\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\mathbb{Z}(\frac{\omega}{2}) \arrow[r] \arrow[d, Rightarrow] & \widetilde{\Pi}^{\circ}_{0,3} \arrow[r, Rightarrow] \arrow[d] & \Pi_{0,3} \arrow[d, hook] & \\
\mathbb{Z}(\omega) \arrow[r] \arrow[d, no head, "\parallel" description] & \widetilde{\Pi}'_{0,3} \arrow[r] \arrow[d] & \Pi'_{0,3} = \Pi_{0,3} \times \{1, \dot\omega\} \arrow[r] \arrow[d] & \Pi_{03} \arrow[d, hook] \\
\mathbb{Z} \arrow[r] \arrow[d, no head, "\parallel" description] & \widetilde{\mathrm{SL}}(2, \mathbb{Z}) = \widetilde{\Pi}^{D}_{0,3} \arrow[r] \arrow[d] & \mathrm{SL}(2, \mathbb{Z}) \arrow[r] \arrow[d] & \Pi^{D}_{0,3} \arrow[d] \\
\mathbb{Z} \arrow[r] \arrow[d, no head, "\parallel" description] & \widetilde{\mathbb{D}}_3 \arrow[r] \arrow[d] & \mathbb{D}'''_3 \arrow[r, "2"] \arrow[d, "3"] & \mathbb{D}''_3 \arrow[d, "3"] \\
\mathbb{Z} \arrow[r, "12"] & \mathbb{Z} \arrow[r] & \mathbb{Z}/12\mathbb{Z} \arrow[r, "2"] & \mathbb{Z}/6\mathbb{Z}
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & 0 & 0 & 0 & 0 \\
0 \arrow[r] & \Phi(X) / (\mathbf{G}_m(X)/U) \arrow[r] \arrow[u] & \Gamma(X, \underline{\Phi}_X / (\underline{\mathcal{O}}^*_X / \underline{U}_X)) \arrow[r] \arrow[u] & \mathrm{Pic}(X) \arrow[r] \arrow[u] & H^1(X, \underline{\Phi}_X) \arrow[u] \\
0 \arrow[r] & \Phi(X) \arrow[r] \arrow[u] & QF(X) \arrow[r] \arrow[u] & \mathrm{Div}(X) \arrow[r] \arrow[u] & H^1(X, \underline{\Phi}_X) \arrow[u, no head, "\Vert" description] \\
0 \arrow[r] & \mathbf{G}_m(X)/U \arrow[r] \arrow[u] & R(X)^*/U \arrow[r] \arrow[u] & \mathrm{Div}_{\ell}(X) \arrow[r] \arrow[u] & 0 \arrow[u] \\
 & 0 \arrow[u] & 0 \arrow[u] & 0 \arrow[u] &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
G \arrow[d] & \Omega S \arrow[l] \arrow[d] & \Omega Y \arrow[l] \arrow[d] &
\Omega G \arrow[l] \arrow[d] & \Omega^{2} S \arrow[l] \arrow[d] &
\Omega^{2} Y \arrow[l] \arrow[d] & \Omega^{2} G \arrow[l] \arrow[d] \\
e \arrow[d] & F \arrow[l] \arrow[d] & F \arrow[l, Rightarrow] \arrow[d] &
e \arrow[l] \arrow[d] & \Omega F \arrow[l] \arrow[d] &
\Omega F \arrow[l, Rightarrow] \arrow[d] & e \arrow[l] \arrow[d] \\
X \arrow[d] & Z \arrow[l] \arrow[d] & G \arrow[l] \arrow[d] &
\Omega X \arrow[l] \arrow[d] & \Omega Z \arrow[l] \arrow[d] &
\Omega G \arrow[l] \arrow[d] & \Omega^{2} X \arrow[l] \arrow[d] \\
S & Y \arrow[l] & G \arrow[l] & \Omega S \arrow[l] & \Omega Y \arrow[l] &
\Omega G \arrow[l] & \Omega^{2} S \arrow[l]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
S \arrow[d] & X' \arrow[l] \arrow[d] & F' \arrow[l] \arrow[d] &
\Omega S \arrow[l] \arrow[d] & \Omega X' \arrow[l] \arrow[d] &
\Omega F' \arrow[l] \arrow[d] & \Omega^{2} S \arrow[l] \arrow[d] \\
F \arrow[d] & G \arrow[l] \arrow[d] & \Phi \arrow[l] \arrow[d] &
\Omega F \arrow[l] \arrow[d] & \Omega G \arrow[l] \arrow[d] &
\Omega\Phi \arrow[l] \arrow[d] & \phantom{e} \\
X \arrow[d] & Z \arrow[l] \arrow[d] & G' \arrow[l] \arrow[d] &
\Omega X \arrow[l] \arrow[d] & \Omega Z \arrow[l] \arrow[d] &
\Omega G' \arrow[l] & \phantom{e} \\
S & X' \arrow[l] & F' \arrow[l] & \Omega S \arrow[l] &
\Omega X' \arrow[l] & \Omega F' \arrow[l] & \phantom{e}
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & & & & e \arrow[d] & \Omega^{3} Z \arrow[l] \arrow[d, "\Omega^{3} g"] & \Omega^{4} Y \arrow[l, "\Omega^{3} h"'] \arrow[d] \\
 & & & e \arrow[d] & \Omega^{3} Y \arrow[l] \arrow[d, "\Omega^{2} h"] & \Omega^{3} X \arrow[l, "\Omega^{3} f"'] \arrow[d] & e \arrow[l] \\
 & & e \arrow[d] & \Omega^{2} X \arrow[l] \arrow[d, "\Omega^{2} f"] & \Omega^{2} Z \arrow[l, "\Omega^{2} g"'] \arrow[d] & e \arrow[l] & \\
 & e \arrow[d] & \Omega Z \arrow[l] \arrow[d, "\Omega g"] & \Omega^{2} Y \arrow[l, "\Omega h"'] \arrow[d] & e \arrow[l] & & \\
e \arrow[d] & \Omega Y \arrow[l] \arrow[d, "h"] & \Omega X \arrow[l, "\Omega f"'] \arrow[d] & e \arrow[l] & & & \\
X \arrow[d, "f"'] & Z \arrow[l, "g"] \arrow[d] & e \arrow[l] & & & & \\
Y & e \arrow[l] & & & & &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
  T \arrow[r, no head, "\mathfrak{z}"] \arrow[d, no head, "T/\mu"'] & \mathfrak{z}(\mu) \arrow[r, no head, "(\mathbb{Z}/h\mathbb{Z})^{*}"] \arrow[d, no head, "T/\mu"] & N(\mu) \arrow[r, no head] \arrow[d, no head, "T/\mu"] & N \arrow[d, no head] & \\
  T \cap N' \arrow[r, no head, "\mathfrak{z}"] \arrow[d, no head, "0"'] & \mathfrak{z}(\mu) \cap N' \arrow[r, no head, "(\mathbb{Z}/h\mathbb{Z})^{*}"] \arrow[d, no head, "0"] & N(\mu) \cap N' \arrow[r, no head, "="] \arrow[d, no head, "(\mathbb{Z}/h\mathbb{Z})^{*}"] & N \cap N' \arrow[r, no head] \arrow[d, no head, "(\mathbb{Z}/h\mathbb{Z})^{*}"] & N' \arrow[d, no head] \\
  T \cap N(D) \arrow[r, no head, "\mathfrak{z}"] \arrow[d, no head, "\mu"'] & \mathfrak{z}(\mu) \cap N(D) \arrow[r, no head, "0"] \arrow[d, no head, "\mu"] & N(\mu) \cap N(D) \arrow[r, no head, "="] \arrow[d, no head, "\mu"] & N \cap N(D) \arrow[r, no head, "T'/\mathfrak{z}"] \arrow[d, no head, "\mu"] & N(D) = \mu \cdot T' \arrow[d, no head, "\mu"] \\
  T \cap T' \arrow[r, no head, "\mathfrak{z}"] & \mathfrak{z}(\mu) \cap T' \arrow[r, no head, "="] & N(\mu) \cap T' \arrow[r, no head, "="] & N \cap T' \arrow[r, no head, "T'/\mathfrak{z}"] & T'
\end{tikzcd}

