Cote n° 105 · pages 4–52
· 41 diagrammes commutatifs · [Champs (stacks) 2] : notes manuscrites (s.d.).
Datation de l’inventaire : [à partir de 1982]
Édition de démonstration
LaTeX source
\begin{tikzcd}
TC \arrow[r, leftrightarrow] \arrow[d, hook] & F \\
C \arrow[r, leftrightarrow] & TF \arrow[u, hook]
\end{tikzcd}LaTeX source
\begin{tikzcd}
A' \times A \arrow[r, hook] \arrow[d, hook] & B' \times A \arrow[d, hook] \\
A' \times B \arrow[r, hook] & B' \times B = B' \times B_0 \times I
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
((A' \times B_0) \times e) \cup (A' \times A_0 \times I) \arrow[r] \arrow[d, "{h(i_0 \times \mathrm{id}_{A'},\, I,\, e)}"] & ((B' \times B_0) \times e) \cup ((B' \times A_0) \times I) \arrow[d, "{h(i_0 \times \mathrm{id}_{B'},\, I,\, e)}"] \\
(A' \times B_0) \times I \arrow[r, hook] & B' \times B_0 \times I
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
A_0 \times \{e\} \arrow[r, hook] \arrow[d, "W"'] & B_0 \times \{e\} \arrow[d, hook, "f" description] \arrow[dr, "W"] & \\
A_0 \times I \arrow[r, hook] & A \arrow[r, hook] & B = B_0 \times I
\end{tikzcd}LaTeX source
\begin{tikzcd}
& B' \arrow[d, "p"] \\
A \arrow[ur, "i"] \arrow[r, hook, "f"'] & B \arrow[u, bend right=40, "s"']
\end{tikzcd}LaTeX source
\begin{tikzcd}
A \arrow[r, hook, "i"] \arrow[d, "="'] & B' \arrow[d, "p"] \\
A \arrow[r, hook, "f"'] & B \arrow[u, bend right=40, "s"']
\end{tikzcd}LaTeX source
\begin{tikzcd}
& X' \arrow[d, "p \in F"] \\
X \arrow[ur, "i \in W \cap C"] \arrow[r, "f"'] & Y \arrow[u, bend right=40, "s"']
\end{tikzcd}LaTeX source
\begin{tikzcd}
A \arrow[r, "f"] \arrow[d, bend left=30, "i"] & B \arrow[d, bend left=30, "j"] \\
A' \arrow[r, "f'"'] \arrow[u, bend left=30, "p"] & B' \arrow[u, bend left=30, "q"]
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
& & X_2 \arrow[r, "C \text{ or } F", "g"'] & Y_2 \arrow[dr, "TF"] & & \\
\varnothing \arrow[r, "C"] & X_1 \arrow[ur, "TC"] \arrow[dr, "TF"'] & & & Y_1 \arrow[r, "F"] & e \\
& & X \arrow[r, "f"'] & Y \arrow[ur, "TC"'] & &
\end{tikzcd}LaTeX source
\begin{tikzcd}
X \arrow[r, "="] \arrow[d, "i"'] & X \arrow[d, "f \in F"] \\
X' \arrow[r, "p"'] \arrow[ur, dashed, "s"] & Y
\end{tikzcd}LaTeX source
\begin{tikzcd}
A \arrow[r, "u"] \arrow[d, "g"'] & X \arrow[d, "f"] \\
B \arrow[r, "v"'] & Y
\end{tikzcd}LaTeX source
\begin{tikzcd}
A \arrow[r, "u"] \arrow[d, "\alpha"'] & X \arrow[d, "i_D"] \arrow[dd, bend left=50, "f"] \\
B \arrow[r] & X_D = X \sqcup_A B \arrow[d, "p_D"] \\
& Y
\end{tikzcd}LaTeX source
\begin{tikzcd}
X \arrow[r, "i_1(f)"] \arrow[d, "f"'] & X_1(f) \arrow[dl, "\Sigma_1(f)"] \\
Y &
\end{tikzcd}LaTeX source
\begin{tikzcd}
X \arrow[r, "i_\alpha(f)"] \arrow[d, "f"'] & X_\alpha(f) \arrow[dl, "\Sigma_\alpha(f)"] \\
Y &
\end{tikzcd}LaTeX source
\begin{tikzcd}
X \arrow[r, "i_{\alpha_0}(f) = i(f)"] \arrow[d] & X_{\alpha_0}(f) = \underline{X}(f) \arrow[dl, "\Sigma_{\alpha_0}(f) = p(f)"] \\
Y &
\end{tikzcd}LaTeX source
\begin{tikzcd}
A \arrow[r, "u"] \arrow[d, "\alpha"'] & X \arrow[d, "\alpha'"] \\
B \arrow[r, "u'"'] & X'
\end{tikzcd}LaTeX source
\begin{tikzcd}
\underline{\mathrm{Fl}}(M) \arrow[rr, "\Sigma"] \arrow[dr, "t"'] & & \underline{\mathrm{Fl}}(M) \arrow[dl, "t"] \\
& M &
\end{tikzcd}LaTeX source
\begin{tikzcd}
X \arrow[r, "i(f)"] \arrow[d] & \underline{X}(f) \arrow[dl, "p(f)"] \\
Y &
\end{tikzcd}LaTeX source
\begin{tikzcd}
A_{i_0} \arrow[r] \arrow[d, "u_0"'] \arrow[rr, bend left=30] & A_J \arrow[r] \arrow[dl, "u_J"] & B \arrow[d] \\
X \arrow[rr, "f"'] & & Y
\end{tikzcd}LaTeX source
\begin{tikzcd}
U \arrow[r, "i_0"] \arrow[d, "q_U"'] & \overline{U} \arrow[d, "q_{\overline{U}}"] \\
U' \arrow[r, "i'_0"'] & \overline{U}'
\end{tikzcd}LaTeX source
\begin{tikzcd}
