaTeX source
\[
(107)\qquad \mathcal V_{J,L}=\bigcup_{\substack{j\in J\\ l\in L}}\underbrace{\mathcal V_j\cap\mathcal V_l}_{\mathcal V_{j,l}}
=\bigcup_{i\in J\cap L}\mathcal V_i\ \cup\bigcup_{\substack{j<l\\ j\in J,\ l\in L}}\mathcal V_{jl}\ \cup\bigcup_{\substack{l<j\\ j\in J,\ l\in L}}\mathcal V_{lj}
\]\[(108)\qquad \mathcal V_{J,L}=\bigcup_{i\in J\cap L}\mathcal V_i\ \cup\bigcup_{\substack{j<l\\ j\in J,\ l\in L}}\mathcal V_{jl}
=\varinjlim_{\substack{d\in\mathrm{Drap}(J\cup L)\\ d\cap J\ne\emptyset,\ d\cap L\ne\emptyset}}\mathcal V_d\]
LaTeX source
\[
(108)\qquad \mathcal V_{J,L}=\bigcup_{i\in J\cap L}\mathcal V_i\ \cup\bigcup_{\substack{j<l\\ j\in J,\ l\in L}}\mathcal V_{jl}
=\varinjlim_{\substack{d\in\mathrm{Drap}(J\cup L)\\ d\cap J\ne\emptyset,\ d\cap L\ne\emptyset}}\mathcal V_d
\]\[(109)\qquad \mathcal V^Z_{J,L}=\varinjlim_{\substack{d\in\mathrm{Drap}(J\cup L)\\ d\cap J\ne\emptyset,\ d\cap L\ne\emptyset}}\mathcal V^Z_d\]
LaTeX source
\[
(109)\qquad \mathcal V^Z_{J,L}=\varinjlim_{\substack{d\in\mathrm{Drap}(J\cup L)\\ d\cap J\ne\emptyset,\ d\cap L\ne\emptyset}}\mathcal V^Z_d
\]\[(110)\qquad \mathcal V^Z_{J,L}\overset{(h)}{\hookrightarrow}\mathcal V_{J,L}\]
LaTeX source
\[
(110)\qquad \mathcal V^Z_{J,L}\overset{(h)}{\hookrightarrow}\mathcal V_{J,L}
\]\[(111)\qquad \mathcal V^L_J\ \Big(\overset{\text{déf}}{=}\mathcal V^{X^{*}_L}_{J,L}\Big)\overset{(h)}{\hookrightarrow}\mathcal V_{J,L}\]
LaTeX source
\[
(111)\qquad \mathcal V^L_J\ \Big(\overset{\text{déf}}{=}\mathcal V^{X^{*}_L}_{J,L}\Big)\overset{(h)}{\hookrightarrow}\mathcal V_{J,L}
\]\[(112)\qquad J_0\prec J_1\prec\cdots\prec J_n\]
LaTeX source
\[ (112)\qquad J_0\prec J_1\prec\cdots\prec J_n \]
\[(113)\qquad \mathcal V_{J_0,\ldots,J_n}=\mathcal V_{J_0}\cap\cdots\cap\mathcal V_{J_n}
=\bigcup_{\substack{j_0\in J_0\\ \cdots\\ j_n\in J_n}}\mathcal V_{j_0 j_1\ldots j_n}
=\varinjlim_{\substack{d\in\mathrm{Drap}(J_0\cup J_1\cup\cdots\cup J_n)\\ d\cap J_0\ne\emptyset,\ldots,\ d\cap J_n\ne\emptyset}}\mathcal V_d\]
LaTeX source
\[
(113)\qquad \mathcal V_{J_0,\ldots,J_n}=\mathcal V_{J_0}\cap\cdots\cap\mathcal V_{J_n}
=\bigcup_{\substack{j_0\in J_0\\ \cdots\\ j_n\in J_n}}\mathcal V_{j_0 j_1\ldots j_n}
=\varinjlim_{\substack{d\in\mathrm{Drap}(J_0\cup J_1\cup\cdots\cup J_n)\\ d\cap J_0\ne\emptyset,\ldots,\ d\cap J_n\ne\emptyset}}\mathcal V_d
\]