\[
  \boxed{Sm\cap Pr \subset S} \qquad
  \boxed{\begin{array}{l} WSC\cap Pr \subset S \\ WSF\cap Sm \subset S \end{array}}
\]

\[
  \boxed{\begin{array}{l} F \subsetneq Sm \\ C \subsetneq Pr \end{array}} \qquad
  \boxed{Sm \cap Pr \subsetneq S} \qquad
  \left\{\begin{array}{l} C = F^{\circ} \\ F = C^{\circ} \end{array}\right. \quad
  S = S^{\circ} \qquad
  \left\{\begin{array}{l} Sm = Pr^{\circ} \\ Pr = Sm^{\circ} \end{array}\right.
\]

\[
  \boxed{T\cap Pr \subset S,\ T\cap Sm \subset S}
\]

\[
  \boxed{WSC \cap Pr = S} \qquad WSF \cap Sm \subset S
\]

\[
  \begin{array}{ccccc}
    & & \text{bifibration} & & \\
    & & \text{Serre} & & \\
    \text{Serre fibration} & \textbf{Fibrations} & + & \textbf{Cofibrations} & \text{Serre cofibration} \\
    & & \text{monoe} & & \\
    \text{open imm.} & & & & \text{closed immersion}
  \end{array}
\]

\[
  Y \xrightarrow[\text{imm}]{\ f^{\natural}\ \text{open}\ } C(f) \xrightarrow{\ \lambda_f\ (\text{epi})\ } X,
  \qquad \sigma_f : X \to C(f) \ \text{closed section}
\]

\[
  Y \xrightarrow[\text{imm}]{\ f^{\natural\prime}\ \text{closed}\ } C'(f) \xrightarrow{\ \lambda'_f\ } X,
  \qquad \sigma'_f : X \to C'(f) \ \text{open section}
\]

\[
  Y \xrightarrow[\text{open imm}]{\ i_Y\ } C(Y) \xrightarrow{\ \varepsilon_Y\ } e
  \quad \text{closed section } (e \text{ strict final object})
\]

\[
  Y \xrightarrow[\text{closed imm}]{\ i'_Y\ } C'(Y) \xrightarrow{\ \varepsilon'_Y\ } e
  \quad \text{open section } (e \text{ strict initial object})
\]

\[
  C'(f)^{\circ} = C(f^{\circ}) \quad \text{i.e.} \quad C'(f) = C(f^{\circ})^{\circ}
\]

\[
  C'(Y)^{\circ} = C(Y^{\circ}) \quad \text{i.e.} \quad C'(Y) = C(Y^{\circ})^{\circ}
\]

\[
  \scriptstyle
  \begin{array}{c|c|c|c|c}
    \text{smooth Serre} & \text{Serre fibration} & \text{Serre} & \text{Serre cofibration} & \text{Serre proper} \\
    \hline
    \text{trivial smooth} & \text{trivial Serre fibration} & \left[\begin{array}{c}\text{Serre}\cap W \\ = \text{triv Serre}\end{array}\right] & \text{trivial cofibration} & \text{trivial proper}
  \end{array}
\]

\[
  \begin{array}{ccccc}
    \text{smooth} & \hookleftarrow & \textit{fibration} \quad + \quad \textit{cofibration} & \hookrightarrow & \text{proper} \\
    & & \text{mono} & & \\
    & & \downarrow & & \\
    & & \text{discrete} & &
  \end{array}
\]

\[
  \left.\begin{array}{l}
    \text{open immersion} \longleftrightarrow X^{\circ} \xrightarrow{\ \varphi\ } \Delta^1 = [0 \to 1] \\
    \phantom{\text{open immersion}} \longleftrightarrow \mathrm{crib}(X) \\
    \phantom{\text{open immersion \ }} Y = \varphi^{-1}(\{1\}) = \psi^{-1}(\{0\}) = X_0
  \end{array}\right\}
\]

