Cote n° 118 · pages 2–40 · 27 diagrammes commutatifs · Cohomology properties of maps in (Cat) : notes manuscrites (s.d.).
Datation de l’inventaire : [à partir de 1983]
Édition de démonstration

batch 1 · p. 2 — lire la page1 / 27
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\begin{tikzcd}[column sep=tiny, row sep=small, nodes={font=\scriptsize}]
 & & & & F\cap C \arrow[dl, "F\cap C'"'] \arrow[dr, "F'\cap C"] & & & & \\
 & & & F\cap Pr \arrow[dl] \arrow[dr, "F'\cap Pr"] & F'\cap C' & Sm\cap C \arrow[dl, "Sm\cap C'"'] \arrow[dr] & & & \\
 & & F\cap S \arrow[dl] \arrow[dr, "F'\cap S"] & & Sm\cap Pr \arrow[dl] \arrow[dr] & & S\cap C \arrow[dl, "S\cap C'"'] \arrow[dr] & & \\
 & F \arrow[dr, "F'"] & & Sm\cap S \arrow[dl] \arrow[dr] \arrow[dddd, dashed] & & S\cap Pr \arrow[dl] \arrow[dr] \arrow[dddd, dashed] & & C \arrow[dl, "C'"] & \\
 & & Sm & & S \arrow[d] & & Pr & & \\
 & & & & \mathrm{Spr}B \arrow[d] & & & & \\
 & & & & \mathrm{Spr}F\cap \mathrm{Spr}C \arrow[dl] \arrow[dr] & & & & \\
 & & & \mathrm{Spr}F & & \mathrm{Spr}C & & &
\end{tikzcd}
batch 1 · p. 2 — lire la page2 / 27
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\begin{tikzcd}[column sep=0pt, row sep=small, nodes={font=\tiny}]
 & & W\cap F\cap C \arrow[dl] \arrow[dr] & & \\
 & W\cap F\cap Pr \arrow[dl] \arrow[dr] & & W\cap Sm\cap C \arrow[dl] \arrow[dr] & \\
 W\cap F\cap S = T\cap F \arrow[dr] & & W\cap Sm\cap Pr \arrow[dl] \arrow[dr] & & W\cap S\cap C = T\cap C \arrow[dl] \\
 & W\cap Sm\cap S = T\cap Sm \arrow[dr] & & W\cap S\cap Pr = T\cap Pr \arrow[dl] & \\
 & & W\cap S = T\cap S = \mathcal{U}W \arrow[d] & & \\
 & & \mathrm{Bias} = W\cap \mathrm{Spr}B \arrow[d] & & \\
 & & \mathrm{As}\cap \mathrm{As}^{\circ} \arrow[dl] \arrow[dr] & & \\
 & \mathrm{As}^{\circ} = W\cap \mathrm{Spr}F \arrow[dr] & & \mathrm{As} = W\cap \mathrm{Spr}C \arrow[dl] & \\
 & & W & &
\end{tikzcd}
batch 1 · p. 2 — lire la page3 / 27
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\begin{tikzcd}[column sep=tiny, row sep=small, nodes={font=\scriptsize}]
 & & \mathrm{strh}\cap F\cap C \arrow[dl, no head] \arrow[dr, no head] & & \\
 & \mathrm{strh}\cap F\cap \mathrm{str}Pr \arrow[dl, no head] \arrow[dr, no head] & & \mathrm{strh}\cap \mathrm{str}Sm\cap C \arrow[dl, no head] \arrow[dr, no head] & \\
 \mathrm{strh}\cap F \arrow[dr, no head] & & \mathrm{strh}\cap \mathrm{str}Sm\cap \mathrm{str}Pr \arrow[dl, no head] \arrow[dr, no head] & & \mathrm{strh}\cap C \arrow[dl, no head] \\
 & \mathrm{strh}\cap \mathrm{str}Sm \arrow[dr, no head] & & \mathrm{strh}\cap \mathrm{str}Pr \arrow[dl, no head] & \\
 & & \mathrm{strh} & &
\end{tikzcd}
batch 1 · p. 4 — lire la page4 / 27
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\begin{tikzcd}[column sep=tiny, row sep=small, nodes={font=\scriptsize}]
 & & & & B \overset{\mathrm{def}}{=} F\cap C \arrow[dl] \arrow[dr] & & & & \\
 & & & F\cap Pr \arrow[dl] \arrow[dr] & & Sm\cap C \arrow[dl] \arrow[dr] & & & \\
 & & F\cap S \arrow[dl, "F\cap WSC"'] \arrow[dr] & & Sm\cap Pr \arrow[dl] \arrow[dr] & & S\cap C \arrow[dl] \arrow[dr, "WSF\cap C"] & & \\
 & (F) \arrow[dr] & & Sm\cap S \arrow[dl, "Sm\cap WSC"'] \arrow[dr] \arrow[dddd] & & S\cap Pr \arrow[dl] \arrow[dr, "WSF\cap Pr"] \arrow[dddd] & & (C) \arrow[dl] & \\
 & & (Sm) & & (S) \arrow[d] & & (Pr) & & \\
 & & & & \boxed{WSB} \arrow[d] & & & & \\
