\[
  \partial \xrightarrow{\ i\ } I
\]

\[
  \left\{
  \begin{array}{l}
    \varnothing \to e \\[2pt]
    \partial \times S^n(N) \longrightarrow I \times S^n(N)
  \end{array}
  \right.
\]

\[
  \underline{\mathrm{Hom}}(T, X)
\]

\[
  \mathrm{Ar}(C) \xrightarrow{\ p\ } C \times C
\]

\[
  \left\{
  \begin{array}{l}
    \text{a) isom. are in } W \\
    \text{b) if } f = vu \text{, and two of } u, v, f \text{ are in } W \text{, so is the third} \\
    \text{c) if } X \underset{g}{\overset{f}{\rightleftarrows}} Y \text{ with } gf = \mathrm{id}_X,\ fg \in W \text{, then } f, g \in W
  \end{array}
  \right.
\]

\[
  (TC)_0 \subset C_0 \ (\subset \mathrm{Fl}(A))
\]

\[
  TF \subset F \ (\subset \mathrm{Fl}(A))
\]

\[
  \begin{array}{ccc}
    TC & \subset & C \quad (\subset \mathrm{Fl}(A)) \\
    \cup & & \cup \\
    (TC)_0 & \subset & C_0
  \end{array}
\]

\[
  TC = C \cap \widetilde{W} \qquad TF = F \cap \widetilde{W}
\]

\[
  \widetilde{W} \subset \mathrm{Fl}(A), \qquad
  \widetilde{W} = \bigl\{\, f = pi \bigm| p \in TF,\ i \in TC \,\bigr\}
\]

\[
  i : A \to B \qquad f : X \to Y
\]

\[
  \underline{\mathrm{Hom}}(B, X) \xrightarrow{\ \alpha(i,f)\ }
  \underline{\mathrm{Hom}}(i, f) \overset{\mathrm{def}}{=}
  \underline{\mathrm{Hom}}(B, Y)
  \times_{\underline{\mathrm{Hom}}(A, Y)}
  \underline{\mathrm{Hom}}(A, X)
\]

\[
  \bigl( \underline{\mathrm{Hom}}(T, f) : \underline{\mathrm{Hom}}(T, X)
  \to \underline{\mathrm{Hom}}(T, Y) \bigr) \in F
\]

\[
  \mathrm{Hom}(A \otimes T, X) \simeq
  \mathrm{Hom}(A, \underline{\mathrm{Hom}}(T, X))
\]

\[
  \alpha(i,f) :
  \underline{\mathrm{Hom}}(B, X) \to \underline{\mathrm{Hom}}(i, f)
  \quad \text{is in } \emph{TF}
\]

\[
  \mathrm{Hom}(B', \underline{\mathrm{Hom}}(B, X)) \longrightarrow
  \mathrm{Hom}(j, \alpha(i,f))
\]

\[
  \mathrm{Hom}(B' \otimes B, X) \longrightarrow
  \mathrm{Hom}(j \boxtimes i, f)
\]

\[
  j \boxtimes i : \ A' \otimes B \amalg_{A' \otimes A} B' \otimes A
  \longrightarrow B' \otimes B
\]

\[
  i \in TC,\ f \in F \Rightarrow \alpha(i,f) \in TF
  \quad\text{and}\quad
  i \in C,\ f \in TF \Rightarrow \alpha(i,f) \in TF .
\]

\[
  \alpha(i', \alpha(i,f)) \simeq \alpha(i' \boxtimes i, f)
\]

\[
  \begin{array}{rcl}
    C \otimes T \subset C & \Longleftrightarrow & TF \text{ stable by } \underline{\mathrm{Hom}}(T, ?) \\
    TC \otimes T \subset TC & \Longleftrightarrow & F \text{ stable by } \underline{\mathrm{Hom}}(T, ?) \\
    C \boxtimes C \subset C & \Longleftrightarrow & \bigl( i \in C,\ f \in TF \Rightarrow \alpha(i,f) \in TF \bigr) \\
    \ \boxtimes TC \subset TC & \Longleftrightarrow & \bigl( i \in TC,\ f \in F \Rightarrow \alpha(i,f) \in TF \bigr) \\
    TC \boxtimes C \subset TC & \Longleftrightarrow & \bigl( i \in C,\ f \in F \Rightarrow \alpha(i,f) \in F \bigr)
  \end{array}
\]