\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}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
  & & & e \arrow[dl, no head, "1"'] \arrow[d, no head, "1"] \arrow[dr, no head, "1"] & \\
  & & t \arrow[dl, no head, "r-1"'] \arrow[d, no head] \arrow[r, no head, "="] & t \arrow[dl, no head] \arrow[dr, no head, "1"] & b_u \arrow[d, no head, "1"] \arrow[dr, no head, "r-1"] & \\
  N_1 \arrow[r, no head, "\circ"] \arrow[d, no head] & T \arrow[d, no head, "\mathbb{Z}/2\mathbb{Z}"] & n \arrow[dr, no head, "2"'] \arrow[dl, no head, "r-1"'] & & b \arrow[dl, no head, "1"] \arrow[dr, no head, "r-1"] & Z(b_u) \arrow[d, no head, "1(\mathbb{G}_m)"] \\
  N \arrow[r, no head] & N_0 & & H & & N(b_u)
\end{tikzcd}

\begin{tikzcd}[column sep=tiny, row sep=small, nodes={font=\scriptsize}]
    1 \arrow[dr] & & 1 \arrow[dr] & & & & & & \\
    1 \arrow[r] & \widehat{\pi}_{g,\nu} \arrow[rr, "\sim"] \arrow[dr] & &
      \widehat{\pi}_{g,\nu} \arrow[rr] \arrow[dr] & & 1 \arrow[dr] & & & \\
    & 1 \arrow[r] & \widehat{\mathcal{T}}_{g,\nu+1} \arrow[rr] \arrow[dr] & &
      \mathrm{Aut}_{\mathrm{lac}}^{\circ}(\widehat{\pi}_{g,\nu})^{!} \arrow[rr] \arrow[dr] & &
      \Gamma \arrow[r] \arrow[dr, no head, "=" description] & 1 & \Sigma_{\nu+1} \\
    & & 1 \arrow[r] & \widehat{\mathcal{T}}_{g,\nu} \arrow[rr] \arrow[dr] & &
      \mathrm{Autext}_{\mathrm{lac}}^{\circ}(\widehat{\pi}_{g,\nu})^{!} \arrow[rr] \arrow[dr] & &
      \Gamma \arrow[r] & 1 \quad \Sigma_\nu \\
    & & & & 1 & & 1 & &
  \end{tikzcd}

\begin{tikzcd}[column sep=tiny, row sep=small, nodes={font=\scriptsize}]
 & \mathrm{SL}(2, \mathbb{Z}) \arrow[r] \arrow[d] & \Pi^{D}_{0,3} \arrow[d] & & \\
\widetilde{E} \arrow[r] \arrow[d] \arrow[dr, dashed] & \mathbb{D}_3 \times_{\mathbb{Z}/12\mathbb{Z}} \mathbb{Z}/12\mathbb{Z} \simeq \mathbb{D}'_3 \times_{\mathbb{Z}/12\mathbb{Z}} \mathbb{Z}/6\mathbb{Z} \arrow[r] & \mathbb{D}_3 \times_{\mathbb{Z}/12\mathbb{Z}} \mathbb{Z}/6\mathbb{Z} = E & & \\
\widetilde{\mathbb{D}}_3 \arrow[dr, Rightarrow] & \mathbb{Z} \arrow[r, dashed] & \mathbb{Z}/12\mathbb{Z} \arrow[r, dashed, "\mathrm{can}"] \arrow[d, dashed] & \mathbb{Z}/6\mathbb{Z} \arrow[d, "\mathrm{can}"] & \\
 & \mathbb{D}'_3 \arrow[r, Rightarrow] \arrow[d] & \mathbb{D}_3 \arrow[r, "\mathrm{can}"] & \mathbb{Z}/2\mathbb{Z} & \\
\mathbb{Z} \arrow[r] & \mathbb{Z}/4\mathbb{Z} \arrow[urr, "\mathrm{can}"'] & & &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
X_{20} \arrow[d] & & & & & & & & & \\
X_{10} \arrow[d] & & & & & & & & & \\
X_{00} & X_{01} \arrow[l] & X_{02} \arrow[l] &
\Omega X_{00} \arrow[l] \arrow[d] & \Omega X_{01} \arrow[l] \arrow[d] &
\Omega X_{02} \arrow[l] \arrow[d] & \Omega^{2} X_{00} \arrow[l] \arrow[d] &
& & \\
& & & X_{20} \arrow[d] & X_{21} \arrow[l] \arrow[d] &
X_{22} \arrow[l] \arrow[d] & \Omega X_{20} \arrow[l] \arrow[d] & & & \\
& & & X_{10} \arrow[d] & X_{11} \arrow[l] \arrow[d] &
X_{12} \arrow[l] \arrow[d] & \Omega X_{10} \arrow[l] \arrow[d] & & & \\
& & & X_{00} & X_{01} \arrow[l] & X_{02} \arrow[l] &
\Omega X_{00} \arrow[l] \arrow[d] & & & \\
& & & & & & X_{20} \arrow[d] & & & \\
& & & & & & X_{10} \arrow[d] & & & \\
& & & & & & X_{00} & X_{01} \arrow[l] & X_{02} \arrow[l] &
\Omega X_{00} \arrow[l]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & & T = \partial M & & t \arrow[ll] \\
 & W & Q \arrow[l, hook'] \arrow[u, "2"'] & & N = D_{1} \sim e \arrow[ll, hook'] \arrow[u, no head] \\
\partial N = S^{1} & P \arrow[l] \arrow[u, leftrightarrow] & S^{1} \times T \arrow[l, hook'] \arrow[u, hook] & & \partial N = S^{1} \arrow[ll, leftrightarrow] \arrow[u, hook] \\
s \arrow[u] & M \arrow[l] \arrow[u, hook] & \partial M = T \arrow[l, hook'] \arrow[u, leftrightarrow] & & (s, t) \arrow[ll] \arrow[u, hook]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
1 \arrow[r] & \overset{\omega' = \varpi^{-1}}{\mathbb{Z}} \arrow[r] \arrow[d, no head, "\parallel" description] & S\Pi^{D}_{0,3} \arrow[r] \arrow[d] & \Pi^{D}_{03} \arrow[r] \arrow[d, hook] & 1 \\
1 \arrow[r] & \mathbb{Z} \arrow[r] & S\Pi^{D}_{03}(4) \arrow[r] & \Pi^{D}_{03} \times \mathbb{Z}/4\mathbb{Z} \arrow[r] & 1 \\
1 \arrow[r] & \underset{\omega^2}{\mathbb{Z}} \arrow[r] \arrow[u] & S\mathcal{T}_{11} \arrow[r] \arrow[u] & \mathcal{T}_{11} \arrow[u, leftrightarrow] &
\end{tikzcd}

