Cote n° 146 · pages 2–132 · 39 commutative diagrams · Groupoïdes de Teichmüller et les S0 Tg,!ν, S0 TG : notes manuscrites (s.d.).
Inventory dating : [à partir de 1978]
Édition de démonstration

batch 1 · p. 2 — read it beside the facsimile1 / 39
LaTeX source
\begin{tikzcd}
\{\pm 1\} \arrow[rr] \arrow[dr, "\mathrm{diag}"'] & & \mathbb{U}^I \\
& \{\pm 1\}^I \arrow[ur, "\text{déduit de l'inclusion } \{\pm1\} \hookrightarrow \mathbb{U}"'] &
\end{tikzcd}
batch 1 · p. 6 — read it beside the facsimile2 / 39
LaTeX source
\begin{tikzcd}
0 \arrow[r] & \mathbb{Z} \arrow[r, "2"] \arrow[d, "\mathrm{diag}"'] & \mathbb{Z} \arrow[r] \arrow[d, "\mathrm{diag}"] & \mathbb{Z}/2\mathbb{Z} \arrow[r] \arrow[d, "\mathrm{diag}"] & 0 \\
0 \arrow[r] & \mathbb{Z}^I \arrow[r, "2"] & \mathbb{Z}^I \arrow[r] & (\mathbb{Z}/2\mathbb{Z})^I \arrow[r] & 0
\end{tikzcd}
batch 1 · p. 7 — read it beside the facsimile3 / 39
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\begin{tikzcd}
0 \arrow[r] & \mathbb{Z} \arrow[r, "2"] \arrow[d] & \mathbb{Z} \arrow[r] \arrow[d] & \mathbb{Z}/2\mathbb{Z} \arrow[r] \arrow[d, "\wr"] & 0 \\
0 \arrow[r] & \mathbb{Z}^I \arrow[r] & \mathcal{E}_I \arrow[r] & \mu_2 \arrow[r] & 0
\end{tikzcd}
batch 1 · p. 14 — read it beside the facsimile4 / 39
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\begin{tikzcd}[column sep=small]
\mathbb{Z}^S \arrow[rr] \arrow[dr, "\text{déduit par contravariance de } \vec{\hat A} \xrightarrow{\mathrm{or.}} S"'] & & \mathbb{Z}^{\hat A} \\
& \mathbb{Z}^{\vec{\hat A}} \arrow[ur, "\text{déduit par cov.\ de } \vec{\hat A} \to \hat A \simeq \vec{\hat A}/\sigma"'] &
\end{tikzcd}
batch 2 · p. 32 — read it beside the facsimile5 / 39
LaTeX source
\begin{tikzcd}[column sep=large]
Q \arrow[r, bend left=40, "\mu'"] \arrow[r, bend left=15, "\mu"'] & Q^\vee \arrow[l, bend left=15, "\check\mu"'] \arrow[l, bend left=40, "\check\mu'"]
\end{tikzcd}
batch 2 · p. 35 — read it beside the facsimile6 / 39
LaTeX source
\begin{tikzcd}[column sep=huge, row sep=large]
x \arrow[r, "\ell_{y,x}"] \arrow[d, leftrightarrow, "L_{x,\varphi_\omega(x)}"'] & y \arrow[d, leftrightarrow, "L_{y,\varphi_\omega(y)}"] \\
\varphi_\omega(x) \arrow[r] & \varphi_\omega(y)
\end{tikzcd}
batch 2 · p. 37 — read it beside the facsimile7 / 39
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\begin{tikzcd}[column sep=huge, row sep=large]
\varphi_\omega(x) \arrow[r, "\varphi_\omega(\ell)"] \arrow[d, "L_{x,\varphi_\omega(x)}"'] & \varphi_\omega(y) \arrow[d, "L_{y,\varphi_\omega(y)}"] \\
x \arrow[r, "\ell"] & y
\end{tikzcd}
batch 2 · p. 39 — read it beside the facsimile8 / 39
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\begin{tikzcd}[column sep=huge, row sep=large]
\varphi_\omega(x) \arrow[r, "\ell_{\varphi_\omega(x)}"] \arrow[d, "L_{x,\varphi_\omega(x)}"'] & -\varphi_\omega(x) \arrow[d, "L_{-x,\varphi_\omega(x)}"] \\
x \arrow[r, "\ell_x"] & -x