\[
  (B_0 \times e) \cup (A_0 \times I)
  \overset{i}{\subset} B_0 \times I ,
\]

\[
  i' \boxtimes \bigl( i = h(i_0, I, e) \bigr) = h(i' \boxtimes i_0, I, e),
\]

\[
  \boxed{TF \subset W}
\]

\[
  \text{①}\qquad \boxed{(L_A \to e) \in \mathrm{U}W}
\]

\[
  f \in TF \Longrightarrow f \text{ is a } I\text{-homotopism}.
\]

\[
  (TC)_0 \subset W .
\]

\[
  \left|
  \begin{array}{l}
    \text{if } S \overset{i}{\hookrightarrow} T \text{ mono, and} \\
    S \overset{g}{\hookrightarrow} S' \text{ mono} \in W \\
    \text{then in } \quad
    \begin{array}{ccc}
      S & \overset{i}{\hookrightarrow} & T \\
      {\scriptstyle g} \downarrow & & \downarrow {\scriptstyle f} \\
      S' & \overset{i'}{\hookrightarrow} & S' \amalg_S T = T'
    \end{array} \\
    f \in W
  \end{array}
  \right.
\]

\[
  A \times e \to A \times I \quad \text{is in } W \text{ for any } A,
\]

\[
  \text{②}\qquad
  \boxed{(I \to e) \in \mathrm{U}W. \text{ for any } I \text{ in the bunch}}
\]

\[
  f : A \overset{TC}{\hookrightarrow} B \qquad f \in TC
\]

\[
  f = pi : \ A \underset{TC}{\hookrightarrow} B' \underset{TF}{\longrightarrow} B ,
\]

\[
  X \overset{i}{\longrightarrow} X'
  \overset{p}{\longrightarrow} Y
  \qquad i \in TC,\ p \in F \ \ (\text{or } i \in C,\ p \in TF)
\]

\[
  \begin{array}{ll}
    F \cap W \subset TF & (\text{necessary for } W \subset \widetilde{W}, \text{ as } TF = F \cap \widetilde{W}) \\
    C \cap W \subset TC & (\text{id})
  \end{array}
\]

\[
  W \subset \widetilde{W} \quad (\text{or equivalently, }
  \widetilde{W} \cap W = W)
\]

\[
  \Phi^* = \{ u \in \mathrm{Fl}(A) \mid u \text{ has the LLP with respect to }
  \Phi \},
\]

\[
  \Phi_* = \{ u \in \mathrm{Fl}(A) \mid u \text{ has the RLP with respect to }
  \Phi \} .
\]

\[
  \forall f \in \mathrm{Fl}(A) \quad \exists \text{ factorization }
  f = pi, \text{ with } i \in \Phi,\ p \in \Psi .
\]

\[
  \begin{array}{ccc}
    C & TF & \\
    \cup & \cap & W \\
    TC & F &
  \end{array}
\]

\[
  a \to Y \times Y, \quad \alpha, \beta \in Y_a, \quad
  \underline{\mathrm{Hom}}_{\alpha,\beta}(I_a, Y_a), \quad
  \underline{\mathrm{Hom}}_{\alpha_b, \beta_b}(I_b, Y_b)
\]

\[
  \begin{array}{cc}
    C & F \cap W = TF \\
    \cup & \cap \\
    TC = C \cap W & F
  \end{array}
\]

\[
  \boxed{\widetilde{TF} \subset W,\ \widetilde{TC} \subset W}
  \quad \text{and this is equivalent with} \quad
  \boxed{\widetilde{TF} = TF,\ \widetilde{TC} = TC}
\]

\[
  \struck{\Sigma_1(f)}\; X_1(f) = \struck{\varinjlim}\;
  \coprod_{\substack{(X) \\ \text{all } D}} X_D
  \qquad \text{(amalgamated sum under } X)
\]

