\[
F_x \longrightarrow F_y
\qquad\qquad
x \leq y
\]

\[
F_0 \overset{s_1}{\underset{t_1}{\leftleftarrows}} F_1
\overset{s_2}{\underset{t_2}{\leftleftarrows}} F_2
\]

\[
F_0 \overset{\sigma_1}{\longrightarrow} F_1 \overset{\sigma_2}{\longrightarrow} F_2
\]

\[
\mu_1^1 \qquad \mu_1^2 \qquad \mu_2^2
\]

\[
a \qquad \ell \qquad r
\]

\[
I_{A\cdot S} \longrightarrow S
\]

\[
\Delta_{n_1} \times \Delta
\qquad\qquad
\Delta_{n_1} \times \Delta_{n_2} \times \cdots \times \Delta_{n_n}
\qquad\qquad
\Delta_n \times \Delta_{n'}
\]

\begin{align*}
F_0 &= \mathrm{Ob}\,\Delta_{n'} \\
F_1 &= \struck{\ill{}}\ \mathrm{Fl}(\Delta_n) \times \mathrm{Ob}\,\Delta_{n'} \\
F_2 &= \mathrm{Fl}(\Delta_n) \times \mathrm{Fl}(\Delta_{n'})
\end{align*}

\[
\begin{array}{l}
\mathrm{Ob}\,C_1 \\
(\mathrm{Fl}\,C_1 \times \mathrm{Ob}\,C_2) \\
\mathrm{Fl}\,C_2 \times \mathrm{Fl}(C_2) \times \mathrm{Ob}\,C_3
\end{array}
\qquad\longrightarrow\qquad \mathrm{Fl}_0(C_1)
\]

\[
F_3 = \mathrm{Fl}\,C_1 \times \mathrm{Fl}\,C_2 \times \mathrm{Fl}\,C_3 \times \mathrm{Ob}\,C_4
\]

\[
F_n = \mathrm{Fl}\,C_1 \times \mathrm{Fl}\,C_2 \times \cdots \times \mathrm{Fl}\,C_n \struck{\times \mathrm{Ob}\,C_4}
\]

\[
O_1 \subset F_1, \quad \struck{B_2}\ O_2 \subset F_2, \quad \cdots \quad O_n \subset F_n
\qquad\qquad \dot{F} \subset F
\]

\[
F_1 \times \cdots \times F_n \times (?)
\qquad\qquad
(F \times I) \sqcup_{\uncertain{\dot{B}} \times I} \dot{F}
\]

\[
(O_1, F_1) * (O_2, F_2) \cdots * (O_3, F_{n+1})
\]

\[
\mathbb{Z}[F_{n-1}] \longleftarrow \mathbb{Z}[F_{n-1}] \overset{t_n - s_n}{\longleftarrow}
\mathbb{Z}[F_n] \longleftarrow \mathbb{Z}[F_{n+1}]
\]

\[
D_n \longrightarrow X \qquad\qquad D_n^{\ill{}}
\qquad\qquad \xi \in F_n
\]

\[
t_{n-1}\bigl( (t_n y) - s_n(y) \bigr) - s_{n-1}\bigl( t_n y - s_n y \bigr)
\qquad (= 0)
\]

\[
\widetilde{D} \qquad\qquad \mathbb{N} \ni n \longmapsto \widetilde{D}_n
\qquad\qquad \widetilde{D}_{n'} \rightrightarrows \widetilde{D}_n
\]

\[
d_n \in \widetilde{D}_n \longrightarrow \struck{\mathbb{N}}\ I
\qquad\qquad d_n \longmapsto n
\]

\[
\widetilde{D}_n \longrightarrow n \in I
\]

\[
1 + n\cdot 2^1 + \frac{n(n-1)}{2}\,2^2 + \cdots + 2^n = (1+2)^n
\]

\[
F_0 \leftleftarrows F_1 \leftleftarrows F_2
\]

\[
\widetilde{D}_{n'} \longrightarrow \widetilde{D}_n
\qquad\qquad
\begin{array}{c} x'' \leq x' \\ y'' \leq y' \end{array}
\ \in \widetilde{D}_{\ill{}}
\]

