Cote n° 151 · pages 2–74 · 51 commutative diagrams · Topos stratifié (1981) : notes manuscrites (s.d.).
Inventory dating : [à partir de 1981-à partir de 1990]
Édition de démonstration

batch 1 · p. 2 — read it beside the facsimile1 / 51
LaTeX source
\begin{tikzcd}[column sep=large]
E_0 \arrow[r, "\delta_0", bend left=18] & E_1 \arrow[l, "p_0"', bend left=18] \arrow[r, "\delta_1", bend left=18] & E_2 \arrow[l, "p_{12}"', bend left=18]
\end{tikzcd}
batch 1 · p. 4 — read it beside the facsimile2 / 51
LaTeX source
\begin{tikzcd}[column sep=large]
E_0 \arrow[r, "\delta_0", bend left=18] & E_1 \arrow[l, "p_0", bend left=18] \arrow[r, "\delta_1"', bend right=18] & E_2 \arrow[l, "p_{12}"', bend right=18]
\end{tikzcd}
batch 1 · p. 6 — read it beside the facsimile3 / 51
LaTeX source
\begin{tikzcd}[column sep=large]
E_1 \arrow[r, "\delta_1", bend left=18] & E_2 \arrow[l, "p_{12}"', bend left=18]
\end{tikzcd}
batch 1 · p. 8 — read it beside the facsimile4 / 51
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
F \times F \times F \arrow[d, "p_{12}", bend left=18] \arrow[d, "p_{13}"', bend right=18] \\
F \times F
\end{tikzcd}
batch 1 · p. 11 — read it beside the facsimile5 / 51
LaTeX source
\begin{tikzcd}[column sep=large, nodes={font=\scriptsize}]
U_{\gamma'} \arrow[r, "\varphi(v)"] & U_\gamma \times U_{\alpha_{n+1}} \arrow[r, "\varphi(u)\times\mathrm{id}"] & (U_{\alpha_0} \times \dots \times U_{\alpha_n}) \times U_{\alpha_{n+1}}
\end{tikzcd}
batch 1 · p. 12 — read it beside the facsimile6 / 51
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
& \gamma \arrow[dl] \arrow[dr] & \\
\alpha_1 \arrow[dr] & & \alpha_2 \arrow[dl] \\
& \alpha_0 &
\end{tikzcd}
batch 1 · p. 16 — read it beside the facsimile7 / 51
LaTeX source
\begin{tikzcd}[column sep=large]
E_n \arrow[r, "\partial_n", bend left=18] & E_{n+1} \arrow[l, "k_n", bend left=18]
\end{tikzcd}
batch 1 · p. 16 — read it beside the facsimile8 / 51
LaTeX source
\begin{tikzcd}[column sep=large]
E^*_n \arrow[r, "d_n = k^*_n"', bend right=18] & E^*_{n+1} \arrow[l, "p_n = \partial^*_n"', bend right=18]
\end{tikzcd}
batch 1 · p. 17 — read it beside the facsimile9 / 51
LaTeX source
\begin{tikzcd}[column sep=large]
C = X \arrow[r, "u = \varphi^*"', bend right=18] & C' = \widehat{A} \arrow[l, "v = \varphi_!"', bend right=18]
\end{tikzcd}
batch 1 · p. 18 — read it beside the facsimile10 / 51
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
I \arrow[r, "\text{épi}"] \arrow[dr, "\text{épi}"'] & J \\
& K
\end{tikzcd}
batch 1 · p. 18 — read it beside the facsimile11 / 51
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
I \arrow[r] \arrow[d] & J \arrow[r] \arrow[d] & J' \arrow[d] \\
K \arrow[r] & L \arrow[r] & M
\end{tikzcd}
batch 1 · p. 18 — read it beside the facsimile12 / 51
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
X_0 & X_1 \arrow[l] & X_2 \arrow[l] \\
X'_0 \arrow[u] & X'_1 \arrow[l] \arrow[u] & X'_2 \arrow[l] \arrow[u]
\end{tikzcd}
batch 1 · p. 18 — read it beside the facsimile13 / 51
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
I \arrow[r, hook] \arrow[d] & J = I \amalg K \arrow[d, no head] \\
I' \arrow[r, hook] & J' = I' \amalg K
\end{tikzcd}
batch 1 · p. 18 — read it beside the facsimile14 / 51