\[
  \begin{cases}
    \Sigma_{\alpha+1}(f) = \Sigma_1(\Sigma_\alpha(f)), \quad
    X_{\alpha+1}(f) = \text{source } \Sigma_{\alpha+1}(f), \\
    i_{\alpha+1}(f) = \text{composition }
    X \xrightarrow{i_\alpha(f)} X_\alpha(f)
    \xrightarrow{i_1(\Sigma_\alpha(f))} X_{\alpha+1}(f) \\[1ex]
    \Sigma_\alpha(f) = \varinjlim_{\alpha' < \alpha} \Sigma_{\alpha'}(f)
    \quad \text{if } \alpha \text{ is a limiting ordinal.}
  \end{cases}
\]

\[
  \struck{\ill{}}\ \mathrm{id}_{\underline{\mathrm{Fl}}(M)} \to \Sigma
\]

\[
  \mathrm{Hom}(B, X) \to \mathrm{Hom}(g_i, f) \quad \text{surjective}, \qquad
  \mathrm{Hom}(g, f) = \varprojlim \mathrm{Hom}(g_i, f)
\]

\[
  \left\{
  \begin{array}{l}
  X_{0,0} = U \cap U', \quad X_{0,1} = U \cap T', \quad X_{1,0} = T \cap U', \quad X_{1,1} = T \cap T' \\
  U = \varphi^{-1}(\{0\} \times \Delta^1), \quad U' = \varphi^{-1}(\Delta^1 \times \{0\})
  \end{array}
  \right.
\]

\[
  X_0 = U \cap U', \quad X_a = U \cap T', \quad X_b = U' \cap T .
\]

\[
  \left\{
  \begin{array}{l}
  \underline{U}_0 = \{0, a\}, \quad \underline{U}'_0 = \{0, b\} \\
  \underline{U}_0 \cup \underline{U}'_0 = \Psi, \quad \underline{U}_0 \cap \underline{U}'_0 = \{0\}
  \end{array}
  \right.
\]

\[
  \left\{
  \begin{array}{l}
  U = \varphi^{-1}(\underline{U}_0), \quad U' = \varphi^{-1}(\underline{U}'_0) \\
  U \cap U' = \varphi^{-1}(\underline{U}_0 \cap \underline{U}'_0) = \varphi^{-1}(\{0\}) = X_0
  \end{array}
  \right.
\]

\[
  U = X_{/a}, \quad U' = X_{/b}, \quad U \cap U' = X_{/0} .
\]

\[
  \Upsilon(X/\underline{\Psi}) = \overline{X} \text{ sur } \Psi
\]

\[
  U = \overline{X}_a, \quad U' = \overline{X}_b, \quad U \cap U' = \overline{X}_0 ,
\]

\[
  \overline{\varphi}_*(\xi_{\overline{X}})(s) =
  \begin{cases}
    H^{\bullet}_{\mathbb{D}}(U \cap U', \xi) & \text{si } s = 0 \\
    H^{\bullet}_{\mathbb{D}}(U, \xi) & \text{si } s = a \\
    H^{\bullet}_{\mathbb{D}}(U', \xi) & \text{si } s = b
  \end{cases}
\]

\[
  \gamma_0 \overset{\text{déf}}{=} \overline{\varphi}_*(\xi_{\overline{X}}) \in \mathbb{D}(\underline{\Psi}) .
\]

\[
  \underline{\Psi} \overset{k}{\subset} \Delta^1 \times \Delta^1 = \text{(le carré ci-dessus)}
\]

\[
  \gamma = \underline{k}_*(\gamma_0) \in \mathbb{D}(\Delta^1 \times \Delta^1)
\]

\[
  \underline{k}^*(\gamma) \simeq \gamma_0
\]

\[
  \psi = \underline{k} \circ \varphi : \overline{X} \longrightarrow \Delta^1 \times \Delta^1 = \underline{Q}
\]

\[
  \gamma = \underline{k}_*(\gamma_0) = \underline{k}_*(\varphi_*(\xi_{\overline{X}})) = \psi_*(\xi_{\overline{X}}) ,
\]

