\[
  \mathrm{Hom}(G, F^{I}) = \mathrm{Hom}\bigl(I, \mathrm{Hom}(G,F)\bigr)
\]

\[
  \mathrm{P}(b,E) \times \mathrm{P}(b',E')
  = \coprod_{(z,z') \in E \times E'} \widetilde{z} \times \widetilde{z}'
\]

\[
  u(F) \subset F' \quad \Longleftrightarrow \quad F \subset u^{-1}(F')
\]

\[
  \coprod_{z \in E} z \ \times \coprod_{z' \in E'} z'
  = \coprod_{(z,z') \in E \times E'} z \times z'
\]

\[
  \mathrm{Hom}_{C}\bigl((b,E),(b',E')\bigr) \hookrightarrow \mathrm{Hom}_{B}(b,b')
\]

\[
  \{\, u : b \to b' \ \mid \ \widetilde{\pi}(u)(E) \subset E' \,\}
\]

\[
  \widetilde{\pi}\backslash B \ \simeq \ B/\widetilde{\pi} \ = \ C,
  \qquad \pi\backslash B \ = \ Z
\]

\[
  C \longrightarrow Z^{\wedge}, \qquad (b,E) \longmapsto \coprod_{z \in E} z
\]

\[
  u^{E,E'}_{*} : \widetilde{E} \longrightarrow \widetilde{E}',
  \qquad \widetilde{E} = \coprod_{z \in E} z, \qquad z \longmapsto \widetilde{z}'
\]

\[
  (b,E) \times (b',E') = \bigl(b \times b',\ p_{1}^{*}(E) \cap p_{2}^{*}(E')\bigr)
\]

\[
  F \in \mathrm{Ob}\,X^{\wedge}/U \simeq (X/U)^{\wedge} = Y^{\wedge},
\]

\[
  \widetilde{\Delta}_{n} = [0,n] \cap \mathbf{Z}
\]

\[
  \pi(I) = \pi_{0}(F^{I}), \qquad \text{i.e.} \qquad
  \pi(\widetilde{\Delta}_{n}) = \pi_{0}(F^{\,n+1}).
\]

\[
  \varphi : (\widetilde{\Delta}_{/\pi})^{\circ} \longrightarrow Y^{\wedge}
  \simeq (X/U)^{\wedge} \simeq X^{\wedge}/U .
\]

\[
  \varphi(\widetilde{\Delta}_{n}, \alpha) = \alpha \subset F^{\,n+1}
\]

\[
  \widetilde{\Delta}^{\wedge} \longrightarrow (X/U)^{\wedge} \simeq X^{\wedge}/U
\]

\[
  f^{*}(G^{\Psi}) = f^{*}(G)^{\,f^{*}(\Psi)}
\]

\[
  \varphi(\rho) : \beta \longrightarrow \alpha,
  \qquad \beta = \varphi(\widetilde{\Delta}_{m}, \beta),
  \qquad \alpha = \varphi(\widetilde{\Delta}_{n}, \alpha).
\]

\[
  \Phi : \bigl((\widetilde{\Delta}_{/\pi})^{\circ}\bigr)^{\wedge}
  \longrightarrow Y^{\wedge},
  \qquad
  \bigl((\widetilde{\Delta}_{/\pi})^{\circ}\bigr)^{\wedge}
  = (\widetilde{\Delta}_{/\pi})^{\wedge} = Z^{\wedge},
\]

\[
  f : Y^{\wedge} \longrightarrow Z^{\wedge}
  \qquad \bigl(\text{où } Z \overset{\text{déf}}{=} (\widetilde{\Delta}_{/\pi})^{\circ}\bigr).
\]

\[
  Z = (A_{/\pi})^{\circ} \simeq \pi\backslash A^{\circ} = \pi\backslash B,
  \qquad \pi : A^{\circ} \longrightarrow (\mathrm{Ens}), \quad B = A^{\circ}.
\]

\[
  \widetilde{E} = \coprod_{a \in E} a
  \qquad \text{(somme prise dans } Z^{\wedge}\text{)}.
\]

\[
  u_{*} = \pi(u) : Z_{b} \longrightarrow Z_{b'},
  \qquad Z_{b} = \pi(b), \quad Z_{b'} = \pi(b'),
\]

\[
  u^{E,E'}_{*} : \widetilde{E} \longrightarrow \widetilde{E}' .
\]

\[
  \widetilde{\pi}(u)(E) \subset E' \qquad \text{(et non } \widetilde{\pi}(u)(E) = E').
\]

