Cote n° 116 · pages 2–9
· 26 displayed formulas · Mapping-cône etc. : notes manuscrites (s.d.).
Inventory dating : [à partir de 1981]
Édition de démonstration
\[\uncertain{{}_{b}\backslash A} \longleftarrow A_{b}\]
LaTeX source
\[
\uncertain{{}_{b}\backslash A} \longleftarrow A_{b}
\]\[\begin{gather*}
i_{!} = \ill{} \\
i^{*} = \ill{} \\
i_{*}
\end{gather*}\]
LaTeX source
\begin{gather*}
i_{!} = \ill{} \\
i^{*} = \ill{} \\
i_{*}
\end{gather*}\[p_{A} : A \longrightarrow A_{0}\]
LaTeX source
\[
p_{A} : A \longrightarrow A_{0}
\]\[\begin{array}{c}
p_{A!}(F) \\
\uparrow \\
p_{A!}\, i_{!}\bigl(\ill{}(F)\bigr)
\end{array}\]
LaTeX source
\[
\begin{array}{c}
p_{A!}(F) \\
\uparrow \\
p_{A!}\, i_{!}\bigl(\ill{}(F)\bigr)
\end{array}
\]\[\begin{gather*}
i_{!}\, i^{*}(F) \xrightarrow{\ \sim\ } F \\
i_{*}\, i^{*} F \xleftarrow{\ \sim\ } F
\end{gather*}\]
LaTeX source
\begin{gather*}
i_{!}\, i^{*}(F) \xrightarrow{\ \sim\ } F \\
i_{*}\, i^{*} F \xleftarrow{\ \sim\ } F
\end{gather*}\[F \longrightarrow \ill{}\ \ill{}\, F \quad \text{induit iso sur } \Gamma, \text{ puis } H^{*} F\]
LaTeX source
\[
F \longrightarrow \ill{}\ \ill{}\, F \quad \text{induit iso sur } \Gamma, \text{ puis } H^{*} F
\]\[\ill{} \quad \mathrm{Ex}_{\alpha} \quad \ill{} \qquad\qquad A_0 \longrightarrow A\]
LaTeX source
\[
\ill{} \quad \mathrm{Ex}_{\alpha} \quad \ill{} \qquad\qquad A_0 \longrightarrow A
\]\[(1) \qquad f : \mathcal{X}_0 \longrightarrow \mathcal{X}_1\]
LaTeX source
\[
(1) \qquad f : \mathcal{X}_0 \longrightarrow \mathcal{X}_1
\]\[(2) \qquad
\begin{array}{l}
(F_0, F_1, u), \qquad F_0 \in \mathrm{Ob}\,\mathcal{F}(\mathcal{X}_0),\ F_1 \in \mathrm{Ob}\,\mathcal{F}(\mathcal{X}_1) \\[4pt]
\qquad u : f^{*}(F_1) \longrightarrow F_0 \quad \text{i.e.} \\[4pt]
\qquad F_1 \longrightarrow f_{*}(F_0)
\end{array}\]
LaTeX source
\[
(2) \qquad
\begin{array}{l}
(F_0, F_1, u), \qquad F_0 \in \mathrm{Ob}\,\mathcal{F}(\mathcal{X}_0),\ F_1 \in \mathrm{Ob}\,\mathcal{F}(\mathcal{X}_1) \\[4pt]
\qquad u : f^{*}(F_1) \longrightarrow F_0 \quad \text{i.e.} \\[4pt]
\qquad F_1 \longrightarrow f_{*}(F_0)
\end{array}
\]\[(3) \qquad \mathcal{X}_1 \xrightarrow{\ i_1\ } \mathcal{X}\]
LaTeX source
\[
(3) \qquad \mathcal{X}_1 \xrightarrow{\ i_1\ } \mathcal{X}
\]\[H^{*}(\mathcal{X}, F) \xrightarrow{\ \sim\ } H^{*}\bigl(\mathcal{X}_1, i_1^{*}(F)\bigr)\]
LaTeX source
\[
H^{*}(\mathcal{X}, F) \xrightarrow{\ \sim\ } H^{*}\bigl(\mathcal{X}_1, i_1^{*}(F)\bigr)
\]\[\varphi : \mathcal{F}(\mathcal{X}_0) \longrightarrow \mathcal{F}(\mathcal{X}_1)\]
LaTeX source
\[
\varphi : \mathcal{F}(\mathcal{X}_0) \longrightarrow \mathcal{F}(\mathcal{X}_1)
\]\[(4) \qquad \varphi = f_{*}\]
LaTeX source
\[
(4) \qquad \varphi = f_{*}
\]\[(5) \qquad \longrightarrow H^{i}_{\mathcal{X}_1}(\mathcal{X}, F) \longrightarrow H^{i}(\mathcal{X}, F) \longrightarrow H^{i}(\mathcal{X}_0, F) \longrightarrow H^{i+1}_{\mathcal{X}_1}(\mathcal{X}, F) \longrightarrow \cdots\]