\[
  H^{\bullet}_{\mathbb{D}}(\Delta^1 \times \Delta^1, \gamma) \simeq H^{\bullet}_{\mathbb{D}}(\overline{X}, \xi) \xrightarrow[\;1\;]{\sim} H^{\bullet}_{\mathbb{D}}(X, \xi)
\]

\[
  \underbrace{H^{\bullet}_{\mathbb{D}}(\underline{Q}, \gamma)}_{H^{\bullet}_{\mathbb{D}}(\overline{X}, \xi)} \simeq \gamma(e) = \gamma(1,1)
\]

\[
  i_0^* \text{ iso} \Longleftrightarrow i^* \text{ iso} ,
\]

\[
  i_0 \in W_{\mathbb{D}} \Longrightarrow i \in W_{\mathbb{D}} .
\]

\[
  i \in W_{\mathbb{D}} \Longrightarrow i_0 \in W_{\mathbb{D}} ,
\]

\[
  \Psi_{\underline{Q}}(X) \mid \underline{\Psi} \simeq \Psi_{\underline{\Psi}}(X) \text{ déjà considéré.}
\]

\[
  \overline{X}_{(1,1)} = X_{/(1,1)} \simeq X
\]

\[
  \underbrace{\Bigl(\int \bigl(V \to U,\ V \to U'\bigr)\Bigr)}_{\overset{\text{déf}}{=}\ \Psi_{\underline{\Psi}}(X)} \longrightarrow X
\]

\[
  \gamma_0 \overset{\text{déf}}{=} \varphi_*(\xi) \in \mathbb{D}(\underline{\Psi}) , \qquad
  \gamma = \underline{k}_*(\gamma_0) \in \mathbb{D}(\underline{Q}) ,
\]

\[
  \begin{array}{l}
  \gamma(0,0) \simeq H^{\bullet}_{\mathbb{D}}(X_0, \xi) \\
  \gamma(0,1) \simeq H^{\bullet}_{\mathbb{D}}(X_a, \xi) \\
  \gamma(1,0) \simeq H^{\bullet}_{\mathbb{D}}(X_b, \xi) \\
  \gamma(1,1) \simeq H^{\bullet}_{\mathbb{D}}(\mathcal{X}, \xi)
  \end{array}
\]

\[
  \psi_*(\xi) \in \mathbb{D}(\underline{Q}) , \qquad \text{où } \psi = \bigl(\mathcal{X} \xrightarrow{\varphi} \underline{\Psi} \xrightarrow{\underline{k}} \underline{Q}\bigr), \quad \underline{Q} = \Delta^1 \times \Delta^1 ,
\]

\[
  \underline{\Psi}^{\mathrm{op}} = \underline{\Phi} = \bigl(a \rightarrow 1 \leftarrow b\bigr)
\]

\[
  \underline{\Phi} \subset \underline{Q}^{\circ} = \text{(le carré ci-dessus)}
\]

\[
  \left\{
  \begin{array}{l}
  i_1 \text{ iso} \Longrightarrow i_0 \text{ iso} , \\
  j_1 \text{ iso} \Longrightarrow j_0 \text{ iso} .
  \end{array}
  \right.
\]

\[
  \xi = \Bigl( \xi_a \xrightarrow{\;j_0\;} \xi_0 \xleftarrow{\;i_0\;} \xi_b \Bigr)
\]

\[
  \rho^* \rho_*(\xi) \longrightarrow \xi \quad (\text{adj.})
\]

\[
  \mathcal{A} = \mathbb{D}(e) \longrightarrow \mathbb{D}^{\mathrm{cc}}(\Delta^1)
\]

\[
  \pi_0(\xi_1) = \mathrm{Im}\, \pi_0(\xi_a) \cup \mathrm{Im}\, \pi_0(\xi_b)
\]

\[
  \begin{array}{l}
  i_a \in W \Longleftrightarrow i'_a \in W \\
  i_b \in W \Longleftrightarrow i'_b \in W
  \end{array}
\]

\[
  \left\{
  \begin{array}{l}
  \text{a) } f \in W \\
  \text{b) } f_0 \in W \\
  \text{c) } f_a \text{ et } f_b \in W
  \end{array}
  \right.
\]

\[
  i_a \in W \Longleftrightarrow i'_a \in W
\]