\end{tikzcd}
batch 3 · p. 51 — read it beside the facsimile9 / 39
LaTeX source
\begin{tikzcd}
  P\Sigma \arrow[r, "\sim", "(23)"'] \arrow[d, "\text{proj.\ can.}"']
    & V^{*} \arrow[d, "\text{proj.\ can.}"] \\
  \Sigma \arrow[r, "\sigma"] & \Sigma \simeq V^{*}/\mathbb{R}^{*}
\end{tikzcd}
batch 4 · p. 66 — read it beside the facsimile10 / 39
LaTeX source
\begin{tikzcd}[column sep=large]
X_2 \arrow[r, "z \mapsto z^2"] & X_1 \arrow[r] & X_0
\end{tikzcd}
batch 4 · p. 66 — read it beside the facsimile11 / 39
LaTeX source
\begin{tikzcd}
X_1 \arrow[r, Rightarrow] & \mathbb{X}_1 \arrow[r, "z \mapsto z^2"] & \mathbb{X}_0 \arrow[r, "\psi"] & \mathbb{X}_0
\end{tikzcd}
batch 4 · p. 67 — read it beside the facsimile12 / 39
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\begin{tikzcd}[column sep=small]
M_{0,5} \arrow[rr, hook] \arrow[dr, hook] & & \hat{M}_{0,5} \\
& \hat{M}'_{0,5} \arrow[ur, hook] &
\end{tikzcd}
batch 4 · p. 67 — read it beside the facsimile13 / 39
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\begin{tikzcd}
\hat{\mathcal{X}}_{0,4} \arrow[d] & \hat{\mathcal{X}}_{0,4} \times \hat{\mathcal{X}}_{0,4} \arrow[l] \arrow[d] \\
\hat{M}_{0,4} \arrow[r, no head, "\supset" description] & \hat{\mathcal{X}}_{0,4} \arrow[u, bend right=40, "{\sigma_i \; (i \leqslant 4), \; \delta}"']
\end{tikzcd}
batch 4 · p. 73 — read it beside the facsimile14 / 39
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\begin{tikzcd}
\mathcal{T}_{0,I} \arrow[d, "p_1"] & M_{0,K} \arrow[d] \\
\mathcal{T}_{0, I \setminus \{i_1\}} & M_{0,I}
\end{tikzcd}
batch 4 · p. 73 — read it beside the facsimile15 / 39
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\begin{tikzcd}
M^!_{0,5} \arrow[d] & \mathcal{U}_{3} \times \mathcal{U}_{0,3} \arrow[d] \\
M^!_{0,4} \simeq \mathcal{U}_{0,3} & \mathcal{U}_{0,3}
\end{tikzcd}
batch 4 · p. 76 — read it beside the facsimile16 / 39
LaTeX source
\begin{tikzcd}[row sep=small, column sep=large]
& \mathfrak{F} \\
\mathcal{R} \arrow[ur] \arrow[r] \arrow[dr] & \mathcal{A} \\
& \mathcal{S}
\end{tikzcd}
batch 5 · p. 84 — read it beside the facsimile17 / 39
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\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\tilde{s}_{i'_n} \arrow[r, "u'_{i'_n}"] & \tilde{t}_{i'_n} \arrow[r, "M_{\alpha,i_n}((a_j))"] & \tilde{s}_{i'_{n-1}} \arrow[r, "u'_{i'_{n-1}}"] & \tilde{t}_{i'_{n-1}} \arrow[r, "M_{\alpha,i_{n-1}}((a_j))"] & \cdots
\end{tikzcd}
batch 5 · p. 84 — read it beside the facsimile18 / 39
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\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\cdots & \tilde{s}_{i'_2} \arrow[r] & \tilde{t}_{i'_2} \arrow[r, "M_{\alpha,2}((a_j))"] & \tilde{s}_{i'_1} \arrow[r, "u'_{i'_1}"] & \tilde{t}_{i'_1}
\end{tikzcd}
batch 5 · p. 90 — read it beside the facsimile19 / 39
LaTeX source
\begin{tikzcd}
C' \arrow[r] \arrow[d, "f"'] & \Pi_1(P) \arrow[d] \\