\begin{tikzcd}[column sep=tiny, row sep=small, nodes={font=\scriptsize}]
    1 \arrow[dr] & & 1 \arrow[dr] & & & & & & \\
    1 \arrow[r] & \pi_1(\overline{X}_\xi \smallsetminus S_\nu) \arrow[rr] \arrow[dr] & &
      \pi_1(\overline{X}_\xi \smallsetminus S_\nu) \arrow[rr] \arrow[dr] & & 1 \arrow[dr] & & & \\
    & 1 \arrow[r] & \pi_1(\overline{M}_{g,\nu+1}) \arrow[rr] \arrow[dr] & &
      \pi_1(M_{g,\nu+1}) \arrow[rr] \arrow[dr] & & \Gamma \arrow[r] \arrow[dr] & 1 & \\
    & & 1 \arrow[r] & \pi_1(\overline{M}_{g,\nu}) \arrow[rr] \arrow[dr] & &
      \pi_1(M_{g,\nu}) \arrow[rr] \arrow[dr] & & \Gamma \arrow[r] \arrow[dr] & 1 \\
    & & & & 1 & & 1 & & 1
  \end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & \prod_{\sigma/\mathbb{Z}_p} \mathbb{G}_m \arrow[r] & T_{\sigma_K} \arrow[r] & \mathbb{Z} \arrow[r] & 0 \\
0 \arrow[r] & g_\sigma \arrow[r] & N \arrow[r] & \mathbb{Z} \arrow[r] & 0 \\
0 \arrow[r] & g_\sigma \arrow[u, Rightarrow] \arrow[r] & \tilde{N} \arrow[u] \arrow[r] & \mathbb{Z}_\sigma \arrow[u] \arrow[r] & 0 \\
0 \arrow[r] & g_K \arrow[r] & \tilde{N}_K \arrow[r] & \mathbb{Z}_K \arrow[r] & 0 \\
 & & g_K \times \mathbb{Z}_K \arrow[u, no head, "\simeq" description] & &
\end{tikzcd}

\begin{tikzcd}[column sep=tiny, row sep=small, nodes={font=\scriptsize}]
(\widehat{B}_{\to}) & & & & & & (\widehat{A}_{\to}) \\
& \underline{\mathrm{Hom}}^{!}(\widehat{B}^{\circ},\widehat{A}) \arrow[rr, bend left=22] \arrow[dd, bend right=22] & & \underline{\mathrm{Hom}}^{!!}(\widehat{A}^{\circ},\widehat{B}^{\circ};\,\mathrm{Ens}) & & \underline{\mathrm{Hom}}^{!}(\widehat{A}^{\circ},\widehat{B}) \arrow[ll, bend right=22] \arrow[dd, bend left=22] \arrow[dl] & \\
& & \underline{\mathrm{Hom}}(B^{\circ},\widehat{A}) \arrow[ul] & & \underline{\mathrm{Hom}}(A^{\circ},\widehat{B}) & & \\
& \underline{\mathrm{Hom}}_{!}^{!}(A^{\vee},\widehat{B}^{\circ};\,\mathrm{Ens}) & & \substack{(A\times B)^{\wedge} \\ =\, \underline{\mathrm{Hom}}(A^{\circ},B^{\circ};\,\mathrm{Ens})} \arrow[ul] \arrow[ur] \arrow[dl] \arrow[dr] \arrow[uu, no head, "\wedge\wedge" description] \arrow[ll, no head, "\vee\wedge" description] \arrow[rr, no head, "\wedge\vee" description] \arrow[dd, no head, "\vee\vee" description] & & \underline{\mathrm{Hom}}^{!}_{!}(\widehat{A}^{\circ},B^{\vee};\,\mathrm{Ens}) & \\
& & \underline{\mathrm{Hom}}(A^{\circ},\widehat{B}) \arrow[dl] & & \underline{\mathrm{Hom}}(B^{\circ},\widehat{A}) \arrow[dr] & & \\
& \underline{\mathrm{Hom}}_{!}(A^{\vee},\widehat{B}) \arrow[uu, bend left=22] \arrow[rr, bend right=22] & & \underline{\mathrm{Hom}}_{!!}(A^{\vee},B^{\vee};\,\mathrm{Ens}) & & \underline{\mathrm{Hom}}_{!}(B^{\vee},\widehat{A}) \arrow[uu, bend right=22] \arrow[ll, bend left=22] & \\
(A^{\vee}_{\to}) & & & & & & (B^{\vee}_{\to})
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\mathbb{Z} \arrow[r] \arrow[d, no head, "\parallel" description] & \widetilde\Pi \arrow[r] \arrow[d] & \Pi' \arrow[r] \arrow[d] \arrow[ddr, bend left=20] & \Pi \arrow[d] \arrow[dr] & \\
\mathbb{Z} \arrow[r] \arrow[d, no head, "\parallel" description] & \widetilde{\mathbb{D}}_3 \arrow[r] \arrow[d] & \mathbb{D}'''_3 \arrow[r] \arrow[d] \arrow[dr] & \mathbb{D}''_3 \arrow[d] \arrow[r, Rightarrow] & \mathbb{D}_3 \arrow[dd] \\
\mathbb{Z} \arrow[r, "12"] & \mathbb{Z} \arrow[r] & \mathbb{Z}/12\mathbb{Z} \arrow[d] & \mathbb{D}'_3 \arrow[ur] \arrow[d, no head] & \\
 & & \mathbb{Z}/4\mathbb{Z} \arrow[rr] & \mathbb{Z}/6\mathbb{Z} \arrow[r] & \mathbb{Z}/2\mathbb{Z}
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & & 0 & & \\
0 \arrow[r, dashed] & \underline{\Phi}_X / \mathrm{Im}\, \underline{\mathcal{O}}^*_X \arrow[r, dashed, "\simeq"] & \underline{\Phi}_X / \mathrm{Im}\, \underline{\mathcal{O}}^*_X \arrow[u] \arrow[r, dashed] & 0 & \\
0 \arrow[r] & \underline{\Phi}_X \arrow[r] \arrow[u, dashed] & \underline{QF}_X \arrow[r] \arrow[u] & \underline{\mathrm{Div}}_X \arrow[u, dashed, "\wr"] & \\
0 \arrow[r] & \underline{\mathcal{O}}^*_X / \underline{U}_X \arrow[r] \arrow[u] & \underline{R}^*_X / \underline{U}_X \arrow[r, dashed] \arrow[u] & \underline{\mathrm{Div}}_X \arrow[r, dashed] \arrow[u, dashed, "\wr"] & 0 \\
 & 0 \arrow[u] & 0 \arrow[u] & 0 \arrow[u, dashed, "\wr"] &
\end{tikzcd}