\[
(*) \qquad \mathrm{Hom}_M(D', D) \longrightarrow \widetilde{D},
\qquad u \longmapsto u(D')
\]

\[
I_n = \widetilde{D}_n = \text{ens.\ des sous-objets de } D_n
\]

\[
I_n \overset{\tau_n}{\longrightarrow} \mathbb{N}
\qquad\qquad (\tau_n, \text{ application « type »})
\]

\[
\tau_n^{-1}(n) = \{d_n\}
\]

\[
\mathrm{Hom}(D_{n'}, D_n) \hookrightarrow \widetilde{D}_n = I_n,
\qquad u \longmapsto u(d_{n'})
\]

\[
\tau_n^{-1}(n') \overset{\text{déf}}{=} I_{n', \struck{n}}
\overset{\varphi_{n,n'}}{\longrightarrow} \mathrm{Hom}(I_{n'}, I_n),
\qquad \xi \longmapsto u_\xi^n
\]

\[
u_\xi(d_{n'}) = \xi.
\]

\[
u_{d_n} = \mathrm{id}_{I_n}
\]

\[
u_\xi^n\, u_y^{n'} = u_\zeta^n
\qquad \text{où } \zeta = u_\xi^n(y) = (u_\xi^n\, u_y^{n'})(d_{n''})
\]

\[
\mathrm{Ob}\,M_0 = \mathbb{N},
\qquad
\mathrm{Hom}_{M_0}(n', n) =
\begin{cases}
\emptyset & \text{si } n' \nleq n \\
\text{ens.\ des homs.\ admissibles } I_{n'} \to I_n & \text{si } n' \leq n
\end{cases}
\]

\[
M \longrightarrow M_0
\]

\[
\varphi_{n,n'} : \tau_n^{-1}(n') \longrightarrow \mathrm{Hom}(I_{n'}, I_n),
\qquad \xi \longmapsto \bigl( u_\xi^n : I_{n'} \to I_n \bigr)
\]

\[
u_\xi^n : I_{n'} \overset{\sim}{\longrightarrow} (I_n)_{\leq \xi}
\qquad (\text{d'où } u_\xi^n(d_{n'}) = \xi)
\]

\[
u_\xi^n \circ u_y^{n'} = u_\zeta^n
\quad \text{où } \zeta = u_\xi^n(y),
\quad \xi \in I_n,\ y \in I_{n'},\ n' = \tau_n(\xi),\ n'' = \tau_{n'}(y).
\]

\[
I_n = (\Delta_n \times \{\pm 1\}) \sqcup \{d_n\},
\]

\[
(P_n)_{\leq \xi} \simeq P_{n'}, \qquad \text{pour } n' \text{ \uncertain{convenable} } \leq n.
\]

\[
K \overset{\tau_K}{\longrightarrow} \struck{\mathbb{N}}\ \mathbf{N}
\qquad (\text{application « type »})
\]

\[
\struck{\varphi_\xi}\ I_n \overset{u_\xi^K}{\underset{\sim}{\longrightarrow}} K_{\leq \xi}
\qquad (\text{application d'épinglage de } K_{\leq \xi})
\]

\[
u_\xi^K(d_n) = \xi \quad (= \text{plus grand élément de } K_{\leq \xi})
\]

\[
(1) \qquad \tau_K \circ u_\xi^K = \tau_n
\]

\[
(2) \qquad \forall\, \xi \in K,\ y \in I_n \ (n = \tau_K(\xi)),
\quad \text{on a, si } n' = \tau_n(y),
\]

\[
\struck{H}\ u_\xi^K \circ u_y^n = u_\zeta^K,
\qquad \text{\add{où} } \zeta = u_\xi^K(y),
\]

\[
K \overset{f}{\longrightarrow} L
\]

\[
K_{\leq \xi} \overset{\sim}{\longrightarrow} L_{\leq y}
\]

\[
u_\alpha^K : I_{\tau_n(\alpha)} \overset{\sim}{\longrightarrow} K_{\leq \alpha}
\qquad (\alpha \in \mathrm{Max}\,K)
\]

\[
I_n \overset{\sim}{\longrightarrow} K_{\leq \xi}
\]