\[
  X \overset{\beta}{\hookrightarrow} M
\]

\[
  \dot{A} = \overline{A} - A^{\circ} \ \struck{\ill{}} \in \mathcal{F}
\]

\[
  A = \textstyle\bigcup A_i \cap \complement B_i \qquad (A_i, B_i \in
  \mathcal{F}_0)
\]

\[
  D - E \subset C = \textstyle\bigcup C_i \subset D
\]

\[
  i \prec j \quad \overset{\text{déf}}{\Longleftrightarrow} \quad
  \overline{X_i} \subset \overline{X_j}
\]

\[
  \mathrm{Mob}^0(X) = X \supset \mathrm{Mob}^1(X) \supset \mathrm{Mob}^2(X)
  \supset \cdots
\]

\[
  F_0(X) \subset F_1(X) \subset F_3(X) \subset \cdots \qquad
  (F_i(X) = X - \mathrm{Mob}^{i+1}(X))
\]

\[
  Y_\nu(Y \times N) = Y \times Y_{\nu - i}(N).
\]

\[
  (Z_j \times Y')_{j \in J} \longrightarrow (X_j)_{j \in J}
\]

\[
  f : X \to Y
\]

\[
  \mathfrak{X} = \bigl( Y \xleftarrow{\;p\;} Z \overset{i}{\hookrightarrow} X
  \bigr), \qquad p^{*}(\mathcal{F}) \xrightarrow{\;\alpha\;}
  i^{*}(\mathcal{G}), \qquad (\mathcal{F}, \mathcal{G}, \alpha)
\]

\begin{align*}
  i'_{*}(\mathcal{F}) &= (\mathcal{F}, 0, 0) \\
  i'^{*}(\mathcal{F}, \mathcal{G}, \alpha) &= \mathcal{F} \\
  p'_{*}(\mathcal{G}) &= (p_{*} i^{*} \mathcal{G}, \mathcal{G},
  \mathrm{can} : p^{*} p_{*}(i^{*}\mathcal{G}) \to i^{*}(\mathcal{G})) \\
  p'^{*}(\mathcal{F}, \mathcal{G}, \alpha) &= \mathcal{G} \\
  q_{!}(\mathcal{G}) &= (0, \mathcal{G}, 0) \\
  q^{!}(\mathcal{G}) &=
\end{align*}

\begin{align*}
  & X_{I,J} = X_I - X_J, \qquad \Sigma \supset I \supset J \\
  & X_{I,J} \xrightarrow{\;\alpha_{IJ/I'J'}\;} X_{I',J'} \\
  & \alpha_{*},\ \alpha^{*},\ \alpha_{!},\ \alpha^{!} ; \qquad
  K_{X,\Lambda} \ \text{complexe dualisant}
\end{align*}

\[
  \partial X_\alpha \xrightarrow{\;p_\alpha\;} \cdots, \qquad
  \partial X_\alpha \xrightarrow{\;i_\alpha\;} X_\alpha
\]

\[
  \partial V_i^\circ = \bigcup_{j \in \Phi_i} \partial_j V_i^\circ
\]

\[
  \partial(\partial_j V_i^\circ) = \bigl[ (\partial_j V_i^\circ) \,|\,
  \partial V_j^\circ \bigr] \cup \partial_{\cdot}(\partial_j V_i^\circ)
\]

\begin{align*}
  \partial(\partial_j V_i^\circ) &= \bigcup_{k \in \Phi_j} (\partial_j
  V_i^\circ \,|\, \partial_k V_j^\circ) \cup \partial_{i,j}(\partial_j
  V_i^\circ) \\
  &= \Bigl( \bigcup_{k < j} (\partial_j V_i^\circ \,|\, \partial_k
  V_j^\circ) \Bigr) \cup \partial_{\emptyset,j} V_i^\circ \\
  \partial(\partial_{j'} V_i^\circ) &= \Bigl( \bigcup_{k < j'}
  \partial_{j'} V_i^\circ \,|\, \partial_k V_j^\circ \Bigr) \cup
  \partial_{i,j'} V_i^\circ
\end{align*}