C \arrow[r] & \Pi_1(T)
\end{tikzcd}
batch 5 · p. 95 — read it beside the facsimile20 / 39
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\begin{tikzcd}
E \arrow[r] \arrow[d] & P \arrow[d, "q"] \\
E/\mu = E_0 \arrow[r] & P_0 = P/\mu
\end{tikzcd}
batch 5 · p. 98 — read it beside the facsimile21 / 39
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
 & P \arrow[d, no head] & U \arrow[d] \\
E_0 \arrow[r] & P/\mu = P_0 \arrow[d] & U_0 = U/\mu \\
 & T &
\end{tikzcd}
batch 6 · p. 103 — read it beside the facsimile22 / 39
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\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
SP_G \arrow[r] \arrow[d, "\mathbb{G}_m^{I}"'] & SP_{G'} \arrow[r] \arrow[d, "\mathbb{G}_m^{I}"'] & SP_{G^{\natural}} \arrow[d, "\mathbb{G}_m^{I}"'] \\
P_G \, (\simeq M_{G,G^{\natural}}) \arrow[r] \arrow[d, "\mathbb{G}_m^{A'}"'] & P_{G'} \simeq M_{G',G^{\natural}} \arrow[r] \arrow[d, "\mathbb{G}_m^{A'}"'] & P_{G^{\natural}} = RP_{G^{\natural}} = M_{G^{\natural}} \\
P_{G,C} \simeq M_{G,G'} \arrow[r] \arrow[d, "\mathbb{G}_m^{C}"'] & RP_{G'} = M_{G'} & \\
RP_G = M_G & &
\end{tikzcd}
batch 6 · p. 103 — read it beside the facsimile23 / 39
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
S\mathfrak{T}_G \arrow[r] \arrow[d, "\mathbb{Z}^{I}"'] & S\mathfrak{T}_{G'} \arrow[r] \arrow[d, "\mathbb{Z}^{I}"'] & S\mathfrak{T}_{G^{\natural}} \arrow[d, "\mathbb{Z}^{I}"'] \\
\mathfrak{T}_G \arrow[r] \arrow[d, "\mathbb{Z}^{A'}"'] & \mathfrak{T}_{G'} \arrow[r] \arrow[d, "\mathbb{Z}^{A'}"'] & \mathfrak{T}_{G^{\natural}} \\
\mathfrak{T}_{G,C} \arrow[r] \arrow[d, "\mathbb{Z}^{C}"'] & R\mathfrak{T}_{G'} & \\
R\mathfrak{T}_G & &
\end{tikzcd}
batch 6 · p. 103 — read it beside the facsimile24 / 39
LaTeX source
\begin{tikzcd}
SRP_G \arrow[d, "\mathbb{G}_m^{I}"'] & SP_G \arrow[l, "\mathbb{G}_m^{A}"'] \arrow[d, "\mathbb{G}_m^{I}"] \\
RP_G & P_G \arrow[l, "\mathbb{G}_m^{A}"]
\end{tikzcd}
batch 6 · p. 103 — read it beside the facsimile25 / 39
LaTeX source
\begin{tikzcd}
SR\mathfrak{T}_G \arrow[d, "\mathbb{Z}^{I}"'] & S\mathfrak{T}_G \arrow[l, "\mathbb{Z}^{A}"'] \arrow[d, "\mathbb{Z}^{I}"] \\
R\mathfrak{T}_G & \mathfrak{T}_G \arrow[l, "\mathbb{Z}^{A}"]
\end{tikzcd}
batch 6 · p. 107 — read it beside the facsimile26 / 39
LaTeX source
\begin{tikzcd}[column sep=small]
1 \arrow[r] & \pi_1(\mathbb{U}^{C}) \arrow[r] \arrow[d, no head, "\wr"'] & \pi_1(\widetilde{M}^C_G) \arrow[r] \arrow[dd] & \pi_1(M_G) \arrow[r] \arrow[d, no head, "\wr"] & 1 \\
 & \mathbb{Z}^{C} & & \pi_1(M^C_G) & \\
 & & \pi_1(M_{G'}) & &
\end{tikzcd}
batch 6 · p. 108 — read it beside the facsimile27 / 39
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
\widetilde{M}^C_G \arrow[rr] \arrow[dd] & & M^C_G \arrow[dd] & \\
 & \widetilde{P}^B_G \arrow[ul] \arrow[rr, dashed] \arrow[dd, dashed] & & P^B_G \arrow[ul, dashed] \arrow[dd, dashed] \\