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
X(J) & X(I \amalg K) \arrow[l] \\
X(J') \arrow[u] & X(I' \amalg K') \arrow[l]
\end{tikzcd}
batch 1 · p. 18 — read it beside the facsimile15 / 51
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
E_n \arrow[r, "\partial_n"] & E_{n+1} \arrow[d, "k_{n-1}"] \\
& E_n
\end{tikzcd}
batch 1 · p. 19 — read it beside the facsimile16 / 51
LaTeX source
\begin{tikzcd}[column sep=large]
X_1 & X_2 \arrow[l, "p_0"', bend left=18] & X_3 \arrow[l, "p_{12}"', bend left=18]
\end{tikzcd}
batch 1 · p. 19 — read it beside the facsimile17 / 51
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
X_3 \arrow[r, "p_1"] & X_2 \\
X_2 \arrow[u, "\delta_3"] & X_1 \arrow[l, "\delta_0"] \arrow[u, "\delta_0"']
\end{tikzcd}
batch 1 · p. 19 — read it beside the facsimile18 / 51
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
X_2 & X_3 \arrow[l, "p_{12}"'] \\
X_1 \arrow[u, "k_1"] & X_2 \arrow[l, "\delta_1"] \arrow[u, "k'_2"']
\end{tikzcd}
batch 1 · p. 19 — read it beside the facsimile19 / 51
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
E_2 \arrow[r, "\partial_2"] \arrow[d, "k_1"'] & E_3 \arrow[d, "k'_2"] \\
E_1 \arrow[r, "\partial_1"'] & E_2
\end{tikzcd}
batch 1 · p. 19 — read it beside the facsimile20 / 51
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
E_3 \arrow[r, "k_2"] \arrow[d, "k'_2"'] & E_2 \arrow[d, "k_1"] \\
E_2 \arrow[r, "k_1"'] & E_1
\end{tikzcd}
batch 1 · p. 19 — read it beside the facsimile21 / 51
LaTeX source
\begin{tikzcd}[column sep=large]
X_2 \arrow[r, "\delta_2", bend left=18] & X_3 \arrow[l, "k_2", bend left=18]
\end{tikzcd}
batch 2 · p. 21 — read it beside the facsimile22 / 51
LaTeX source
\begin{tikzcd}[column sep=large]
X_1 & X_2 \arrow[r, "d", bend left=18] & X_3 \arrow[l, "p", bend left=18]
\end{tikzcd}
batch 2 · p. 22 — read it beside the facsimile23 / 51
LaTeX source
\begin{tikzcd}[column sep=large]
X^{*}_3 \arrow[r, "p^{*} = p|X^{*}_3"] & X_2
\end{tikzcd}
batch 2 · p. 24 — read it beside the facsimile24 / 51
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
X_3 \arrow[r, "\delta_1"] & X_2 \arrow[d, "\delta_0"] \\
X_2 \arrow[u, "\delta_3"] \arrow[r, "\delta_0"'] & X_1 = \{e_1\}
\end{tikzcd}
batch 2 · p. 24 — read it beside the facsimile25 / 51
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
X_2 & X_3 \arrow[l, "p = \mathrm{pr}_2"'] \\
X_1 = \{e_1\} \arrow[u, "\kappa_1"] & X_2 \arrow[l, "p_1"] \arrow[u, "\kappa'_2"']
\end{tikzcd}
batch 2 · p. 26 — read it beside the facsimile26 / 51
LaTeX source
\begin{tikzcd}[column sep=large]
Y' \arrow[r, hook] \arrow[d, "\mathrm{can}\,\wr"'] & X' \arrow[d, "\wr"] \\
Y'\times\{z\} \arrow[r, hook, "\mathrm{id}_{Y'} \times i_{z,Z}"'] & Y'\times Z
\end{tikzcd}
batch 2 · p. 28 — read it beside the facsimile27 / 51
LaTeX source
\begin{tikzcd}[column sep=large]
\mathcal{V}^{*}_{Y,X} \arrow[r, "\text{bouchage de trous le long de } Y"] \arrow[d] & \mathcal{V}_{Y,X} \arrow[d] \\
X^{*} \arrow[r] & X
\end{tikzcd}
batch 2 · p. 28 — read it beside the facsimile28 / 51
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
\mathcal{V}^{*}_{Y,X} \arrow[r, hook] \arrow[d] & \mathcal{V}_{Y,X} \\
X^{*} &
\end{tikzcd}
batch 2 · p. 29 — read it beside the facsimile29 / 51