\[
  \left\{\begin{array}{l}
    \text{closed immersion} \longleftrightarrow X \xrightarrow{\ \varphi\ } \Delta^1 = [\emptyset \to \{\emptyset\}] \\
    \phantom{\text{closed immersion}} \longleftrightarrow \mathrm{Cocrib}(X) \\
    \phantom{\text{closed immersion \ }} Y = \varphi^{-1}(\{1\}) = X_1
  \end{array}\right.
\]

\[
  Y \xrightarrow[\text{open imm}]{\ f^{\natural}\ } C(f) \xrightarrow{\ \lambda_f\ \text{epi}\ } X,
  \qquad \sigma_f : \text{closed section and \textit{homotopism}}
\]

\[
  Y \xrightarrow[\text{closed imm.}]{\ f^{\natural\prime}\ } C'(f) \xrightarrow{\ \lambda'_f\ } X,
  \qquad \sigma'_f : \text{open section and \textit{homotopism}}
\]

\[
  (Y \longrightarrow X) \Longrightarrow X \qquad Y_x \longrightarrow X \qquad 0 \longrightarrow 1
\]

\[
  X \longrightarrow \text{cofibers of } f^{\sharp}\,? \qquad
  Y \longrightarrow \text{cofibers of } \sigma_f\,?
\]

\[
  \begin{array}{c} \text{bifibration} \\ \uparrow \\ \text{Serre} \end{array}
\]

\[
  \begin{array}{c} \text{discrete} \\ \downarrow \\ \text{mono} \end{array}
\]

\[
  X \xrightarrow{\ \varphi^{\circ}\ } [\emptyset, \{\emptyset\}] = \Delta_1, \qquad
  Y = \varphi^{-1}(\{1\}) = \varphi^{-1}(\{0\})
\]

\[
  (\psi : X \xrightarrow{\ \varphi^{\circ}\ } \Delta_1^{\circ} \simeq \Delta_1)
\]

\[
  X \xrightarrow{\ \varphi'\ } [\emptyset, \{\emptyset\}] = \Delta_1 : [0 \longrightarrow 1], \qquad
  Y = \varphi'^{-1}(\{1\})
\]

\[
  \begin{array}{ccc}
    Y & \xrightarrow{\ f_0\ } & X_0 \\
    {\scriptstyle f_1}\downarrow & & \downarrow{\scriptstyle \alpha_1} \\
    X_1 & \xrightarrow{\ \alpha_0\ } & \Sigma
  \end{array}
  \qquad \Sigma \searrow X_0 \amalg_Y X_1
\]

\[
  \int \left(\begin{smallmatrix} Y & \to & X_0 \\ \downarrow & & \\ X_1 & & \end{smallmatrix}\right) \longrightarrow X
\]

\[
  \begin{array}{ccc}
    Y & \xrightarrow{\ f_0\ } & X_0 \\
    {\scriptstyle f_1}\downarrow & \text{cart} & \downarrow{\scriptstyle q_1} \\
    X_1 & \xrightarrow{\ q_0\ } & X
  \end{array}
\]

\[
  \begin{array}{ccc}
    Y & \overset{i}{\hookrightarrow} & X \\
    {\scriptstyle f}\downarrow & & \downarrow{\scriptstyle g} \\
    Y' & \underset{i'}{\hookrightarrow} & X' = X \amalg_Y Y'
  \end{array}
  \qquad i \text{ immersion ouverte ou fermée}
\]

\[
  \begin{array}{ccc}
    Y' & \overset{i}{\hookrightarrow} & X \\
    {\scriptstyle g}\downarrow & & \downarrow{\scriptstyle f} \\
    Y' & \underset{i'}{\hookrightarrow} & X' = X \amalg_Y Y'
  \end{array}
\]

\[
  i \in \underline{W}_A \Longrightarrow i' \in \underline{W}_A, \qquad
  g \in \underline{W}_A \Longrightarrow f \in \underline{W}_A
\]