\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}

\begin{tikzcd}[column sep=small, row sep=normal, nodes={font=\scriptsize}]
  0 \arrow[r] & \Gamma^{n}\Phi_{*} \arrow[r] \arrow[d, "u_{0}"] & \Gamma^{n-1}\Phi_{*} \otimes \Psi_{*} \arrow[r] \arrow[d, "u_{1}"] & \Gamma^{n-2}\Phi_{*} \otimes \Lambda^{2}\Psi_{*} \arrow[r] \arrow[d, "u_{2}"] & \cdots \arrow[r] & \Lambda^{n}\Psi_{*} \arrow[r] \arrow[d, "u_{n}"] & 0 \arrow[d] \\
  0 \arrow[r] & \Phi_{*} \arrow[r, "\wedge T"'] & \Lambda^{2}\Phi_{*} \arrow[r, "\wedge T"'] & \Lambda^{3}\Phi_{*} & \cdots \arrow[r, "\wedge T"'] & \Lambda^{n+1}\Phi_{*} \arrow[r, "\wedge T"'] & \Lambda^{n+2}\Phi_{*}
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
H^1(X, \mu_n) \arrow[r, "\alpha"] \arrow[d] & {}_n\mathrm{Pic}(X) \arrow[d] & & \\
H^1(X_\eta, \mu_n)/(\mathbf{Z}/n\mathbf{Z}) \arrow[r, "\alpha_\eta"] \arrow[d] & {}_n\mathrm{Pic}(X_\eta) \arrow[r, "\beta_\eta"] \arrow[d] & {}_nP(K) \arrow[r, "\simeq"] \arrow[d] & {}_nP(\bar K)^I \arrow[dl, bend left=20] \\
H^1(X_{\bar\eta}, \mu_n) \arrow[r, "\alpha_{\bar\eta}"] & {}_n\mathrm{Pic}(X_{\bar\eta}) \arrow[r, "\beta_{\bar\eta}"] & {}_nP(\bar K) &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & & \mathbb{Z}/2\mathbb{Z} \arrow[d] & & \\
1 \arrow[r] & \Pi_{0,3} \arrow[r] \arrow[d, no head, "=" description] & \mathrm{GL}(2,\mathbb{Z}) \arrow[r] \arrow[d, "2"] & \mathbb{D}'_3 \times \{1, \tau\} \arrow[r] \arrow[d, "2"] & 1 \\
1 \arrow[r] & \Pi_{0,3} \arrow[r] & \mathrm{GL}(2,\mathbb{Z})' \arrow[r] \arrow[d, no head, "\wr" description] & \mathbb{D}_3 \times \{1, \tau\} \arrow[r] & 1 \\
 & & \pi_1(\uncertain{U}_{0,3}, \mathbb{D}_3 \times \{1, \tau\}; P_+) & &
\end{tikzcd}