\[
  \widetilde{\pi}\backslash B \longrightarrow (\pi\backslash B)^{\wedge} = Z^{\wedge},
  \qquad
  (b,E) \longmapsto \widetilde{E} = \coprod_{a \in E} a \ \subset
  \coprod_{a \in Z_{b}} a,
\]

\[
  \widetilde{\pi}\backslash B \ \simeq \ \pi\backslash B \ \simeq \ B,
\]

\[
  u_{*} = \widetilde{\pi}(u) : \widetilde{\pi}(b) \longrightarrow
  \widetilde{\pi}(b'),
  \qquad
  \mathfrak{P}^{*}(\pi(b)) \longrightarrow \mathfrak{P}^{*}(\pi(b')),
  \qquad E \longmapsto \pi(u)(E),
\]

\[
  \Delta(E) = \text{groupe engendré par les } [x,y], \quad (x,y) \in E^{2},
\]

\[
  [x,x] = 1, \qquad [x,y][y,x] = 1, \qquad [x,y][y,z] = [x,z].
\]

\[
  g_{i} = [i, i+1] \qquad \text{pour } i \in [1, n-1].
\]

\[
  \begin{cases}
    [i,j] = g_{i}\,g_{i+1} \cdots g_{j-1} \\
    [j,i] = [i,j]^{-1} = g_{j-1}^{-1} \cdots g_{i+1}^{-1} g_{i}^{-1}
  \end{cases}
\]

\[
  [\alpha v, \alpha u] = [\beta v, \beta u],
  \qquad \text{i.e.} \qquad [\alpha v, \alpha u]\,[\beta v, \beta u]^{-1} = 1 .
\]

\[
  H_{m,n} \times H_{n,p} \longrightarrow H_{m,p}
\]

\[
  fu = gv, \qquad gv' = gw, \qquad fuv_{1} = gwv_{1} = gv'v'_{1} = gw\,v'_{1}
\]

\[
  H_{x} = x/R_{x}
\]

\[
  M = \mathrm{End}_{X}(e) = \mathrm{End}_{X_{0}}(e)
\]

\[
  uu' = vv',
\]

\[
  [v,u] \overset{\text{déf}}{=} v'u'^{-1} \in G = \pi_{1}(X_{0}, e).
\]

\[
  [v,u] = v_{1}^{-1} u_{1} .
\]

\[
  G = \pi_{1}(X_{0}, e) \longrightarrow \pi_{1}(X, e)
\]

\[
  [w, gv]\,[v, fu]\,[w, gfu]^{-1} .
\]

\[
  \pi_{1}(X_{0}, e) \ \xrightarrow{\ \sim\ }\ G
  \ \xrightarrow{\ \sim\ }\ \pi_{1}(X,e) \qquad \text{iso !}
\]

\[
  u \sim_{R} v \quad \Longleftrightarrow \quad \exists\, p, q \in M
\]

\[
  Y = B_{M}, \qquad \varphi \in M, \qquad \varphi f u = \varphi g v ,
\]

\[
  X = Y/F .
\]

\[
  \mathrm{Hom}(x,y) \simeq \{\, u \in M \mid x = yu \,\},
\]

\[
  x = \dot{u} = au .
\]

\[
  M_0 = \mathrm{End}_X(e) = \{\, f \in M \mid ef = e,\ \text{i.e. } a\psi f = a\psi,
  \ \text{i.e. } \psi f \overset{R}{\sim} \psi \,\}.
\]

\[
  f \in M_0 \iff \forall\, p,q \in \mathbb{N} \quad \varphi^{p}\psi f = \varphi^{q}\psi .
\]

\[
  \delta(f) \overset{\text{déf}}{=} q - p
\]

\[
  \begin{cases}
    \varphi^{p}\psi f = \varphi^{q}\psi \\
    \varphi^{p'}\psi f = \varphi^{q'}\psi
  \end{cases}
\]

\[
  \varphi^{p+p'}\psi f =
  \begin{cases}
    \varphi^{p'+q}\psi \\
    \varphi^{p+q'}\psi
  \end{cases}
\]

\[
  \varphi^{p'+q}\psi = \varphi^{p+q'}\psi .
\]

\[
  M_0 \xrightarrow{\ \delta\ } \mathbb{Z} ,
\]

\[
  \begin{cases}
    \varphi^{p}\psi f = \varphi^{q}\psi \\
    \varphi^{p'}\psi g = \varphi^{q'}\psi
  \end{cases}
\]