\[ H_X(u, y) \longrightarrow \mathrm{Hom}_Z(f_U(u), f_Y(y)) \]

\[
  T \times U \subset T \times X \supset T \times Y \qquad
  H_{T \times X}((t, u), (t', y)) \simeq \mathrm{Hom}_T(t, t') \times H_X(u, y)
\]

\[
  h_{U'} : \Delta \times U' \to U, \qquad h_{Y'} : \Delta \times Y' \to Y
\]

\[
  \mathrm{Hom}(s, t) \times H_{X'}(u', y') \longrightarrow H_X(h_{U'}(s, u'), h_{Y'}(t, y'))
\]

\[
  \begin{array}{ccccc}
    Y & \overset{i}{\hookrightarrow} & X & \hookleftarrow & U \\
    {\scriptstyle g}\downarrow & & {\scriptstyle f}\downarrow & & \downarrow\wr \\
    Y' & \underset{i'}{\hookrightarrow} & X' & \hookleftarrow & U'
  \end{array}
\]

\[
  \mathrm{Hom}_{X'}(u', y') \overset{?}{\longleftarrow}
  \varinjlim_{y \in Y_{/y'}} \mathrm{Hom}(u, y)
\]

\[
  \varinjlim_{Y_{y'}} \mathrm{Hom}(u, Y_{y'}) .
\]

\[
  \int \left(\begin{smallmatrix} Y & \to & X \\ \downarrow & & \\ Y' & & \end{smallmatrix}\right)
  \longrightarrow X' = \varinjlim \left(\begin{smallmatrix} Y & \to & X \\ \downarrow & & \\ Y' & & \end{smallmatrix}\right) = X \amalg_Y Y'
\]

\[
  \begin{array}{ccc}
    \int(i, g) & \longrightarrow & X' \\
    \uparrow & & \uparrow \\
    \int(i_1, g_1) & \longrightarrow & X'_1
  \end{array}
\]

\[
  \simeq e_{u'} \ \text{pour } u' \in U', \qquad
  \simeq C(Y_{y'}) \ \text{pour } x = y' \in Y'
\]

\[
  \begin{array}{ccc}
    Y & \xrightarrow{\ i\ } & X \\
    {\scriptstyle g}\downarrow & & \downarrow{\scriptstyle f} \\
    Y' & \xrightarrow{\ i'\ } & X'
  \end{array}
\]

\[
  \int \left(\begin{smallmatrix} Y & \to & X \\ \downarrow & & \\ Y' & & \end{smallmatrix}\right) \longrightarrow X' .
\]

\[
  \int \left(\begin{smallmatrix} Y_{x'} = Y'_{x'} \times X_{x'} & \xrightarrow{\ \mathrm{pr}_2\ } & X_{x'} \\ \downarrow{\scriptstyle \mathrm{pr}_1} & & \\ Y'_{x'} & & \end{smallmatrix}\right) ,
\]

\[
  \int \left(\begin{smallmatrix} A & \xrightarrow{\mathrm{id}_A} & A \\ \downarrow & & \\ e & & \end{smallmatrix}\right)
  \qquad \left(\text{ou } \int \left(\begin{smallmatrix} A & \to & e \\ \downarrow{\scriptstyle \mathrm{id}_A} & & \\ A & & \end{smallmatrix}\right)\right),
\]

\[
  i \in \underline{W} \Longrightarrow i' \in \underline{W}, \qquad
  g \in \underline{W} \Longrightarrow f \in \underline{W}
\]

\[
  \mathrm{Hot}_A \simeq \overline{\underline{W}_A}^{\,-1}\, \overline{A^{\wedge}}
\]

\[
  X' \xrightarrow{\;s\;} X \overset{f}{\underset{g}{\rightrightarrows}} Y
  \quad \text{de } \overline{A^{\wedge}}
\]