\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}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
(A \otimes TA') \otimes (E \otimes TE') \arrow[r, "u \otimes v"]
\arrow[d, "a"'] &
(B \otimes TB') \otimes (F \otimes TF') \arrow[d, "a"] \\
((A \otimes TA') \otimes E) \otimes TE'
\arrow[d, "a^{-1} \otimes \mathrm{id}"'] &
((B \otimes TB') \otimes F) \otimes TF'
\arrow[d, "a^{-1} \otimes \mathrm{id}"] \\
(A \otimes (TA' \otimes E)) \otimes TE'
\arrow[d, "(\mathrm{id} \otimes c) \otimes \mathrm{id}"'] &
(B \otimes (TB' \otimes F)) \otimes TF'
\arrow[d, "(\mathrm{id} \otimes c) \otimes \mathrm{id}"] \\
(A \otimes (E \otimes TA')) \otimes TE'
\arrow[d, "a \otimes \mathrm{id}"'] &
(B \otimes (F \otimes TB')) \otimes TF'
\arrow[d, "a \otimes \mathrm{id}"] \\
((A \otimes E) \otimes TA') \otimes TE' \arrow[d, "a^{-1}"'] &
((B \otimes F) \otimes TB') \otimes TF' \arrow[d, "a^{-1}"] \\
(A \otimes E) \otimes (TA' \otimes TE')
\arrow[d, "\mathrm{id} \otimes \check{T}"'] &
(B \otimes F) \otimes (TB' \otimes TF')
\arrow[d, "\mathrm{id} \otimes \check{T}"] \\
(A \otimes E) \otimes T(A' \otimes E') \arrow[r, "W"'] &
(B \otimes F) \otimes T(B' \otimes F')
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
1 \arrow[r] & \mathbb{Z} \arrow[r] \arrow[dd, no head, "=" description] & S\mathcal{T}_{11} \arrow[rr, Rightarrow] \arrow[dd] & & \mathcal{T}_{11} \arrow[rr, Rightarrow] \arrow[dd] \arrow[dr, leftrightarrow] & & \Pi^{\mathbb{D}'}_{03} \arrow[dr, "\mathrm{can}"] \arrow[dd] & \\
 & & & & & \mathbb{D}'_3 \arrow[rr] \arrow[dd] & & \mathbb{D}_3 \arrow[dd, "\mathrm{can}"] \\
1 \arrow[r] & \mathbb{Z} \arrow[r, "12"] & \mathbb{Z} \arrow[rr, "\mathrm{can}"] & & \mathbb{Z}/12\mathbb{Z} \arrow[rr, "\mathrm{can}"] \arrow[dr, "\mathrm{can}"'] & & \mathbb{Z}/6\mathbb{Z} \arrow[dr, "\mathrm{can}"] & \\
 & & & & & \mathbb{Z}/4\mathbb{Z} \arrow[rr, "\mathrm{can}"] & & \mathbb{Z}/2\mathbb{Z}
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & M^{\pi} \otimes \mathbb{Q}_{\ell}/\mathbb{Z}_{\ell} \arrow[d, no head, "\wr" description] & & & \\
0 \arrow[r] & H^{0}(S, A) \otimes \mathbb{Q}_{\ell}/\mathbb{Z}_{\ell} \arrow[r] \arrow[d] & H^{1}(S, {}_{\ell^{\infty}}A) \arrow[r] \arrow[d] & H^{1}(S, A)(\ell) \arrow[r] \arrow[d] & 0 \\
0 \arrow[r] & (H^{0}(\overline{S}, \overline{A}) \otimes \mathbb{Q}_{\ell}/\mathbb{Z}_{\ell})^{\pi} \arrow[r] \arrow[d, no head, "\wr" description] & H^{1}(\overline{S}, {}_{\ell^{\infty}}\overline{A})^{\pi} \arrow[r] & H^{1}(\overline{S}, \overline{A})(\ell)^{\pi} \arrow[r] & 0 \\
 & (M \otimes \mathbb{Q}_{\ell}/\mathbb{Z}_{\ell})^{\pi} & & &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & 0 \arrow[d] & 0 \arrow[d] & 0 \arrow[d] & \\
0 \arrow[r] & \Phi \arrow[r] \arrow[d] & P^0 \arrow[r] \arrow[d] & Q^0 \arrow[r] \arrow[d] & 0 \\
0 \arrow[r] & E \arrow[r] \arrow[d] & P' \arrow[r] \arrow[d] & Q' \arrow[r] \arrow[d] & 0 \\
0 \arrow[r] & E/\Phi \arrow[r] \arrow[d] & E^* \arrow[r] \arrow[d] & \Psi \arrow[r] \arrow[d] & 0 \\
 & 0 & 0 & 0 &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & & e \arrow[dl] & & e \arrow[dl] & & e \arrow[dl] & & e \arrow[dl] & & \\
Y & X \arrow[l] & Z \arrow[l] \arrow[dl] & \Omega Y \arrow[l] \arrow[ul] & \Omega X \arrow[l] \arrow[dl] & \Omega Z \arrow[l] \arrow[ul] & \Omega^{2} Y \arrow[l] \arrow[dl] & \Omega^{2} X \arrow[l] \arrow[ul] & \Omega^{2} Z \arrow[l] \arrow[dl] & \Omega^{3} Y \arrow[l] \arrow[ul] & \cdots \arrow[l] \\
 & e \arrow[ul] & & e \arrow[ul] & & e \arrow[ul] & & e \arrow[ul] & & &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & \Psi^{i-1} \arrow[r] \arrow[d, "u\circ d"] & \Phi^{i} \arrow[r] \arrow[d] & \Psi^{i} \arrow[d, no head, "="] & \\
0 \arrow[r] & \Psi^{i-1} \arrow[r, "D"] \arrow[d, "\wr"] & \Phi(i,-) \arrow[r] \arrow[d, "\alpha=?"] & \Psi^{i} \arrow[r] \arrow[d, "\wr"] & 0 \\
0 \arrow[r] & \Psi^{i-1} \arrow[r] & \Phi(i,m) \arrow[r, "u"] & \Psi^{i} \arrow[r] & 0
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=normal, nodes={font=\scriptsize}]
    1 \arrow[r] & \pi_1(\overline{M}_{g,\nu;\xi}) \arrow[r] \arrow[d, "\wr"'] &
      \pi_1(M_{g,\nu;\xi}) \arrow[r] \arrow[d] & \Gamma \arrow[r] \arrow[drr] & 1 & \\
    1 \arrow[r] & \widehat{\mathcal{T}}_{g,\nu} \arrow[r] &
      \mathrm{Autext}_{\mathrm{lac}}^{\circ}\bigl(\pi_1(X \smallsetminus \{x_1, \ldots, x_\nu\})\bigr)
      \arrow[rrr] & & & \mathrm{Out}\ldots \arrow[d] \\
    & & & & & 1
  \end{tikzcd}