\[
  \varphi^{p+p'}\psi fg = \varphi^{q+p'}\psi g
  = \varphi^{q}\bigl(\underbrace{\varphi^{p'}\psi g}_{\varphi^{q'}\psi}\bigr)
  = \varphi^{q+q'}\psi
\]

\[
  M_0 \xrightarrow{\ \mathrm{can}\ } G_0 \xrightarrow{\ \delta_0\ } \mathbb{Z} .
\]

\[
  \begin{cases}
    z = a\rho \\
    y = a\rho g \\
    x = a\rho g f
  \end{cases}
  \qquad\text{donc}\qquad
  \begin{cases}
    e = xu = a\rho g f u \\
    e = yv = a\rho g v \\
    e = zw = a\rho w
  \end{cases}
\]

\[
  (\ast\ast) \qquad
  \boxed{\ \rho g f u \overset{R}{\sim} \rho g v \overset{R}{\sim} \rho w \overset{R}{\sim} \psi\ }
\]

\[
  \boxed{\ \varphi^{p}(\rho g f u) = \varphi^{q}(\rho g v) = \varphi^{r}(\rho w) = \varphi^{s}\psi\ }
\]

\[
  \boxed{\ [w,\, gv]\,[v,\, fu]\,[w,\, gfu]^{-1} = 1\ } ,
\]

\[
  \boxed{\ \sigma\lambda \overset{R}{\sim} \sigma\mu \overset{R}{\sim} \psi\ } ,
\]

\[
  \eta =
  \begin{cases}
    e\lambda' = a\sigma\lambda\lambda' = a\sigma\pi \\
    e\mu' = a\sigma\mu\mu' = a\sigma\pi
  \end{cases}
\]

\[
  \boxed{\ \sigma\pi\nu \overset{R}{\sim} \psi\ }
  \qquad\text{où } \pi = \lambda\lambda' = \mu\mu' .
\]

\[
  \begin{cases}
    \lambda'\nu,\ \mu'\nu \in M_0 \\
    \lambda(\lambda'\nu) = \mu(\mu'\nu)
  \end{cases}
\]

\[
  [\mu,\lambda] = (\mu'\nu)(\lambda'\nu)^{-1} .
\]

\[
  [\mu,\lambda] = \bigl(\mu'\,\theta(\sigma\pi)\bigr)\bigl(\lambda'\,\theta(\sigma\pi)\bigr)^{-1}
\]

\[
  u\,\theta(u) = \psi \qquad \text{(plus seulement } \overset{R}{\sim} \psi\text{)} ;
\]

\[
  [\mu,\lambda] = \bigl(\underbrace{\theta(\mu)\,\theta(\sigma\psi)}_{\in M_0}\bigr)
  \bigl(\underbrace{\theta(\lambda)\,\theta(\sigma\psi)}_{\in M_0}\bigr)^{-1}
\]

\[
  \begin{array}{l}
    [w,\, gv] = \bigl(\theta(w)\,\theta(\rho\psi)\bigr)\bigl(\theta(gv)\,\theta(\rho\psi)\bigr)^{-1} \\
    {}[v,\, fu] = \bigl(\theta(v)\,\theta(\rho g\psi)\bigr)\bigl(\theta(fu)\,\theta(\rho g\psi)\bigr)^{-1} \\
    {}[w,\, gfu] = \bigl(\theta(w)\,\theta(\rho\psi)\bigr)\bigl(\theta(gfu)\,\theta(\rho\psi)\bigr)^{-1}
  \end{array}
\]

\[
  \boxed{\ \bigl(\theta(gv)\,\theta(\rho\psi)\bigr)^{-1}
  \bigl(\theta(v)\,\theta(\rho g\psi)\bigr)
  \bigl(\theta(fu)\,\theta(\rho g\psi)\bigr)^{-1}
  \bigl(\theta(gfu)^{-1}\,\theta(\rho\psi)\bigr) = 1\ }
\]

\[
  \rho g f u \overset{R}{\sim} \rho g v \overset{R}{\sim} \rho w \overset{R}{\sim} \psi
\]

\[
  \rho g f u \overset{R}{\sim} \rho g v \overset{R}{\sim} \rho w \overset{R}{\sim} 1
\]

\[
  \boxed{\ \rho g f u,\ \rho g v,\ \rho w \in M_0\ }
\]

