\[
  \{X_i\} \in \mathfrak{P}(Y) \qquad X = \bigcup X_i
\]

\[
  \mathrm{int}(X_i) = X_i \smallsetminus \partial X_i
\]

\[
  F \subset \mathfrak{P}_f(X) \qquad \text{i.e.} \qquad F \in \mathfrak{P}(\mathfrak{P}_f(X))
\]

\[
  \mathfrak{F} \subset \mathfrak{P}(\mathfrak{P}_f) \qquad \text{i.e.} \qquad \mathfrak{F} \in \mathfrak{P}(\mathfrak{P}(\mathfrak{P}_f(X)))
\]

\[
  \Sigma = \coprod_{F \in \mathfrak{F}} \Sigma_F \longrightarrow \mathfrak{F}
\]

\[
  |F'| \subset |F|
\]

\[
  j \leq i \Longrightarrow j \in \Phi
\]

\[
  \Sigma'_F \simeq \Sigma_F, \qquad \Sigma'_F \subset \mathfrak{F}_{\mathrm{él}}
\]

\[
  U_i \cap U'_j = \emptyset \quad \text{\emph{ou}} \quad U_i = U'_j .
\]

\[
  X_i = X'_j .
\]

\[
  \mathcal{R} \subset \mathfrak{F}_{\mathrm{él}} \times \mathfrak{F}_{\mathrm{él}}
\]

\[
  S \subset \mathfrak{P}_2(\mathfrak{F}_{\mathrm{él}})
\]

\[
  \Sigma'_F = \Phi \subset \mathfrak{F}_{\mathrm{él}}
\]

\[
    \mathrm{Conf}_{\mathrm{l.f.}} \subset \mathfrak{P}(\mathfrak{F}_{\mathrm{él}})
  \]

\[
    \struck{\ill{}}\ \forall\, X \in F, \quad \exists\, X' \in F' \text{ telle que } X \ll X'.
  \]

\[
  \partial V_{i\alpha} \overset{\mathrm{déf}}{=} \overline{V_{i\alpha}} \smallsetminus V_{i\alpha} \quad \text{est réunion de strates}
\]

\[
  (\Longleftrightarrow [\text{si } X', X'' \ll X, \text{ sont compatibles à } X \text{, ils sont compatibles entre eux}])
\]

\[
  X_0 < X_1 < \cdots < X_{n-1} \ll X_n \in F .
\]

\[
  \ll, \quad \text{compatibilité sur } \mathfrak{F}_{\mathrm{él}} \struck{\ill{}}.
\]

\[
  \struck{\text{Pour tout}}\ \mathcal{L} \subset \mathcal{M} \quad (= \mathfrak{F}_{\mathrm{él}} = \text{ensemble des multistrates})
\]

\[
  \widehat{\mathcal{L}}_X \subset \mathfrak{P}(X) \qquad \text{i.e.} \qquad \widehat{\mathcal{L}}_X \in \mathfrak{P}(\mathfrak{P}(X))
\]

\[
  X \longmapsto \widehat{\mathcal{L}}_X \qquad \mathcal{M} \longrightarrow \mathfrak{P}(\mathfrak{P}(X)) .
\]

\[
  \struck{\ill{}}\ \mathcal{L}_{\mathcal{F}} = \text{ens. des lieux qui raffinent } \mathcal{F} \quad \text{i.e.} \quad \mathcal{L}_{\mathcal{F}} = \bigcup_{X \in \mathcal{F}} \mathcal{L}_X \in \mathcal{L}
\]

\[
  \widehat{\mathcal{L}}_{\mathcal{F}} = \{\mathcal{L}_X \mid X \in \mathcal{F}\} \in \mathfrak{P}(\mathcal{L}) \qquad \text{i.e.} \in \mathfrak{P}(\mathfrak{P}(\mathcal{L}))
\]

\[
  \mathcal{F} \longmapsto \widehat{\mathcal{L}}_{\mathcal{F}} \qquad \mathcal{F} \longrightarrow \mathfrak{P}(\mathfrak{P}(X))
\]