\[
  X \overset{f}{\underset{g}{\rightrightarrows}} Y \xrightarrow{\;t\;} Y'
\]

\[
  (\mathrm{Cat}) \to \mathrm{Hot}(\underline{W}), \qquad
  A^{\wedge} \to \mathrm{Hot}_A \to \mathrm{Hot}(\underline{W})
\]

\[
  \begin{matrix} f = (f_j)_{j \in J} \\ g = (g_j)_{j \in J} \end{matrix}
  \; : \; \underbrace{\textstyle\coprod_J X_j}_{X}
  \overset{f}{\underset{g}{\rightrightarrows}} Y
\]

\[
  I \times X \simeq \coprod_J I \times X_j
\]

\[
  \begin{matrix} f_n \\ g_n \end{matrix} \Big\} \; X_n \rightrightarrows Y
  \qquad (n \in \mathbb{N})
\]

\[
  Y = I_\infty = (\;\underbrace{\to\leftarrow}\,\underbrace{\to\leftarrow}\;\cdots\;)
\]

\[
  \mathrm{Hom}_{\mathrm{Hot}}(X, Y) \to \prod_{j \in J} \mathrm{Hom}_{\mathrm{Hot}}(X_j, Y)
\]

\[
  s_j i_j = \mathrm{id}_{\overline{Y}'_j}
\]

\[
  (\mathrm{Cat}) \to \mathrm{Hot}(\underline{W}), \qquad
  A^{\wedge} \to \mathrm{Hot}(\underline{W}) \quad (A \text{ catégorie test
  stricte})
\]

\[
  \mathrm{Hom}_{\underline{W}_A^{-1}A^{\wedge}}\Bigl(X, \prod Y_j\Bigr)
  \to \prod \mathrm{Hom}_{\underline{W}_A^{-1}A^{\wedge}}(X, Y_j)
\]

\[
  f_j : X \to Y'_j \qquad \text{où } Y_j \to Y'_j \text{ équiv.\ faible.}
\]

\[
  f = (f_j) : X \to \prod Y'_j = Y'
\]

\[
  \beta = \prod \beta_j : Y = \prod Y_j \to Y'' = \prod Y'_j
\]

\[
  A_{/X \times Y} \to A_{/X} \times A_{/Y}
\]

\[
  (A_{/X \times Y})_{/\alpha} \simeq A_{/a \times b}
\]

\[
  f : X \to Y
\]

\[
  X \xrightarrow{\;\varphi\;} I
\]

\[
  \varphi(x) = \mathop{\mathrm{Inf}}_{i \in I(x)} i ,
  \qquad I(x) = \{\, i \in I \mid x \in X_i \,\}
\]

\[
  \alpha : x \to y ,
\]

\[
  X_i = \varphi^{-1}(I_i) \qquad \text{où } I_i = \{\, j \in I \mid j
  \leq i \,\}
\]

\[
  I \to \mathrm{Ouv}(X), \qquad i \mapsto X_i
\]

\[
  X_{/i} = X_i
\]

\[
  f_i : X_i \to Y_i
\]

\[
  f_{/i} : X_{/i} \to Y_{/i}
\]

\[
  \gamma(gf) = \gamma(\alpha), \quad \gamma(f'g) = \gamma(\beta), \quad
  \beta f = f' \alpha .
\]

\[
  \alpha, \beta \in \underline{W}, \qquad gf = \alpha, \quad f'g = \beta
  \quad \text{(quitte à factoriser)}
\]

\[
  X_0 \xrightarrow{f_0 = f} X_1 \xrightarrow{f_1 = g} X_2
  \xrightarrow{f_2} \cdots
\]

\[
  X_0 = \varinjlim (X_0 \to X_0 \to X_0 \to \cdots X_0 \to \cdots)
\]

\[
  X_\infty = \varinjlim_n (X_0 \to X_2 \to X_4 \to \cdots X_{2n} \to
  \cdots)
\]