LaTeX source
\begin{tikzcd}[column sep=large, nodes={font=\scriptsize}]
\Pi_1\mathcal{V}^{*}_{Y,X} \arrow[r, "p"] \arrow[d, "i"'] & \Pi_1\mathcal{V}_{Y,X}\ (\simeq \Pi_1 Y) \arrow[d, dashed] \\
\Pi_1 X^{*} \arrow[r, dashed] & (\Pi_1 X)
\end{tikzcd}
batch 2 · p. 30 — read it beside the facsimile30 / 51
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
\Pi_1(\mathcal{V}^{*}_{Y,X}) \arrow[r] \arrow[d] & \Pi_1(Y) \\
\Pi_1(X^{*}) &
\end{tikzcd}
batch 2 · p. 32 — read it beside the facsimile31 / 51
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\Pi_2(Y) \arrow[r] & \Pi_1(\mathcal{V}^{*}_{z,Z}) \arrow[r] & \Pi_1(\mathcal{V}^{*}_{Y,X}) \arrow[r] & \Pi_1(Y) \arrow[r] & \Pi_0(\mathcal{V}^{*}_{z,Z}) \arrow[d] \\
& & & & \Pi_0(\mathcal{V}^{*}_{Y,X}) \arrow[d] \\
& & & & \Pi_0(Y)
\end{tikzcd}
batch 2 · p. 33 — read it beside the facsimile32 / 51
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
1 \arrow[r] & \Pi_1(\mathcal{V}^{*}_{z,Z}) \arrow[r] & \Pi_1(\mathcal{V}^{*}_{Y,X}) \arrow[r] \arrow[d] & \Pi_1(Y) \arrow[r] & 1 \\
& & \Pi_1(X^{*}) & &
\end{tikzcd}
batch 2 · p. 36 — read it beside the facsimile33 / 51
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
D^{*}_a \arrow[r, hook, "\text{« bouchage de trou en } a\text{ »}"] \arrow[d] & D_a \\
X^{*} = X \setminus \{a\} &
\end{tikzcd}
batch 2 · p. 36 — read it beside the facsimile34 / 51
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
\mathbb{Z}_{a,X} \arrow[r] \arrow[d] & \{1\} \\
\Pi_1(X^{*}) &
\end{tikzcd}
batch 2 · p. 36 — read it beside the facsimile35 / 51
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
D^{*}_0 \arrow[r] \arrow[d] & D_0 \\
\mathbb{C}^{*} &
\end{tikzcd}
batch 2 · p. 38 — read it beside the facsimile36 / 51
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
D^{*}_{\infty} \arrow[r, hook] \arrow[d] & D_{\infty} \arrow[d, dashed] \\
\mathbb{C} \arrow[r, dashed] & \mathbb{P}^1_{\mathbb{C}} \simeq S^2
\end{tikzcd}
batch 2 · p. 39 — read it beside the facsimile37 / 51
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
\mathbb{Z} \arrow[r] \arrow[d] & 1 \\
1 &
\end{tikzcd}
batch 3 · p. 45 — read it beside the facsimile38 / 51
LaTeX source
\begin{tikzcd}[column sep=large, row sep=small]
& X_{\Delta_1} \arrow[dl, "\sigma"'] \arrow[dr, "b"] & \\
X_{\Delta_0} & & X_{\Delta_0}
\end{tikzcd}
batch 3 · p. 46 — read it beside the facsimile39 / 51
LaTeX source
\begin{tikzcd}[column sep=large, row sep=small, nodes={font=\scriptsize}]
X_{\Delta_2} \underset{\text{déf}}{=} (X_{\Delta_1}, b) \times_{X_{\Delta_0}} (X_{\Delta_1}, \sigma) \arrow[r] \arrow[d, hook] & X_{\Delta_1} \arrow[d, hook] \\
X_{\Delta_0} \times X_{\Delta_0} \times X_{\Delta_0} & X_{\Delta_0} \times X_{\Delta_0}
\end{tikzcd}
batch 3 · p. 50 — read it beside the facsimile40 / 51
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
& X_{\Delta_1} \arrow[dl, "\sigma"'] \arrow[dr, "b"] & \\
X_{\Delta_0} \arrow[dr, dashed] & & X_{\Delta_0} \arrow[dl, dashed] \\
& X &
\end{tikzcd}
batch 3 · p. 57 — read it beside the facsimile41 / 51
LaTeX source
\begin{tikzcd}[column sep=large, row sep=small]
& X_i^{*} \arrow[d, "\text{« âme »}"] \\
\mathcal{V}^{*}_{ij} \arrow[r] \arrow[d] & \mathcal{V}_{i,j} \\
X_j^{*} &
\end{tikzcd}