\[
  \widehat{\mathcal{M}} = \operatorname{Im}\bigl(\underbrace{\mathcal{M} \xrightarrow{X \mapsto \widehat{\mathcal{L}}_X} \mathfrak{P}(\mathfrak{P}(\mathcal{L}))}_{\text{application injective}}\bigr) \in \mathfrak{P}(\mathfrak{P}(\mathfrak{P}(\mathcal{L})))
\]

\[
  \partial X = \bigcup_{Y \in F,\ Y \subsetneq X} Y, \qquad X^{\circ} = X \smallsetminus \partial X
\]

\[
  X \cap Y = \bigcup_{Z \in F,\ Z \leq X, Y} Z
\]

\[
  C \ \text{donnée de}\ C = (\mathcal{M}, \ll, R_{\mathcal{M}}, \mathfrak{F} \ \text{ens. de parties de}\ \mathcal{M})
\]

\[
  \underbrace{\sup_{X' \ll X} \dim \mathrm{comb}\, \mathcal{M}_{\leq X'}}_{\text{dimension (géom.) de } X} < \infty
\]

\[
  \mathcal{L}^{\circ}_X = \mathcal{L}_X \smallsetminus \bigcup_{Y < X} \mathcal{L}_Y .
\]

\[
  X \longmapsto \mathcal{L}_X = \widetilde{X}, \qquad F \longrightarrow \mathfrak{P}(\mathcal{L})
\]

\[
  X \longmapsto \mathcal{L}^{\circ}_X = \widetilde{X}^{\circ}, \qquad F \longrightarrow \mathfrak{P}(\mathcal{L})
\]

\[
    F' \longmapsto \widetilde{F}'
  \]

\[
  (\mathcal{M}', \ll_{\mathcal{M}'}, F_{\mathcal{M}'})
\]

\[
  C_F \cap C_{\complement G}
\]

\[
  F' \to F
\]

\[
  \widetilde{X}'^{\circ} \subset \widetilde{X}^{\circ}
\]

\[
  X' \ll X
\]

\[
  \begin{array}{ccc}
    Y & \ll & X \\
    \vee & & \vee \\
    Y' & \ll & X'
  \end{array}
\]

\[
  \widetilde{Y} \cap \widetilde{X}' = \bigcup_{Y' \in R_{X'}} \widetilde{Y}'
\]

\[
  (\mathcal{M}, \mathfrak{F}, \Sigma)
\]

\[
  X \leq Y \overset{\mathrm{def}}{\Longleftrightarrow} F_X \subset F_Y \quad \text{i.e. } X \in F_Y
\]

\[
  \boxed{G' \cap F \subset G}
\]

\[
  F \ll G
\]

\[
  G \subset F' \cap H \overset{\text{C 4}}{\subset} F
\]

\[
  \begin{array}{c}
    F \subset H = F \cup G \\
    \uparrow{\scriptstyle \text{subd.}} \\
    F' \\
    \cup \\
    G
  \end{array}
\]

\[
  X \in F \Longleftrightarrow
  \begin{cases}
    X \text{ comp. à } F \\
    X \text{ raffine } F
  \end{cases}
\]

\[
  X \leq Y \Longleftrightarrow
  \begin{cases}
    X \text{ comp. à } Y \\
    X \ll Y
  \end{cases}
\]

\[
  G' \subset F'
\]

\[
  F'_i \,|\, F_i \cap F_j = F'_j \,|\, F_i \cap F_j
\]

\[
  G' = \lbrace X \in F' \mid X \ll G \rbrace
\]

\[
  \begin{array}{ccc}
    F' \subset \overline{F}' & \xrightarrow[s]{\sim} & F \\
    \cup & & \cup \\
    \overline{G}' & \xrightarrow[s]{\sim} & G
  \end{array}
\]

\[
  X \in \overline{F}' \cap \widetilde{G} = \overline{G}' , \quad \text{qed,}
\]