 & & & & WSF\cap WSC \arrow[dl] \arrow[dr] & & & & \\
 & & & WSF \arrow[dr] & & WSC \arrow[dl] & & & \\
 & & & & \mathrm{Fl}(\mathrm{Cat}) & & & &
\end{tikzcd}
batch 1 · p. 4 — lire la page5 / 27
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\begin{tikzcd}[column sep=0pt, row sep=small, nodes={font=\tiny}]
 & & \mathrm{str}\ W\cap F\cap C \arrow[dl] \arrow[dr] & & \\
 & \mathrm{str}\ W\cap F\cap Pr \arrow[dl] \arrow[dr] & & W\cap Sm\cap C \arrow[dl] \arrow[dr] & \\
 W\cap F\cap S = T\cap F \arrow[dr] & & W\cap Sm\cap Pr \arrow[dl] \arrow[dr] & & W\cap S\cap C = T\cap C \arrow[dl] \\
 & W\cap Sm\cap S = T\cap Sm \arrow[dr] & & W\cap S\cap Pr = T\cap Pr \arrow[dl] & \\
 & & W\cap S = T\cap S = UW \arrow[d, "\mathrm{str}"] & & \\
 & & \mathrm{Tot}\,\mathrm{As} = W\cap WSB \arrow[dl, "\mathrm{str}"'] \arrow[dr] & & \\
 & \mathrm{As}^{\circ} = W\cap WSF \arrow[dr, "\mathrm{str}"'] & & \mathrm{As} = W\cap WSC \arrow[dl] & \\
 & & W & &
\end{tikzcd}
batch 1 · p. 10 — lire la page6 / 27
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\begin{tikzcd}[column sep=large]
  Y & \mathrm{Fib}(f) \arrow[l, "f^{\flat}"'] & X \arrow[l, hook', "\mu_f"'] \\
  & \mathrm{Fib}(f) \arrow[r, bend right=20, "\rho_f"'] & X
\end{tikzcd}
batch 1 · p. 10 — lire la page7 / 27
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\begin{tikzcd}[column sep=large]
  Y & \mathrm{Cofib}(f) \arrow[l, "f^{\flat\prime}"'] & X \arrow[l, hook', "\mu'_f"'] \\
  & \mathrm{Cofib}(f) \arrow[r, bend right=20, "\rho'_f"'] & X
\end{tikzcd}
batch 1 · p. 11 — lire la page8 / 27
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\begin{tikzcd}
  Y & \mathrm{Fl}(Y) \arrow[l] \arrow[d, bend right=15] & \mathrm{Fib}(f) \arrow[l] \arrow[ll, bend right=30, "f^{\flat}"'] \arrow[d, "\rho_f"] \\
  & Y' \arrow[u, dashed, bend right=15] & X \arrow[l, "f"] \arrow[u, dashed, bend right=40, "\mu_f"']
\end{tikzcd}
batch 1 · p. 11 — lire la page9 / 27
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\begin{tikzcd}[column sep=large]
  Y & \mathrm{Fib}(f) \arrow[l, "f^{\flat}"'] \arrow[r, bend right=25, "\rho_f"'] & X \arrow[l, hook', "\mu_f\ \mathrm{mono}"']
\end{tikzcd}
batch 2 · p. 21 — lire la page10 / 27
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\begin{tikzcd}
Y \arrow[r, hook] \arrow[d] & X \arrow[d] \\
Y' \arrow[r, hook] & X'
\end{tikzcd}
batch 2 · p. 21 — lire la page11 / 27
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\begin{tikzcd}
A_{/Y} \arrow[r, hook] \arrow[d] & A_{/X} \arrow[d] \\
A_{/Y'} \arrow[r, hook] & A_{/X'}
\end{tikzcd}
batch 2 · p. 22 — lire la page12 / 27
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\begin{tikzcd}
Y \arrow[r, hook, "i"] \arrow[d, "g"'] & X \arrow[d, "f"] \\
Y' \arrow[r, hook, "i'"'] & X'
\end{tikzcd}
batch 2 · p. 22 — lire la page13 / 27
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\begin{tikzcd}
Z \arrow[r, hook] \arrow[d, hook, "\alpha"'] & \check{C}(Z)\ [\supset e_{u'_0}] \arrow[d] \\
Y = C(Z) \arrow[r, hook, "\text{imm. f. (surj.)}"'] \arrow[d, "g \in W"'] & \Sigma(Z) = X\ [\supset e_{w_0}] \arrow[d, "f"] \\
Y' = e \arrow[r, hook, "\text{imm. f.}"'] & \Theta(Y) = X'\ [\supset e_{u'_0}]
\end{tikzcd}
batch 2 · p. 23 — lire la page14 / 27
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\begin{tikzcd}
Y \arrow[r, "i"] \arrow[d, "g"'] & X \arrow[d, "f"] \\
Y' \arrow[r, "i'"'] & X'