LaTeX source
\[
(5) \qquad \longrightarrow H^{i}_{\mathcal{X}_1}(\mathcal{X}, F) \longrightarrow H^{i}(\mathcal{X}, F) \longrightarrow H^{i}(\mathcal{X}_0, F) \longrightarrow H^{i+1}_{\mathcal{X}_1}(\mathcal{X}, F) \longrightarrow \cdots
\]\[(5') \qquad \longrightarrow H^{i}_{\mathcal{X}_1}(\mathcal{X}, F) \longrightarrow H^{i}(\mathcal{X}_1, F_1) \longrightarrow H^{i}(\mathcal{X}_0, F_0) \longrightarrow H^{i+1}_{\mathcal{X}_1}(\mathcal{X}, F) \longrightarrow \cdots\]
LaTeX source
\[
(5') \qquad \longrightarrow H^{i}_{\mathcal{X}_1}(\mathcal{X}, F) \longrightarrow H^{i}(\mathcal{X}_1, F_1) \longrightarrow H^{i}(\mathcal{X}_0, F_0) \longrightarrow H^{i+1}_{\mathcal{X}_1}(\mathcal{X}, F) \longrightarrow \cdots
\]\[H^{i}_{\mathcal{X}_1}(\mathcal{X}, F) \overset{\text{déf}}{=} H^{i}\bigl(\mathcal{X}_1 \bmod \mathcal{X}_0,\ F = (F_0, F_1, u)\bigr)\]
LaTeX source
\[
H^{i}_{\mathcal{X}_1}(\mathcal{X}, F) \overset{\text{déf}}{=} H^{i}\bigl(\mathcal{X}_1 \bmod \mathcal{X}_0,\ F = (F_0, F_1, u)\bigr)
\]\[H^{i}\bigl(\{\mathcal{X}_1 \bmod \mathcal{X}_0\},\ F_1\bigr) \qquad \text{si } F_0 = f^{*}(F_1),\ u \text{ tautolog.}\]
LaTeX source
\[
H^{i}\bigl(\{\mathcal{X}_1 \bmod \mathcal{X}_0\},\ F_1\bigr) \qquad \text{si } F_0 = f^{*}(F_1),\ u \text{ tautolog.}
\]\[(6) \qquad \mathcal{X}_0 \subset \mathcal{X},\]
LaTeX source
\[
(6) \qquad \mathcal{X}_0 \subset \mathcal{X},
\]\[\pi : \mathcal{X} \longrightarrow \mathrm{Top}(\Delta_1)\]
LaTeX source
\[
\pi : \mathcal{X} \longrightarrow \mathrm{Top}(\Delta_1)
\]\[(\mathcal{X}_0, \mathcal{X}_1, \varphi) \qquad
\varphi : \mathcal{F}(\mathcal{X}_0) \longrightarrow \mathcal{F}(\mathcal{X}_1)\]
LaTeX source
\[
(\mathcal{X}_0, \mathcal{X}_1, \varphi) \qquad
\varphi : \mathcal{F}(\mathcal{X}_0) \longrightarrow \mathcal{F}(\mathcal{X}_1)
\]\[\mathcal{X}_0 \longrightarrow \mathcal{X} \longleftarrow \mathcal{X}_1\]
LaTeX source
\[
\mathcal{X}_0 \longrightarrow \mathcal{X} \longleftarrow \mathcal{X}_1
\]\[X_1 \longrightarrow \widehat{X}_0\]
LaTeX source
\[
X_1 \longrightarrow \widehat{X}_0
\]\[\widehat{X}_1 \xrightarrow{\ f^{*}\ } \widehat{X}_0\]
LaTeX source
\[
\widehat{X}_1 \xrightarrow{\ f^{*}\ } \widehat{X}_0
\]\[(x_0, x_1) \mapsto h(x_1, x_0), \qquad X_0^{\mathrm{op}} \times X_1 \longrightarrow (\mathrm{Ens}).\]
LaTeX source
\[
(x_0, x_1) \mapsto h(x_1, x_0), \qquad X_0^{\mathrm{op}} \times X_1 \longrightarrow (\mathrm{Ens}).
\]\[A_{/b} \longleftrightarrow A_{b}\]
LaTeX source
\[
A_{/b} \longleftrightarrow A_{b}
\]\[\varprojlim_{A_{/b}} \bigl(F \mid A_{/b}\bigr) \overset{\text{iso}}{\Longrightarrow} \varprojlim_{A_b} F \mid A_{b}\]
LaTeX source
\[
\varprojlim_{A_{/b}} \bigl(F \mid A_{/b}\bigr) \overset{\text{iso}}{\Longrightarrow} \varprojlim_{A_b} F \mid A_{b}
\]