\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}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\pi_2(Z) \arrow[r] & \pi_2(Y) \arrow[r] \arrow[d] & \pi_1(F) \arrow[r] \arrow[d, "\text{épi}"] & \pi_1(Z) \arrow[r] \arrow[d] & \pi_1(Y) \arrow[r] \arrow[d, "\wr"] & \pi_0(F) \arrow[r] \arrow[d, "\wr"] & \pi_0(Z) \arrow[r] \arrow[d] & 1 \\
& 1 \arrow[r] & \pi_1(\Phi) \arrow[r] & \pi_1(\widetilde{Z}) \arrow[r] & \pi_1(\widetilde{Y}) \arrow[r] & \pi_0(\Phi) \arrow[r] & \pi_0(\widetilde{Z}) \arrow[r] & 1
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
& \operatorname{Im}(\pi_1(\bar{X}_s) \to \pi_1(X)) \arrow[d, no head] & & & \\
e \arrow[r] & \operatorname{Ker}[\pi_1(X) \to \pi_1(S)] \arrow[r, "\sim"] & \pi_1(X) \times_{\pi_1(S)} \pi_1(Y) \arrow[r] & \pi_1(Y) \arrow[r] & e \\
e \arrow[r] & \operatorname{Ker}[\pi_1(X \times_S Y) \to \pi_1(Y)] \arrow[u] \arrow[d, no head] \arrow[r] & \pi_1(X \times_S Y) \arrow[u] \arrow[r] & \pi_1(Y) \arrow[u] \arrow[r] & e \\
& \operatorname{Im}(\pi_1(\bar{X}_s) \to \pi_1(X \times_S Y)) & & &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
Y & X_0^0 \arrow[l, dashed] & X_1^1 \arrow[l, bend left=12]
  \arrow[l, bend right=12] & X_2^2 \arrow[l, bend left=20] \arrow[l]
  \arrow[l, bend right=20] \\
\xi \arrow[u] & X_0^{\xi} \arrow[l] \arrow[u] & X_1^{\xi}
  \arrow[l, bend left=12] \arrow[l, bend right=12] \arrow[u] & X_2^{\xi}
  \arrow[l, bend left=20] \arrow[l] \arrow[l, bend right=20] \arrow[u] \\
& K_0^0 & K_1^1 \arrow[l, bend left=12] \arrow[l, bend right=12] & K_2^2
  \arrow[l, bend left=20] \arrow[l] \arrow[l, bend right=20] \\
& k_0^0 \arrow[u, "i_0"] & k_1^1 \arrow[l] \arrow[u, "i_1"] & k_2^2
  \arrow[l, bend left=12] \arrow[l, bend right=12] \arrow[u, "i_2"]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\mathrm{Br}(\bar{Y}) \arrow[r] \arrow[d, "\alpha"] &
\mathrm{Br}(\bar{X}) \arrow[r] \arrow[d, "\beta"] &
H^{1}(\bar{Y},\bar{P}) \arrow[r] \arrow[d, "\gamma"] &
H^{3}(\bar{Y},\underline{\mathbb{G}}_{m}) \arrow[r] \arrow[d, "\alpha'"] &
H^{3}(\bar{X},\underline{\mathbb{G}}_{m}) \arrow[d, "\beta'"] \\
\mathrm{Br}(y) \arrow[r] & \mathrm{Br}(X_{y}) \arrow[r] &
H^{1}(y,P_{0}) \arrow[r] & H^{3}(y,\underline{\mathbb{G}}_{m}) \arrow[r] &
H^{3}(X_{y},\underline{\mathbb{G}}_{m})
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
1 \arrow[r] & \mathcal{T}^{!}_{0,4} \arrow[r] \arrow[d, no head, "\wr"'] & \mathcal{T}_{0\,4} \arrow[r] \arrow[d] & \mathfrak{S}_4 \arrow[r] \arrow[d] & 1 \\
1 \arrow[r] & \pi_{0,3} \arrow[r] & \mathcal{T}'_{0\,4} \arrow[r] & \mathfrak{S}_3 \arrow[r] & 1 \\
1 \arrow[r] & \pi_{0,3}(2,\mathbf{Z}) \arrow[r] \arrow[u, no head, "\wr"] & \mathcal{T}_{1,1} \arrow[r] \arrow[u] & \widetilde{\mathfrak{S}}_3 \arrow[u]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
  & X \times_{S} Y \arrow[d] & \pi^{(n)}(Y/X/S) \arrow[l] \arrow[d]
    & Y \arrow[l, "\varepsilon^{n}(Y/X/S)"'] \arrow[d] \\
  X & X \times_{S} X \arrow[l, "pr_{1}"']
    & \Delta^{(n)}_{X/S} \arrow[l, "i^{(n)}_{X/S}"'] \arrow[d]
    & X \arrow[l, "\Delta_{X/S}"'] \arrow[d] \\
  X' & X' \times_{S} X \arrow[l] & (\Gamma_{\varphi})^{(n)} \arrow[l]
    & X' \arrow[l]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & L^\bullet \overset{\mathbf{L}}{\otimes}_{\Lambda} \mathbb{R}\Gamma(S^1) \simeq L^\bullet \times L^\bullet[-1] \arrow[dl, "{(\alpha',\ \beta')}"'] & \\
\mathbb{R}\mathrm{Hom}_{\Lambda}(K^{\bullet\,\pi}, \Lambda)[-2n-1] \arrow[rr] & & K^{\bullet\,\pi} \simeq \mathbb{R}\Gamma(P) \simeq \mathbb{R}\Gamma(W - W_0) \arrow[ul, "{(\alpha,\ \beta)}"']
\end{tikzcd}

\begin{tikzcd}[column sep=large]
 & C \times S & \\
C \arrow[d, hook, "\operatorname{codim} j+1"'] & P(N) \arrow[l] \arrow[u, no head, "\simeq" description] \arrow[d, hook, "\alpha"] & C_{y} \arrow[l, hook'] \arrow[d, hook, "\alpha_{y}"] \\
X & X' \arrow[l, "f"] \arrow[d] & Y(y) \arrow[l, hook', "i'_{y}"] \arrow[d] \arrow[ll, bend left=35, "i_{y}"] \\
 & S & y \arrow[l]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & \pi \arrow[d, "u"] & & & \\
 & G(\mathbb{Q}_{p}) \arrow[d, no head, "p"'] & & & \\
 & G(\mathbb{C}) \arrow[d, no head, "j_{1}"'] & & & \\
1 \arrow[r] & G'(\mathbb{Q}) \arrow[r] \arrow[d, no head] & E \arrow[r] & \pi \arrow[r] \arrow[l, bend left=30, "\sigma_{1}"] \arrow[l, bend right=30, "\sigma_{2}"'] & 1 \\
 & G'(\mathbb{Q}_{p}) & & & \\
 & G_{m}(\mathbb{Q}_{p}) = \mathbb{Q}_{p}^{*} \arrow[u, "j_{2}"] & & & \\
 & \pi \arrow[u, "\chi"] & & &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & 0 \arrow[d] & & & \\
k^{*} \arrow[r] \arrow[dr] & U(k) \arrow[dr] & & & \\
0 \arrow[r] & K^{*} \arrow[r] & \mathbf{V}(k) \arrow[r] \arrow[dr] & J(k) \arrow[r] \arrow[d] & 0 \\
 & & & \mathcal{D}(k) \arrow[r] \arrow[d] & J_C(k) \arrow[r] & \mathrm{Hom}(\mathbb{G}_m, \mathbb{G}_m) = \mathbb{Z} \\
 & & & 0 & &
\end{tikzcd}

\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}

\begin{tikzcd}[column sep=small]
  H^{0}(\widetilde{T}') \arrow[r] & H^{0}(\widetilde{T}) \arrow[r] & H^{1}_{!}(\widetilde{Y}) \arrow[r] & H^{1}(\widetilde{T}') \\
  H^{0}(\mathbb{P}^{2}) \arrow[r] \arrow[u] & H^{0}(T') \arrow[r] \arrow[u] & H^{1}_{!}(X) \arrow[r] \arrow[u] & H^{1}(\mathbb{P}^{2}) \arrow[u] \\
  H^{0}(\mathbb{P}^{2}) \arrow[r] & H^{0}(T) \arrow[r] \arrow[u] & H^{1}_{!}(U) \arrow[r] \arrow[u] & H^{1}(\mathbb{P}^{2}) \arrow[u, no head, "\Vert" description]
\end{tikzcd}