\[
  \underbrace{\rho g f u}_{\overset{\text{déf}}{=}\ \pi},\qquad
  \underbrace{\rho g v}_{\overset{\text{déf}}{=}\ \pi'} \ \in M_0 ,
\]

\[
  \pi\bigl(\underbrace{\theta(gv)\,\theta(\rho\psi)}_{c_1}\bigr)
  = \pi\bigl(\underbrace{\theta(v)\,\theta(\rho g\psi)}_{c_2}\bigr) = \psi
\]

\[
  \pi'\bigl(\underbrace{\theta(fu)\,\theta(\rho g\psi)}_{c_3}\bigr)
  = \pi'\bigl(\underbrace{\theta(gfu)\,\theta(\rho\psi)}_{c_4}\bigr) = \psi
\]

\[
  \begin{array}{l}
    \delta(c_1) = \delta(c_2) = \delta(\psi) - \delta(\pi) \\
    \delta(c_3) = \delta(c_4) = \delta(\psi) - \delta(\pi')
  \end{array}
\]

\[
  \delta_0\bigl(c_1 c_2^{-1} c_3 c_4^{-1}\bigr)
  = \underbrace{\bigl(\delta(c_1) - \delta(c_2)\bigr)}_{0}
  + \underbrace{\bigl(\delta(c_3) - \delta(c_4)\bigr)}_{0} = 0
\]

\[
  \delta'_0 : \pi_1(X,e) \longrightarrow \mathbb{Z} .
\]

\[
  \bigl(u \overset{R_\varphi}{\sim} v \iff \exists\, p,q \in \mathbb{N}
  \ \ill{}\ \varphi^{p}u = \varphi^{q}v\bigr),
\]

\[
  \pi_1(X,e) \longrightarrow \mathbb{Z} ,
\]

\[
  uv = 1 ;
\]

\[
  p \overset{\text{déf}}{=} vu = 1 ,
\]

\[
  \bigl[\ \bar{u}\bar{f} = \bar{v}\bar{f} \Longrightarrow \bar{u} = \bar{v}\ \bigr],
\]

\[
  R_0 : \qquad u \overset{R_0}{\sim} v \iff u\psi = v\psi
\]

\[
  \begin{array}{rcl}
    M &\longrightarrow& \mathrm{Ep}(M_0) \\
    u &\longmapsto& u_{M_0} = \bigl(\lambda \mapsto u\lambda : M_0 \to M_0\bigr)
  \end{array}
\]

\[
  u \overset{R_\varphi}{\sim} v \iff \exists\, p,q \in \mathbb{N}
  \ \text{avec}\ \varphi^{p}u = \varphi^{q}v ,
\]

\[
  M_0 = \{\, f \in M \mid \psi f \overset{R_\varphi}{\sim} \psi \,\} ,
\]

\[
  \pi_1(M_0) = G_0 \xrightarrow{\ \delta_0\ } \mathbb{Z}
\]

\[
  \delta(f) = q - p \qquad \text{si } \varphi^{p}\psi f = \varphi^{q}\psi .
\]

\[
  \mathbb{N} \times e \longrightarrow e \times e ,\qquad
  (n,u) \longmapsto (\varphi^{n}u,\, u)
\]

\[
  \pi_1(X_0, e) \;\simeq\; \pi_1(M) \twoheadrightarrow \Gamma .
\]

\[
  (*) \qquad u p \;\struck{= up}\; = p , \qquad
  \text{(en particulier } p^2 = p\text{)},
\]

\[
  p \;=\; u_i v_i .
\]

\[
  (**) \qquad p \;=\; u\, v(u) ,
\]

\[
  \mathrm{Im}\, p \;\subset\; \mathrm{Im}\, u \qquad \forall u \in M,
\]

\[
  u(E_0) \;=\; u\, p(E) \;=\; \mathrm{Im}(up) \;\supset\; E_0 ,
\]

\[
  u \longmapsto u_{E_0} \;:\; M \longrightarrow \mathrm{Aut}(E_0)
\]

\[
  E_0 \;=\; \text{ens. des pts fixes de } M \text{ opérant dans } E,
\]

\[
  u\, p(x) \;=\; p(x) \quad \forall x \in E, \qquad \text{i.e.}
\]

\[
  up \;=\; p \qquad \forall u \in M,
\]

\[
  p \;:\; e \longrightarrow a .
\]

\[
  e \xrightarrow{\;p_b\;} b .
\]

\[
  u \;:\; b \longrightarrow a
\]