\end{tikzcd}
batch 2 · p. 24 — lire la page15 / 27
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\begin{tikzcd}[column sep=small]
 & I \times X \arrow[rr, dashed] & & Y' \\
I \times X' \arrow[r, hook, "{[\in \underline{W}_A]}"'] & M \arrow[u, "{[\in \underline{W}_A]}"] \arrow[rr] & & Y \arrow[u, dashed, "t"'] \\
X' \sqcup X' \arrow[u] \arrow[r, hook, "\in \underline{W}_A"'] & X \sqcup X \arrow[u] \arrow[urr] & &
\end{tikzcd}
batch 2 · p. 24 — lire la page16 / 27
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\begin{tikzcd}
Y \arrow[r, "i"] \arrow[d, "f"'] & X \arrow[d, "g"] \\
Y' \arrow[r] & X'
\end{tikzcd}
batch 2 · p. 25 — lire la page17 / 27
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\begin{tikzcd}[column sep=small]
X \arrow[rr] \arrow[dr] & & Y \arrow[dl] \\
 & J &
\end{tikzcd}
batch 2 · p. 27 — lire la page18 / 27
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\begin{tikzcd}
I_N \times X \arrow[r] & X \\
I_N \times X_j \arrow[u, hook] \arrow[r] & X_j \arrow[u, hook]
\end{tikzcd}
batch 2 · p. 30 — lire la page19 / 27
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\begin{tikzcd}
Y \arrow[r, hook, "\overline{\alpha}_j"] \arrow[dr, hook, bend right=30, "\alpha"'] & \overline{Y}'_j \arrow[r, "i_j"] \arrow[d, "\text{comm.}"'] & Y'_j \arrow[l, dashed, bend left=40, "s_j"] \arrow[dl, dashed, "\beta_j"] \\
 & Y' &
\end{tikzcd}
batch 2 · p. 32 — lire la page20 / 27
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\begin{tikzcd}
 & Y \arrow[rr, "\mathrm{pr}_j"] \arrow[d, "\alpha\ \in W"'] & & Y_j \arrow[d, "\beta_j\ \in \underline{W}"] \\
f, g : X \arrow[r, bend left=15] \arrow[r, bend right=15] & Y' \arrow[rr, "\varphi_j"'] & & Y'_j
\end{tikzcd}
batch 2 · p. 33 — lire la page21 / 27
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\begin{tikzcd}
 & Y \arrow[d, "\alpha\ \in W"'] \arrow[dr, "\beta = \prod \beta_j\ \in W"] & \\
X \arrow[r, bend left=15, "f"] \arrow[r, bend right=15, "g"'] & Y' \arrow[r, "\varphi = (\varphi_j)"'] & Y'' = \prod Y'_j
\end{tikzcd}
batch 2 · p. 33 — lire la page22 / 27
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\begin{tikzcd}
X \times a \arrow[r] \arrow[d] & Y \times a \arrow[d] \\
X \arrow[r, "\in \underline{W}_A"'] & Y
\end{tikzcd}
batch 2 · p. 34 — lire la page23 / 27
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\begin{tikzcd}[column sep=small, row sep=small]
x \arrow[r] & X \times Y \\
x \arrow[r] & X \\
x \arrow[r] & Y
\end{tikzcd}
batch 2 · p. 39 — lire la page24 / 27
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\begin{tikzcd}
X \arrow[r, "f"] \arrow[d, "\alpha \in \underline{W}"'] & Y \arrow[dl, "g"] \arrow[d, "\beta \in \underline{W}"] \\
X' \arrow[r, "f'"'] & Y'
\end{tikzcd}
batch 2 · p. 39 — lire la page25 / 27
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\begin{tikzcd}
X \arrow[r, "f"] \arrow[d, "\alpha"'] & Y \arrow[dl, "g"'] \arrow[d, "\beta"] \\
X' \arrow[r, "f'"'] & Y'
\end{tikzcd}
batch 2 · p. 39 — lire la page26 / 27
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\begin{tikzcd}[column sep=small]
X \arrow[rr, hook] & & Y \arrow[dl, "g_1"'] \\
 & X'_1 \arrow[dl] \arrow[dr] & \\
X' \arrow[rr] & & Y'
\end{tikzcd}
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\begin{tikzcd}
X_0 \arrow[rr, "f = f_0"] \arrow[dr, "i_0"'] & & X_1 \arrow[dl, "i_1"] \\
 & X_\infty &
\end{tikzcd}