\begin{tikzcd}[column sep=tiny, row sep=small, nodes={font=\scriptsize}]
 & H^{i-1}(R, F) \arrow[dl, no head] \arrow[dr] \arrow[d] & \\
H^{n-i}(P, \check{F} \otimes \mathcal{T}_W)^\vee \simeq H^{i}_{!}(\mathring{P}, F) \arrow[r] & H^{i}(W, F) \arrow[dl] \arrow[dr] & H^{i}_{!}(\mathring{Q}, F) \simeq H^{n-i}(Q, \check{F} \otimes \mathcal{T}_W)^\vee \arrow[l] \\
H^{i}(P, F) \arrow[r] & H^{i}(R, F) & H^{i}(Q, F) \arrow[l]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
  & & A_{\mathrm{num}} & \\
  & & A_{\tau} \arrow[u, "\overset{?}{\simeq}"'] \arrow[r, "\overset{?}{\simeq}"] & A_{\mathbb{Q}_{\ell}\text{-hom}} \\
  & & A_{\mathrm{alg}} \arrow[u] \arrow[r, "\overset{?}{\simeq}"] & A_{\mathbb{Z}_{\ell}\text{-hom}} \arrow[u] \\
  & & A_{\mathrm{pic}} \arrow[u] & \\
  G_{\mathrm{top}}K \arrow[uuurr] \arrow[uurrr, dashed, "?"'] & & A_{\mathrm{alb}} \arrow[u, "\simeq ?"'] & \\
  G_{\mathrm{alg}}K \arrow[u] & & A_{\mathrm{lin}} \arrow[u, "\simeq ?"'] \arrow[ull, "\text{surj}"'] & \\
  & CK \arrow[ul, "\text{surj}"] \arrow[ur] & &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\alpha_0 \arrow[r, two heads] \arrow[d, "\xi_0"'] & \alpha'_0 \arrow[d, dashed, "\xi'_0"'] & \alpha_1 \arrow[l, hook'] \arrow[r] \arrow[d, "\xi_1"'] & \alpha_2 \arrow[r] \arrow[d, "\xi_2"'] & \cdots \arrow[r] & \alpha_{n-1} \arrow[r, two heads] \arrow[d, "\xi_{n-1}"'] & \alpha'_{n-1} \arrow[d, dashed, "\xi'_{n-1}"'] & \alpha_n \arrow[l, hook'] \arrow[d, "\xi_n"] \\
X_0 \arrow[r, two heads] & X'_0 & X_1 \arrow[l, hook'] \arrow[r] & X_2 \arrow[r] & \cdots \arrow[r] & X_{n-1} \arrow[r, two heads] & X'_{n-1} & X_n \arrow[l, hook']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
  T \arrow[r, no head] \arrow[d, no head] & T \cdot \mathfrak{z} \arrow[r, no head, "(\mathbb{Z}/h\mathbb{Z})^{*}"] \arrow[d, no head] & N(\mu) \arrow[r, no head] \arrow[d, no head] & N \arrow[d, no head] & \\
  \mu \arrow[r, no head] \arrow[d, no head] & \mu \times \mathfrak{z} \arrow[r, no head, "(\mathbb{Z}/h\mathbb{Z})^{*}"'] \arrow[d, no head] & N(\mu) \cap N' \arrow[r, no head, "="] & N \cap N' \arrow[r, no head] & N' \\
  0 \arrow[r, no head] & \mathfrak{z} & & &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
  T \arrow[r, no head, "\mathfrak{z}"] \arrow[d, no head] & \mathfrak{z}(\gamma) \arrow[r, no head] \arrow[d, no head] & N(T) \arrow[r, no head] \arrow[d, no head] & G \arrow[d, no head] \\
  T \cap N(T') \arrow[r, no head, "\mathfrak{z}"] \arrow[d, no head, "\mathbb{Z}/h\mathbb{Z}"] & \mathfrak{z}(\gamma) \cap N(T') = N'^{\gamma} \arrow[r, no head] \arrow[d, no head, "\mathbb{Z}/h\mathbb{Z}"] & N(T) \cap N(T') \arrow[r, no head] \arrow[d, no head] & N(T') \arrow[d, no head, "W'"] \\
  T \cap T' \arrow[r, no head, "\mathfrak{z}"] & \mathfrak{z}(\gamma) \cap T' = T'^{\gamma} \arrow[r, no head, "="] & N(T) \cap T' \arrow[r, no head] & T'
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\cdots \arrow[r] & H^1(X, \underline{\mathbb{C}}^{*}) \arrow[r] & H^1(X, \underline{\mathrm{Gl}}(n)) \arrow[r] & H^1(X, \mathrm{GP}(n-1)) \arrow[r] & H^2(X, \underline{\mathbb{C}}^{*}) \\
\cdots \arrow[r] & H^1(X, \mathbb{Z}/n\mathbb{Z}) \arrow[r] \arrow[u] & H^1(X, \underline{\mathrm{Sl}}(n)) \arrow[r] \arrow[u] & H^1(X, \mathrm{GP}(n-1)) \arrow[r] \arrow[u, no head] & H^2(X, \mathbb{Z}/n\mathbb{Z}) \arrow[u]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
A \otimes T(A'_{1} \otimes C'_{1})
\arrow[r, "\mathrm{id} \otimes \check{T}^{-1}"]
\arrow[d, "\mathrm{id} \otimes Tu'"'] &
A \otimes (TA'_{1} \otimes TC'_{1}) \arrow[r, "a"] &
(A \otimes TA'_{1}) \otimes TC'_{1} \arrow[r, "u_{1} \otimes \mathrm{id}"]
& (B \otimes TB'_{1}) \otimes TC'_{1} \arrow[d, "a^{-1}"] \\
A \otimes T(A'_{2} \otimes C'_{2})
\arrow[d, "\mathrm{id} \otimes \check{T}^{-1}"'] & & &
B \otimes (TB'_{1} \otimes TC'_{1}) \arrow[d, "\mathrm{id} \otimes \check{T}"] \\
A \otimes (TA'_{2} \otimes TC'_{2}) \arrow[d, "a"'] & & &
B \otimes T(B'_{1} \otimes C'_{1}) \arrow[d, "\mathrm{id} \otimes Tv'"] \\
(A \otimes TA'_{2}) \otimes TC'_{2}
\arrow[r, "u_{2} \otimes \mathrm{id}"'] &
(B \otimes TB'_{2}) \otimes TC'_{2} \arrow[r, "a^{-1}"'] &
B \otimes (TB'_{2} \otimes TC'_{2})
\arrow[r, "\mathrm{id} \otimes \check{T}^{-1}"'] &
B \otimes T(B'_{2} \otimes C'_{2})
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
A \otimes T(A' \otimes B'')
\arrow[r, "\mathrm{id} \otimes \check{T}^{-1}"]
\arrow[dddd, "\omega"'] &
A \otimes (TA' \otimes TB'') \arrow[r, "a"] &
(A \otimes TA') \otimes TB'' \arrow[r, "u \otimes \mathrm{id}"] &
(B \otimes TB') \otimes TB'' \arrow[d, "a^{-1}"] \\
& & & B \otimes (TB' \otimes TB'') \arrow[d, "\mathrm{id} \otimes c"] \\
& & & B \otimes (TB'' \otimes TB') \arrow[d, "a"] \\
& & & (B \otimes TB'') \otimes TB' \arrow[d, "v \otimes \mathrm{id}"] \\
C \otimes T(B' \otimes C'') &
C \otimes (TB' \otimes TC'') \arrow[l, "\mathrm{id} \otimes \check{T}"] &
C \otimes (TC'' \otimes TB') \arrow[l, "\mathrm{id} \otimes c"] &
(C \otimes TC'') \otimes TB' \arrow[l, "a^{-1}"]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
& T = \tilde X_{/\!/(\tilde a,\tilde b)} \arrow[ld] \arrow[rd] & \\
\tilde X_{/\!/\tilde b} \arrow[rd, "W\text{-cocart.}"'] & & \tilde X_{/\!/\tilde a} \arrow[ld] \\
& \simeq \Sigma^{1}(T) \arrow[ld] \arrow[rd] & \\
\tilde X_{/\!/\tilde\xi} \arrow[rd, "W\text{-cocart.}"'] & & \tilde X_{/\!/\tilde\xi'} \arrow[ld] \\
& \tilde X \simeq \Sigma^{1}_{W}(T) \arrow[ld] \arrow[rd] & \\
X \arrow[rd] & & e \arrow[ld] \\
& X' &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
H^{i+p}(X, \mathbb{Z}) \arrow[r] & H^{i+p}(X, \mathbb{C}) \arrow[r, "\sim"] & H^{i+p}(X, \Omega^{\bullet}_{X}) \\
H^{i}(Y, \mathbb{Z}) \arrow[u, "P_{\mathrm{top}}"] \arrow[r] & H^{i}(Y, \mathbb{C}) \arrow[u, "P_{\mathrm{top}}"] \arrow[r, "\frac{1}{(2i\pi)^{p}}\,\mathrm{can}"] & H^{i}(Y, \Omega^{\bullet}_{Y}) \arrow[u, "P_{\mathrm{an}}"]
\end{tikzcd}