\[
  uq = p, \qquad \struck{\ill{}} \text{ avec } q \text{ comme } p,
  \qquad \text{donc } uq = q
\]

\[
  \underbrace{u\,q}_{q} \;=\; \underbrace{v\,q'}_{q'} \;=\; p
\]

\[
  up = q, \qquad p = v\,\overline{v}, \qquad p = q\,q', \qquad
  q = v\,\struck{\ill{}}
\]

\[
  H_y \;=\; f \cdot H_x .
\]

\[
  \lambda \longmapsto f\lambda \;:\; H_x \xrightarrow{\ \sim\ } H_y
\]

\[
  u \longmapsto fu \;:\; H_e \longrightarrow H_x
\]

\[
  fu \;=\; gu \qquad \text{pour } u \in H_e, \quad
  f, g : e \rightrightarrows x .
\]

\[
  M \;=\; \mathrm{Ep}(E)
\]

\[
  \struck{p}\, w \;=\; u\, w_u
\]

\[
  w \;=\; u\, w_u \qquad \text{avec } w, w_u \text{ minimaux}
\]

\[
  M \;=\; \mathrm{Mon}(E)
\]

\[
  w \;=\; w\, u , \qquad
  \mathrm{card}\bigl(E - w(E)\bigr) \;=\; \pi
\]

\[
  R(a,b) \;:\; X_{/\!/a} \cap X_{/\!/b} \neq \emptyset
\]

\[
  \mathrm{Ob}\, X \longrightarrow \pi_0(X) .
\]

\[
  \varphi u = \varphi v \quad \text{avec } u, v, \varphi \ \uncertain{surj.}
  \quad \Longrightarrow \quad u = v
\]

\[
  \varphi u = \varphi \quad \Longrightarrow \quad u = \mathrm{id}\ ?
  \qquad \uncertain{\mathrm{Mon}}
\]

\[
  M \;=\; \mathrm{Ep}(E)
\]

\[
  u f = v f \quad \Longrightarrow \quad u = v .
\]

\[
  f u = f v , \qquad u \neq v
\]

\[
  \mathrm{Hom}(e,x) \;\simeq\; \pi_0\bigl(e \backslash \Psi / x\bigr)
  \;\simeq\; \pi_0\bigl(S/x\bigr)
\]

\[
  \varinjlim_{x \in X} \pi_0\bigl(S/x\bigr)
\]

\[
  \pi_{0}(S_{/x}) \rightrightarrows \pi_{0}(S_{/a})
\]

\[
  \pi_{0}(E_{/x}) \simeq
\]

\[
  B\times C = \emptyset \quad\text{dans } X^{\wedge},
\]

\[
  \beta\times\gamma = \emptyset \quad\text{dans } X^{\vee},
\]

\[
  \mathrm{Hom}(a,x) = \emptyset
\]

\[
  (u,\beta) \in \mathrm{Hom}(a,x)\times E
\]

\[
  \pi_{0}(S_{/x}) \simeq I(x) \amalg \bigl(J(x)\times E\bigr)
\]

\[
  I(x) = \mathrm{Im}\bigl(\mathrm{Hom}(\beta,x)\amalg\mathrm{Hom}(\gamma,x)
  \longrightarrow \mathrm{Hom}(a,x)\bigr)
\]

\[
  J(x) = \mathrm{Hom}(a,x)\setminus I(x)
\]

\[
  \pi_{0}(S_{/x}) = \emptyset \quad\text{si } \mathrm{Hom}(a,x) = \emptyset
\]

\[
  X' \sim \Sigma^{1} Z
\]

\[
  \beta/\!/Z \simeq {}_{a}\backslash X
\]

\[
  T = b_{0}/\!/X_{0} \ \cup\ c_{0}/\!/X_{0}
\]

\[
  X_{0} \simeq \beta/\!/\tilde X \simeq \gamma/\!/\tilde X
\]

\[
  \tilde X \sim \Sigma^{1}T \qquad X' \sim \Sigma^{2}T \qquad
  X' \simeq \Sigma^{1}\tilde X
\]

\[
  \Sigma^{1}X
\]

\[
  X\times Y \qquad X \longrightarrow X\times I \qquad x_{0}\times I
\]

\[
  \Longrightarrow\ \exists\, X \longrightarrow X' \text{ cofinal},
\]

\[
  H^{2}(X') \simeq H^{2}(S^{2}) ;
\]

\[
  f^{*} : H^{2}(T,\underline{k}) \xrightarrow{\ \sim\ } H^{2}(S,\underline{k})
\]