\begin{align*}
  \partial(\partial_0 V_2^\circ) &\simeq \partial(\partial_1 V_2^\circ) =
  \partial_1 V_2^\circ \,|\, \partial V_1^\circ \\
  \partial_j V_i^\circ \,|\, \partial_k V_j^\circ &= \partial_{j,k}
  V_i^\circ \quad \text{fibré sur } V_k^\circ \\
  \partial_{j > k > l} &= \partial_{j,k} V_i^\circ \,|\, \partial_l
  V_k^\circ \quad \text{fibré sur } V_l^\circ \\
  & \partial_{j > k > l > m} V_i^\circ
\end{align*}

\[
  U \subset V \subset W
\]

\[
  U_n \to U_n^{\circ} \leftarrow \partial U_n^{\circ} ; \qquad
  \partial_{n-1} U_n^{\circ},\ \partial_{n-2} U_n^{\circ},\ \ldots,\
  \partial_0 U_n^{\circ} \longrightarrow U_{n-1}^{\circ},\
  U_{n-2}^{\circ},\ \ldots,\ U_0^{\circ} = U_0 .
\]

\[
  \partial_j V_i^{\circ} \to V_j^{\circ} ; \qquad
  \partial(\partial_j V_i^{\circ}) = \bigcup_{k} \partial_j
  V_i^{\circ}|\partial_k V_j^{\circ} \ \cup \ \cdots
\]

\[
  \left\lbrace
  \begin{array}{l}
  \mathrm{L.P.M.} \\
  \mathrm{SA.P.M.} \\
  \mathrm{Alg}_{\mathbb{C}},\ \mathrm{An}_{\mathbb{C}}
  \end{array}\right.
\]

\[
  X^1 = \bigcup_{\alpha} \dot{T}_\alpha
\]

\[
  \overline{F_i} \subset \overline{F_j} \ \overset{?}{\Longleftrightarrow}\
  \overline{F_i} \cap U \subset \overline{F_j} \cap U
\]

\[
  X = X^0 \supset X^1 \supset X^2 \supset \cdots \supset X^n \supset
  \cdots
\]

\[
  X - F_i = \bigcup_{j \neq i} \overline{F_j} \quad \text{(fermé)}
\]

\[
  F_i \text{ ouvert} \Longleftrightarrow X - F_i \ \bigl(=
  \textstyle\bigcup_{j \neq i} F_j\bigr) \text{ fermé} \Longleftrightarrow
  \forall j \neq i,\ \overline{F_j} \subset X - F_i
\]

\[
  F_i = \Phi_i - \bigcup_{\Phi_j \subsetneq \Phi_i} \Phi_j
\]

\[
  X \mapsto |X| = \varphi : \mathcal{M} \longrightarrow (\text{Espaces
  compacts})
\]

\[
  \mathrm{SsObj}(X) \xrightarrow{\ \varphi\ } \mathrm{SsObj}(\varphi(X))
\]

\[
  X_1 \to X_2 \to \cdots \to X_n \to \cdots
\]

\[
  \mathfrak{X} \longmapsto |\mathfrak{X}| = \varinjlim_i |X_i|
\]

\[
  \mathbb{R} = \text{``}\varinjlim\text{''}\, I_n \simeq \varinjlim I_1 ,
\]

\[
  \Gamma_{fg} = p^{-1}(\Gamma_f) \cap q^{-1}(\Gamma_g),
\]

\[
  (K \times K' \times K) \cap \mathrm{Im}\bigl(I^{n+n'} \to
  I^{n+n'+n}\bigr), \qquad (x, x') \mapsto (x, x', x)
\]

\[
  n + n' = \nu, \qquad p + n + n' = N, \qquad p' + n + n' = N'
\]

\[
  F_n = \mathrm{Hom}_M(I^n, I) \wr \ \mathrm{Hom}_M(I_0^n, I_0)
\]