batch 4 · p. 61 — read it beside the facsimile42 / 51
LaTeX source
\begin{tikzcd}
  \mathcal{V}^{*}_{\Delta_1} \arrow[r, hook] \arrow[d]
    & \mathcal{V}_{\Delta_1} \arrow[d, dashed]
    & X^{*}_{\Delta_0} \arrow[l] \arrow[dl, dashed] \\
  X^{*}_{\Delta_0} & X &
\end{tikzcd}
batch 4 · p. 61 — read it beside the facsimile43 / 51
LaTeX source
\begin{tikzcd}
  \mathcal{V}^{*}_{i,j} \arrow[r] \arrow[d]
    & \mathcal{V}_{i,j} \arrow[d, dashed]
    & X^{*}_{i} \arrow[l] \arrow[dl, dashed] \\
  X^{*}_{j} & X &
\end{tikzcd}
batch 4 · p. 63 — read it beside the facsimile44 / 51
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
  & i' \arrow[r, no head] & j' & & X_{i'} \arrow[r, no head] & X_{j'} \\
  i \arrow[ur, no head] \arrow[dr, no head] & & & X_i \arrow[ur, no head] \arrow[dr, no head] & & \\
  & j & & & X_j &
\end{tikzcd}
batch 4 · p. 64 — read it beside the facsimile45 / 51
LaTeX source
\begin{tikzcd}
  {[i,j]}^{*} \arrow[r] \arrow[d] & {[i,j]} & i^{*} \arrow[l] \\
  {[j]}^{*} & &
\end{tikzcd}
batch 4 · p. 65 — read it beside the facsimile46 / 51
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
  X_0^{*} \arrow[r] & \mathcal{V}_{0,1} & \mathcal{V}^{*}_{0,1} \arrow[l] \arrow[d] & & & & & & \\
  & & X_1^{*} \arrow[r] & \mathcal{V}_{1,2} & \mathcal{V}^{*}_{1,2} \arrow[l] \arrow[d] & & & & \\
  & & & & X_2^{*} \arrow[r] & \mathcal{V}_{2,3} & \mathcal{V}^{*}_{2,3} \arrow[l] \arrow[d] & & \\
  & & & & & & X_3^{*} & & \\
  & & & & & & \cdots \arrow[d] & & \\
  & & & & & & X^{*}_{n-1} \arrow[r] & \mathcal{V}_{n-1,n} & \mathcal{V}^{*}_{n-1,n} \arrow[l] \arrow[d] \\
  & & & & & & & & X_n^{*}
\end{tikzcd}
batch 4 · p. 65 — read it beside the facsimile47 / 51
LaTeX source
\begin{tikzcd}[column sep=small, nodes={font=\scriptsize}]
  \mathcal{V}^{*}_{0,b} \arrow[r] \arrow[d] & \mathcal{V}_{0,b}
    & X_0^{*} \arrow[l] \arrow[r] & \mathcal{V}_{0,a}
    & \mathcal{V}^{*}_{0,a} \arrow[l] \arrow[d] \\
  X_b^{*} & & & & X_a^{*}
\end{tikzcd}
batch 4 · p. 65 — read it beside the facsimile48 / 51
LaTeX source
\begin{tikzcd}
  X_a^{*} \arrow[d] & & X_b^{*} \arrow[d] \\
  \mathcal{V}_{a,c} & & \mathcal{V}_{b,c} \\
  \mathcal{V}^{*}_{a,c} \arrow[u] \arrow[r] & X_c^{*} & \mathcal{V}^{*}_{b,c} \arrow[u] \arrow[l]
\end{tikzcd}
batch 4 · p. 66 — read it beside the facsimile49 / 51
LaTeX source
\begin{tikzcd}
  X_m^{*} \arrow[r] & \mathcal{V}_{m,m+1} & \mathcal{V}^{*}_{m,m+1} \arrow[l] \arrow[d] \\
  & & X^{*}_{m+1}
\end{tikzcd}
batch 4 · p. 67 — read it beside the facsimile50 / 51
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
  X_n^{*} \arrow[r] & \mathcal{V}_{n,n-1} & \mathcal{V}^{*}_{n,n-1} \arrow[l] \arrow[d] & & & & \\
  & & X^{*}_{n-1} \arrow[r] & \mathcal{V}_{n-1,n-2} & \mathcal{V}^{*}_{n-1,n-2} \arrow[l] \arrow[d] & & \\
  & & & & X^{*}_{n-2} & & \\
  & & & & \cdots \arrow[d] & & \\
  & & & & X_1^{*} \arrow[r] & \mathcal{V}_{1,0} & \mathcal{V}^{*}_{1,0} \arrow[l] \arrow[d] \\
  & & & & & & X_0^{*}
\end{tikzcd}
batch 4 · p. 74 — read it beside the facsimile51 / 51
LaTeX source
\begin{tikzcd}
  \mathcal{V}_{i',j',X'} \arrow[r, hook] & \mathcal{V}_{i',j'} \\
  \mathcal{V}^{*}_{i',j',X'} \arrow[r, hook] \arrow[u, hook]
    & \mathcal{V}^{*}_{i',j'} \arrow[u, hook]
\end{tikzcd}