\[
  \xi \qquad \xi' \qquad H^{p}(S',\xi') \longrightarrow H^{p}(S,\xi)
\]

\[
  S\times T \longleftarrow S'\times T
\]

\[
  X_{/e} \simeq X_{/F}
\]

\[
  X \longrightarrow Y \quad \text{cocofinal},
\]

\[
  \tilde X \sim \Sigma(T), \quad\text{si } \beta/\!/\tilde X,\ \gamma/\!/\tilde X
  \ \text{asph. (à vérifier)}
\]

\[
  X' \sim \Sigma(\tilde X)
\]

\[
  (vu_{1},\alpha) \sim (vu_{2},\beta)
\]

\[
  (vu_{1},\alpha) \sim (vu_{2},\beta)
\]

\[
  (v,\alpha) \sim (w,\beta)
\]

\[
  S = \bigl(\, a \to \xi \to b \ ;\ a \to \xi' \to b \,\bigr)
\]

\[
  B = (u_{1},\alpha) \sim (u_{2},\beta)
\]

\[
  C = (u_{2},\alpha) \sim (u_{1},\beta)
\]

\[
  B \longrightarrow \xi
\]

\[
  \xi = (vu_{1},\alpha) = (vu_{2},\beta)
\]

\[
  \xi = (vu_{2},\alpha) = (vu_{1},\beta)
\]

\[
  \xi = (v,\alpha) = (w,\beta)
\]

\[
  \mathrm{Hom}_{X'\supset X}(x,e) \simeq \pi_{0}\bigl({}_{x}\backslash S_{/e}\bigr)
  \simeq \pi_{0}\bigl({}_{x}\backslash S\bigr)
\]

\[
  \simeq \varinjlim \Bigl(\mathrm{Hom}(x,a) \to \mathrm{Hom}(x,\xi)
  \leftarrow \mathrm{Hom}(x,b) \to \mathrm{Hom}(x,\xi') \leftarrow \Bigr)
\]

\[
  \simeq \underbrace{\bigl(\mathrm{Hom}(x,\xi)\amalg\mathrm{Hom}(x,\xi')\bigr)}_{E(x)}
  \big/ R_{(x)},
\]

\[
  (*) \quad
  \begin{cases}
    \lambda v \sim \lambda' v & v : x \to a \\
    \mu w \sim \mu' w & w : x \to b
  \end{cases}
\]

\[
  \mathrm{Hom}(b,\xi) \rightrightarrows \mathrm{Hom}(a,\cdot)
\]

\[
  \tilde X = \tilde X_{/\!/\tilde\xi} \ \cup\ \tilde X_{/\!/\tilde\xi'}
\]

\[
  \tilde X_{/\!/\tilde\xi} \ \cap\ \tilde X_{/\!/\tilde\xi'}
  = \tilde X_{/\!/\tilde a} \ \cup\ \tilde X_{/\!/\tilde b}
\]

\[
  \mathrm{Hom}(e,x) \simeq \pi_{0}\bigl({}_{e}\backslash D_{/x}\bigr)
  \simeq \pi_{0}\bigl(D_{/x}\bigr)
  \simeq \varinjlim \bigl(\mathrm{Hom}(b,x) \rightrightarrows \mathrm{Hom}(a,x)\bigr)
\]

\[
  u \in \mathrm{Hom}(x,\xi) \subset \mathrm{Hom}(x,\xi)\amalg\mathrm{Hom}(x,\xi')
  \overset{\text{déf}}{=} E(x)
\]

\[
  u' \in \mathrm{Hom}(x,\xi') \subset E(x)
\]

\[
  f_{0} = u,\ f_{1},\ f_{2},\ \ldots\ f_{n} = u'
\]

\[
  X' \simeq \Sigma^{1}_{W}(X) \simeq \Sigma^{2}_{W}(\tilde X)
  \simeq \Sigma^{3}(T)
\]

\[
  T \neq \emptyset .
\]

\[
  \underbrace{\lambda v,\; \mu w}_{\mathrm{Hom}(m,\xi)}
  \qquad
  \underbrace{\lambda' v,\; \mu' w}_{\mathrm{Hom}(a,\xi')}
\]

\[
  v^{*}(\tilde a) = w^{*}(\tilde b)
\]

\[
  \boxed{\;\lambda v \sim \mu w\;}
\]

\[
  \widetilde{X}_{/\!/(\tilde a,\tilde b)} \neq \emptyset
\]