\[
  \mathcal{C} = (\mathfrak{F}, \leq, \preccurlyeq)
\]

\[
  F = \operatorname{Sup}_{i \in I} F_i \Longrightarrow \exists\, i \in I, \ F_i = F .
\]

\[
  \struck{\ill{}} \quad \mathcal{M} \subset \mathfrak{F}
\]

\[
  F \mapsto \overline{F} \qquad \mathfrak{F} \to \mathfrak{P}(\mathcal{M}) .
\]

\[
  F = \operatorname*{Sup}_{X \in \overline{F}} X
\]

\[
  \mathfrak{F} \to \mathfrak{P}(\mathcal{M})
\]

\[
    X \in A \Longleftrightarrow A_X \subset A \quad \text{(tautologique)}
  \]

\[
    \overline{A} = \lbrace A_X \mid A_X \subset A \rbrace \quad \text{i.e. } X \in A
  \]

\[
  \underset{\text{ss-figures de } F}{\mathfrak{F}_{\leq F}} \times \underset{\text{ss-fig. de } G}{\mathfrak{F}_{\leq G}}
  \xrightarrow{\ \sim\ }
  \underset{\text{ss-fig. de } F \cup G = H}{\mathfrak{F}_{\leq F \cup G}}
\]

\[
  \boxed{\mathfrak{P}_f(\mathfrak{F}) \underset{\psi}{\overset{\varphi}{\rightleftarrows}} \mathfrak{P}_f(\mathcal{M})}
  \qquad
  \varphi(\mathfrak{F}') = \mathfrak{F}' \cap \mathcal{M} , \quad
  \psi(\mathcal{M}') = \lbrace F \in \mathfrak{F} \mid \overline{F} \subset \mathcal{M}' \rbrace
\]

\[
  F = F' \cup F'' \Longrightarrow F' = F \ \text{ou}\ F'' = F .
\]

\[
  F = \Bigl(\coprod F_i\Bigr) \amalg H .
\]

\[
  F_i = \operatorname{Sup}_{X \in \mathcal{M}_i \cap \mathcal{M}_{\leq F}} X .
\]

\[
  \overline{F_i} = \overline{F} \cap \mathcal{M}_i , \qquad \overline{F} = \bigcup \overline{F_i} \quad (\text{réunion disj.}) .
\]

\[
  \mathfrak{F} \to \prod \mathfrak{F}_i , \qquad F \mapsto (F_i)
\]

\begin{align*}
  \mathfrak{F}' &= \operatorname{cosupp}_{\mathfrak{F}}(\Phi) = \lbrace F \in \mathfrak{F} \mid F \ \text{disjoint des}\ H_i \in \Phi \rbrace \quad (\subset \mathfrak{F}) \\
  \mathcal{M}' &= \operatorname{cosupp}_{\mathcal{M}}(\Phi) = \lbrace X \in \mathcal{M} \mid X \ \text{disjoint des}\ H_i \in \Phi \rbrace \\
  &= \mathcal{M} \cap \operatorname{cosupp}_{\mathfrak{F}}(\Phi) \subset \mathcal{M}
\end{align*}

\[
  \left\lbrace
  \begin{array}{l}
    F \in \mathfrak{F}' \Longleftrightarrow \overline{F} \subset \mathcal{M}' \\
    \mathcal{M}' = \mathfrak{F}' \cap \mathcal{M}
  \end{array}
  \right.
\]

\begin{align*}
  \mathfrak{F}'' &= \operatorname{cosupp}_{\mathfrak{F}}(\mathfrak{F}') \overset{\text{déf}}{=} \operatorname{supp}_{\mathfrak{F}}(\Phi) \\
  \mathcal{M}'' &= \operatorname{cosupp}_{\mathcal{M}}(\mathfrak{F}') \overset{\text{déf}}{=} \operatorname{supp}_{\mathcal{M}}(\Phi)
\end{align*}