\begin{tikzcd}[column sep=huge]
0 \arrow[r] & \mathbb{Z} \arrow[r, "{\lambda=(d_i)}"] & \mathbb{Z}^r \arrow[r, "\alpha"] & E_{(s)} \arrow[r] \arrow[d, dashed, "u"] & 0 \\
 & \mathbb{Z} & \mathbb{Z}^r \arrow[l, "{\mu=(d_i\delta_i)=(d_i^t)}"'] & E^*_{(s)} \arrow[l, "\beta"'] & 0 \arrow[l]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & 0 \arrow[d] & & & \\
G(C) \arrow[r] \arrow[dr] & U_G(k) \arrow[dr] & & & \\
0 \arrow[r] & G(K) \arrow[r] & \mathbf{V}_G(k) \arrow[r] \arrow[dr] & J_G(k) \arrow[r] \arrow[d] & 0 \\
 & & & \mathcal{D}_G(k) \arrow[r] \arrow[d] & J_{G,C} \arrow[r, dashed] & \mathrm{Hom}(\mathbb{G}_m, G) \\
 & & & 0 & &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
H^i(L) \arrow[r] \arrow[d, "\alpha"'] & H^i(G) \arrow[r] \arrow[d, "\beta"] & H^{i+1}(F) \arrow[r] \arrow[d, "\varphi"] & H^{i+1}(L) \arrow[r] \arrow[d, "\alpha'"] & H^{i+1}(G) \arrow[d, "\beta'"] \\
H^i(\hat{L}) \arrow[r] & H^i(\hat{G}) \arrow[r] & H^{i+1}(\hat{F}) \arrow[r] & H^{i+1}(\hat{L}) \arrow[r] & H^{i+1}(\hat{G})
\end{tikzcd}

\begin{tikzcd}[column sep=small]
0 & I \arrow[l] & \varprojlim_g gI \arrow[l, "\alpha^{0*}"'] & D(H^2_x(F)) \arrow[l, "\partial^{*}"'] & A \arrow[l, "\alpha^{1*}"'] \arrow[dl, dashed] & 0 \arrow[l] \\
 & & & A_f \arrow[u, "\varphi"] \arrow[ul, dashed, "d"] & &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
  & & \underline{\mathrm{Isom}}(G_0, G) \arrow[d, no head, "\wr"] \arrow[r, dashed] & \underline{\mathrm{Hom}}_{\mathrm{gr}}(\mathrm{Sl}(2), G) & \\
  & & \text{SSPEP} \arrow[dl, no head] \arrow[d, no head] \arrow[dr, no head] & & \\
  & \text{SSPTor}_0 \arrow[r, no head, "\simeq"] \arrow[d, no head] \arrow[dl, no head] & \text{SSPKill} \arrow[dl, no head] \arrow[dr, no head] & \text{SSPBorep} \arrow[d, no head] \arrow[r, no head] & \text{SAP}_0 \simeq \text{EUP} \arrow[d, no head] \\
  \text{Kill} \simeq \text{ST}_0\text{P} \arrow[d, no head] & \text{SSPTor} \arrow[dr, no head] \arrow[dl, no head] & & \text{SSPBor} \arrow[dl, no head] \arrow[r, no head] & \text{SAP} \\
  \text{STP} & & \text{SSP} & &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\pi_2(X) \arrow[r] & \pi_2(S) \arrow[r] & \pi_1(F) \arrow[r] & \pi_1(X) \arrow[r] \arrow[d, no head, "\wr"] & \pi_1(S) \arrow[r] \arrow[d, no head, "\wr"] & \pi_0(F) \arrow[d, no head, "\wr"] \\
& 1 \arrow[r] & \pi_1(\widetilde{F}) \arrow[r] & \pi_1(\widetilde{X}) \arrow[r] & \pi_1(\widetilde{S}) \arrow[r] & \pi_0(\Phi) \arrow[r] & 1
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
0 \arrow[r] & \underline{\operatorname{Ext}}^1(A, G) \arrow[r, "i"] & R^1f_*(G_A) \arrow[r, "j"] \arrow[dl, "k"'] & \underline{\operatorname{Tors}}\,\underline{\operatorname{birig}}\,\underline{\operatorname{sym}}(A, A; G) \arrow[d, no head, "\Vert" description] \\
0 \arrow[r] & \underline{NS}(A) \arrow[rr, "j'"] & & \underline{\operatorname{Tors}}\,\underline{\operatorname{birig}}\,\underline{\operatorname{sym}}(A, A; G)
\end{tikzcd}