\[
  \mathcal{M}_n \subset I^n \qquad n \in \mathbb{N} \qquad (I = [-1, +1])
\]

\[
  \Delta \quad \bigl\lbrace (x_0, \ldots, x_n, y) \bigm| x_i \geqslant 0,\ \
  \textstyle\sum x_i = 1,\ \ y - \sum_{0 \leqslant i \leqslant n} a_i x_i
  = 0 \bigr\rbrace \in \mathcal{M}_{n+2}
\]

\[
  \Longrightarrow \quad S \xleftarrow{\ \sim\ } Y \sqcup_X Z
\]

\[
  (Z \sqcup Y) \times_S (Z \sqcup Y) = Z \sqcup X \sqcup X \sqcup (Y
  \times_S Y)
\]

\[
  Z \sqcup X \sqcup X \sqcup (Y \times_S Y) \rightrightarrows Z \sqcup Y
  \to S
\]

\[
  Z \sqcup X \sqcup X \sqcup \bigl[ (X \times_Z X) \sqcup \Delta_Y \bigr]
  \rightrightarrows Z \sqcup Y
\]

\[
  \begin{array}{c} E \\ \mid \\ S \end{array} \qquad
  \mathrm{Grass}_n(E) \leftarrow \Sigma_{n,\sigma}(E) \to D_\sigma(E),
  \qquad \mathrm{Drap}(N) \to \mathrm{Grass}_n(E)
\]

\[
  \begin{array}{ccc} E & & E' \\ \mid & & \mid \\ S & \text{---} & S' =
  D_\sigma(E) \end{array}
\]

\[
  H(S)\bigl[ c_i(M), c_j(N) \bigr] \Big/ \Bigl( \textstyle\sum
  c_i(M) \Bigr) \Bigl( \sum c_j(N) \Bigr) = \sum_{h} c_h(E)
\]

\[
  \xi_1, \ldots, \xi_n \qquad c_1(N), \ldots, c_d(N)
\]

\[
  \Bigl( \prod_{1 \leqslant i \leqslant n} (1 + \xi_i) \Bigr)
  \sum_{0 \leqslant j \leqslant d} c_j(N) = \sum_{0 \leqslant h \leqslant
  N} c_h(E)
\]

\[
  Y \hookrightarrow I^p, \qquad \psi = (\psi_1, \ldots, \psi_p)
\]

\[
  (\varphi_i, \psi_j) : Y \longrightarrow (I \times I)^{n \times p}
\]

\[
  Y \times Y \leftarrow Y \times_S Y = \underbrace{\bigl( Y \times_{Y'} Y
  \bigr)}_{\Delta_{Y/Y'}} \cup (X \times X)
\]

\[
  Y' \times Y' \supset Y' \times_S Y' \neq \Delta_{Y'} \cup (X' \times X')
\]

\[
  I^p \text{ --- } I^{n+p}
\]

\[
  Z \overset{\sigma}{\longrightarrow} I^q
\]

\[
  Y \times_{I^q} Y = \Delta_Y \cup (X \times X)
\]

\[
  \begin{array}{ccc}
    Y \times_S Y & \hookrightarrow & S \times_S S \\
    \updownarrow & & \updownarrow \delta \\
    Y \times_{S'} Y & \hookrightarrow & S \times_{S'} S \\
    \parallel & & \\
    \Delta_Y \cup X \times X & &
  \end{array}
\]

\[
    R^n \xrightarrow{\ \varphi_{p_{*}}\ } R^{p_1 + \cdots + p_n}, \quad
    (x_1, \ldots, x_n) \mapsto (\underbrace{x_1, \ldots,
    x_1}_{p_1\ \text{fois}}, \ldots, \underbrace{x_n, \ldots,
    x_n}_{p_n\ \text{fois}})
  \]

\[
  \text{(5)} \qquad R^n = \bigcup_{X \in C_n} X
\]

\[
  (C_{*}\text{-modèles}) \longrightarrow \text{espaces compacts.}
\]