\begin{align*}
  \operatorname{cosupp}_{\mathfrak{F}}(\Phi) &= \operatorname{cosupp}_{\mathfrak{F}}(\mathcal{M}_{\overline{\Phi}}) \\
  \operatorname{cosupp}_{\mathcal{M}}(\Phi) &= \lbrace X \in \mathcal{M} \mid X \ \text{disjoint de tt}\ Y \in \mathcal{M}_{\overline{\Phi}} \rbrace
\end{align*}

\[
  \mathcal{M}_0 = \mathfrak{F}_0 \cap \mathcal{M} , \qquad \mathfrak{F}_0 = \lbrace F \in \mathfrak{F} \mid F \subset \mathcal{M}_0 \rbrace .
\]

\[
  \mathcal{M}_\sigma \subset \mathcal{M}, \qquad \mathfrak{F}_\sigma \subset \mathfrak{F} .
\]

\[
  \operatorname{cosupp}_{\mathcal{M}}(\mathcal{M}') = \lbrace X \in \mathcal{M} \mid (X, X') \in R \ \forall X' \in \mathcal{M}' \rbrace
\]

\[
  F' \preccurlyeq F , \quad G' \preccurlyeq G , \qquad G' \leq F' , \quad G \leq F
\]

\[
  \begin{array}{ccc}
    H & \succcurlyeq & H' \\
    & & \geq \\
    G & \succcurlyeq & G' \\
    \geq & & \\
    F & &
  \end{array}
  \qquad \text{on complète par C6} \qquad
  \begin{array}{ccccc}
    H & \succcurlyeq & H' & & \\
    \geq & & \geq & & \\
    \overline{H} & \succcurlyeq & G & \succcurlyeq & G' \\
    \geq & & \geq & & \geq \\
    K & \succcurlyeq & \overline{G} & \succcurlyeq & F
  \end{array}
  \quad (*)
\]

\[
  F \preccurlyeq K \leq \underset{\overset{\shortparallel}{H}}{F}
\]

\[
  \begin{array}{ccc}
    G & \ll & F \\
    & & \geq \\
    & & F'
  \end{array}
\]

\[
  \begin{array}{ccc}
    G & \ll & F \\
    \geq & & \geq \\
    G' & \ll & F'
  \end{array}
\]

\[
  H = \operatorname*{Sup}_{\substack{X \in \mathcal{M} \\ X \leq H}} X \leq \operatorname*{Sup}_{\substack{X \in \mathcal{M} \\ X \in G'}} X = G' .
\]

\[
  G \leq \overline{F} \preccurlyeq F
\]

\[
  \begin{array}{ccccc}
    G & \leq & \overline{F} & \preccurlyeq & F \\
    \geq & & \geq & & \geq \\
    G' & \leq & \overline{F}' & \preccurlyeq & F' \\
    \shortparallel & & & & \\
    G \cap \overline{F}' & & & &
  \end{array}
\]

\[
  \begin{array}{ccc}
    G & \preccurlyeq & F \\
    \geq & & \geq \\
    G' & \preccurlyeq & F'
  \end{array}
  \qquad (\text{cf C6})
\]

\[
  \begin{array}{ccccccc}
    K & \leq & \overline{G} & \preccurlyeq & G & \preccurlyeq & F \\
    & & \geq & & \geq & & \geq \\
    & & \overline{G}' & \preccurlyeq & G' & \preccurlyeq & F'
  \end{array}
\]

\[
  \begin{array}{ccc}
    \overline{G} & \preccurlyeq & F \\
    \geq & & \geq \\
    \overline{G}' & \preccurlyeq & F'
  \end{array}
\]

\[
  \begin{array}{ccc}
    G & \leq & F \\
    \geq & & \geq \\
    G' & \leq & F' \\
    \shortparallel & & \\
    G \cap F' & &
  \end{array}
\]

\[
  \partial Y = \operatorname*{Sup}_{\substack{Y' \in \mathcal{M} \\ Y' \leq Y}} Y'
\]