P_C \arrow[rr] & & M_G & \\
 & SP_G \arrow[ul] \arrow[rr] & & P_B \arrow[ul]
\end{tikzcd}
batch 6 · p. 109 — read it beside the facsimile28 / 39
LaTeX source
\begin{tikzcd}
\widetilde{P}^B_G \arrow[r] \arrow[d, "\text{$\mathbb{U}^B$-tors. sur $\widetilde{M}^C_G$}"'] & SP_{G'} \arrow[d] \\
\widetilde{M}^C_G \arrow[r] & M_{G'}
\end{tikzcd}
batch 6 · p. 109 — read it beside the facsimile29 / 39
LaTeX source
\begin{tikzcd}
\pi_1(\widetilde{P}^B_G) \arrow[r] \arrow[d, no head, "\wr"'] & \pi_1(SP_{G'}) \arrow[d, no head, "\wr"] \\
S\mathfrak{T}_G & S\mathfrak{T}_{G'}
\end{tikzcd}
batch 6 · p. 110 — read it beside the facsimile30 / 39
LaTeX source
\begin{tikzcd}[column sep=small]
\mu_{\vec{C}} \times \widetilde{M}^C_G \arrow[rr] \arrow[dr, dashed] & & M_{G'} \\
 & \widetilde{M}^C_G \arrow[ur, dashed, no head] &
\end{tikzcd}
batch 6 · p. 110 — read it beside the facsimile31 / 39
LaTeX source
\begin{tikzcd}
\mu_{\vec{C}} \times P^B_G \arrow[r] \arrow[d, "\sim"'] & SP_{G'} \arrow[d] \\
\mu_{\vec{C}} \times M^C_G \arrow[r] & M_{G'}
\end{tikzcd}
batch 6 · p. 110 — read it beside the facsimile32 / 39
LaTeX source
\begin{tikzcd}
SP_G \times \mu_C \arrow[r] \arrow[d] & SP_{G'} \arrow[d] \\
P_C \times \mu_C \arrow[r] & M_{G'}
\end{tikzcd}
batch 6 · p. 115 — read it beside the facsimile33 / 39
LaTeX source
\begin{tikzcd}
\xi \arrow[r, "u_{\alpha,\xi}"] \arrow[rr, bend right=30, "u_{\alpha\beta,\xi}\, n{(\alpha,\beta)}"'] & \alpha.\xi = \xi' \arrow[r, "u_{\beta,\xi'}"] & \beta\xi' = \beta\alpha\xi = \xi''
\end{tikzcd}
batch 6 · p. 115 — read it beside the facsimile34 / 39
LaTeX source
\begin{tikzcd}
\xi \arrow[r, "u_{a,\xi}"] \arrow[d, "u_{b,\xi}"'] & \sigma_a \xi \arrow[d, "u_{b,\sigma_a\xi}"] \\
\sigma_b \xi \arrow[r, "u_{a,\sigma_b\xi}"'] & \sigma_{ab}
\end{tikzcd}
batch 6 · p. 116 — read it beside the facsimile35 / 39
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\begin{tikzcd}
\xi \arrow[r, "u_{a,\xi}"] & a\xi \arrow[r, "u_{a,a\xi}"] & a^2 \xi = \xi
\end{tikzcd}
batch 6 · p. 116 — read it beside the facsimile36 / 39
LaTeX source
\begin{tikzcd}
S_0\mathfrak{T}_G \arrow[r] \arrow[d] & S\mathfrak{T}_{G^{\natural}} \arrow[d] \\
{}_0\mathfrak{T}_G \arrow[r] & \mathfrak{T}_{G^{\natural}}
\end{tikzcd}
batch 7 · p. 130 — read it beside the facsimile37 / 39
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
S \arrow[rr] \arrow[dr, "g"'] & & S' \arrow[dl, "g'"] \\
& \mathbb{N} &
\end{tikzcd}
batch 7 · p. 130 — read it beside the facsimile38 / 39
LaTeX source
\begin{tikzcd}
\hat{\vec{A}}{}' \arrow[r, "u_1"] \arrow[d, "\sigma'"'] & \hat{\vec{A}} \arrow[d, "\sigma"] \\
S' \arrow[r, "u_0"'] & S
\end{tikzcd}
batch 7 · p. 132 — read it beside the facsimile39 / 39
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\begin{tikzcd}[column sep=small, row sep=small]
S' \arrow[rr, "u_0"] \arrow[dr, "g'"'] & & S \arrow[dl, "g"] \\
& \mathbb{N} &
\end{tikzcd}