& & a = (0,1) \\
X \arrow[r] & \underline{\Psi} = 0 \arrow[ur] \arrow[dr] & \\
& & b = (1,0)
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
& U \\
U \cap U' \arrow[ur, "i'_0"] \arrow[dr, "i_0"'] & \\
& U'
\end{tikzcd}LaTeX source
\begin{tikzcd}
X \arrow[rr, "\alpha"] \arrow[dr, "\varphi"', no head] & & \overline{X} \arrow[dl, "\overline{\varphi}", no head] \\
& \underline{\Psi} &
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small]
H^{\bullet}_{\mathbb{D}}(U, \xi) \arrow[dr] & & H^{\bullet}_{\mathbb{D}}(U', \xi) \arrow[dl] \\
& H^{\bullet}_{\mathbb{D}}(U \cap U', \xi) &
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
& (0,1) \arrow[dr, no head] & \\
(0,0) \arrow[ur, no head] \arrow[dr] & & (1,1) \\
& (1,0) \arrow[ur] &
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small]
& H^{\bullet}_{\mathbb{D}}(X, \xi) \arrow[dl, "i^*"'] \arrow[dr, "i'^*"] & \\
H^{\bullet}_{\mathbb{D}}(U, \xi) \arrow[dr, "i_0'^*"'] & & H^{\bullet}_{\mathbb{D}}(U', \xi) \arrow[dl, "i_0^*"] \\
& H^{\bullet}_{\mathbb{D}}(U \cap U', \xi) &
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
& U \arrow[dr] & \\
U \cap U' \arrow[ur] \arrow[dr] & & X \\
& U' \arrow[ur] &
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
& U \arrow[dr, no head] & \\
U \cap U' \arrow[ur, no head] \arrow[dr] & & X \\
& U' \arrow[ur, no head] &
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
& X_a \\
X_0 \arrow[ur, no head] \arrow[dr] & \\
& X_b
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small]
& H^{\bullet}_{\mathbb{D}}(\mathcal{X}, \xi) \arrow[dl] \arrow[dr] & \\
H^{\bullet}_{\mathbb{D}}(X_a, \xi) \arrow[dr] & & H^{\bullet}_{\mathbb{D}}(X_b, \xi) \arrow[dl] \\
& H^{\bullet}_{\mathbb{D}}(X_0, \xi) &
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
& X_a \arrow[dr] & \\
X_0 \arrow[ur] \arrow[dr] & & \mathcal{X} \\
& X_b \arrow[ur] &
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
& X_{/a} \arrow[dr, no head] & \\
X_{/0} \arrow[ur, no head] \arrow[dr] & & \mathcal{X} \\
& X_{/b} \arrow[ur, no head] &
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
& X_0 \\
X_a \arrow[ur, no head] \arrow[dr] & \\
& X_b
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
& a \arrow[dl] & \\
1 & & 0 \arrow[ul] \arrow[dl] \\
& b \arrow[ul] &
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
& \gamma(0,1) \arrow[dl, "j_0"'] & \\
\gamma(0,0) & & \gamma(1,1) \arrow[ul, "i_1"'] \arrow[dl, "j_1"] \\
& \gamma(1,0) \arrow[ul, "i_0"] &
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
& \xi_0 \arrow[dl] \arrow[dr] & \\
\xi_a \arrow[dr] & & \xi_b \arrow[dl] \\
& \xi_1 &
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
& \xi_a \arrow[dr] \arrow[ddd, "f_a"] & \\
\xi_0 \arrow[ur] \arrow[dr] \arrow[ddd, "f_0"'] & & \xi_1 \arrow[ddd, "f_1"] \\
& \xi_b \arrow[ur] \arrow[ddd, "f_b"'] & \\
& \eta_a \arrow[dr] & \\
\eta_0 \arrow[ur] \arrow[dr] & & \eta_1 \\
& \eta_b \arrow[ur] &
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
& X_a \\
X_0 \arrow[ur, no head] \arrow[dr] & \\
& X_b
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
& X_0 \arrow[dl, "i_a"'] \arrow[dr, "i_b"] & \\
X_a \arrow[dr, "i'_b"'] & & X_b \arrow[dl, "i'_a"] \\
& \mathcal{X} &
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
& X_a \arrow[dr, dashed] \arrow[ddd, "f_a"] & \\
X_0 \arrow[ur] \arrow[dr] \arrow[ddd, "f_0"'] & & \mathcal{X} \arrow[ddd, "f"] \\
& X_b \arrow[ur, dashed] \arrow[ddd, "f_b"'] & \\
& X'_a \arrow[dr, dashed] & \\
X'_0 \arrow[ur] \arrow[dr] & & \mathcal{X}' \\
& X'_b \arrow[ur, dashed] &
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
& X & \\
X_a \arrow[ur, "i'_b"] & & X_b \arrow[ul, "i'_a"'] \\
& X_0 \arrow[ul, "i_a"] \arrow[ur, "i_b"'] &
\end{tikzcd}