\[
  \underset{\overset{\shortparallel}{X}}{F_Y} \cap \underset{\overset{\shortparallel}{X}}{F_{Y'}} = F_{Y \cap Y'}
\]

\[
  \varphi : \overline{F} \to \overline{G}
\]

\[
  \begin{array}{ccccc}
    G & \leq & F & \geq & X \\
    \succcurlyeq & & \succcurlyeq & & \\
    \overline{G} & \leq & \overline{F} & & \\
    \geq & & & & \\
    X & & & &
  \end{array}
\]

\[
  \begin{array}{ccccc}
    X \ll & G & \leq & F \\
          & \geq & & \geq \\
          & G_X & \leq & X
  \end{array}
\]

\[
  \begin{array}{ccc}
    G_X & \leq & X \\
        &      & \succcurlyeq \\
    \overline{G}_X & \leq & F_X \\
    \geq & & \\
    X & &
  \end{array}
\]

\[
  \begin{array}{lll}
    (a) & X \leq \Phi \preccurlyeq X & \Phi \in \mathfrak{F},\ X \in \mathcal{M} \\
    (b) & X \preccurlyeq \Psi \leq X & \Psi \in \mathfrak{F},\ X \in \mathcal{M}
  \end{array}
\]

\[
  X \leq F_X \preccurlyeq X
\]

\[
  X \leq \overline{G}_X \leq F_X = X ,
\]

\[
  \overline{G}_X = X \preccurlyeq G_X \leq X
\]

\[
  \begin{array}{ccccc}
    F & \ll & G & \ll & F \\
    \geq & & \geq & & \geq \\
    F_X & \ll & G_X & \ll & X \\
    \geq & & & & \\
    X & & & &
  \end{array}
\]

\[
  \begin{array}{ccccc}
    F & \leq & G' & \preccurlyeq & G \\
      &      &    &              & \leq \\
      &      &    &              & F' \\
      &      &    &              & \preccurlyeq \\
      &      &    &              & F
  \end{array}
\]

\[
  \begin{array}{ccccc}
    F & \leq & G' & \preccurlyeq & G \\
      &      & \geq &            & \leq \\
      &      & H & \preccurlyeq & F' \\
      &      &   &              & \preccurlyeq \\
      &      &   &              & F
  \end{array}
\]

\[
  (*) \qquad F \leq H \preccurlyeq F
\]

\[
  G' = F , \quad F' = F , \quad \text{d'où}
\]

\[
  (**) \qquad F \preccurlyeq G \leq F
\]

\[
    \begin{array}{ccc}
      F & \geq & G \\
      \succcurlyeq & & \\
      F' & &
    \end{array}
  \]

\[
    (*) \qquad
    \begin{array}{ccc}
      F & \geq & G \\
      \succcurlyeq & & \succcurlyeq \\
      F' & \geq & G'
    \end{array}
  \]

\[
    \begin{array}{ccc}
      G & \leq & F \\
      \succcurlyeq & & \\
      G' & &
    \end{array}
  \]

\[
    \begin{array}{ccc}
      G & \leq & F \\
      \succcurlyeq & & \succcurlyeq \\
      G' & \leq & F'
    \end{array}
  \]

\[
    G' = \operatorname{Inf}_{\ll}(F', G)
  \]

\[
    \begin{array}{ccc}
      F & \geq & G \\
      \geq & & \geq \\
      F' & \geq & G'
    \end{array}
  \]

\[
  \operatorname{Subdiv}(F) \simeq \prod_i \operatorname{Subdiv}(F_i)
\]

\[
  \widehat{F} = \lbrace\, |X| \mid X \in \overline{F} \ \text{i.e.}\ X \in \mathcal{M},\ X \leq F \,\rbrace
\]

\[
  |F| \subset \mathcal{L} , \qquad \widehat{F} \subset \mathfrak{P}(\mathcal{L})
\]

\[
  |F| = \bigcup_{X \in \overline{F}} |X| = \bigcup_{A \in \widehat{F}} A .
\]