\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & & \widetilde{\Pi}{}^{\circ}_{0,3} \arrow[dr, "\mathbb{Z}/2\mathbb{Z}"] & & \widetilde{\mathrm{SL}}(2,\mathbb{Z}) \arrow[dr, "\mathbb{Z}/2\mathbb{Z}"] & \\
1 \arrow[r, "\mathbb{Z}"] & \mathbb{Z}\omega^2 \arrow[ur, "\Pi_{03}"] \arrow[dr, "\mathbb{Z}/2\mathbb{Z}"'] & & \widetilde{\Pi}_{03} \arrow[ur, "\mathbb{D}_3"] \arrow[dr, "\mathbb{Z}/2\mathbb{Z}"'] & & \widetilde{\mathrm{GL}}(2,\mathbb{Z}) \\
 & & \mathbb{Z}\omega \arrow[ur, "\Pi_{03}"'] & & \widetilde{\Pi}{}^{\tau}_{0,3} \arrow[ur, "\mathbb{D}_3"'] &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
1 \arrow[r] & \mathbb{Z}/2\mathbb{Z} \times \Pi_{0,3} \arrow[r] \arrow[d, no head, "\wr" description] & \mathrm{SL}(2,\mathbb{Z}) \arrow[r] \arrow[d, hook] & \mathbb{D}_3 \arrow[r] \arrow[d, hook] & 1 \\
1 \arrow[r] & \mathbb{Z}/2\mathbb{Z} \times \Pi_{0,3} \arrow[r] & \mathrm{GL}(2,\mathbb{Z}) \arrow[r] & \mathbb{D}_3 \times \{1, \tau\} \arrow[r] & 1
\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=small, row sep=small, nodes={font=\scriptsize}]
 & \mathbb{Z} \arrow[d, "2n"] \\
\Pi^{D}_n \arrow[r] \arrow[dr] & \mathbb{Z} \arrow[d] \\
 & (\Pi^{D}_n)_{\mathrm{ab}} \simeq \mathbb{Z}/2n\mathbb{Z}
\end{tikzcd}

\begin{tikzcd}
\mathbb{D}_3 = \mathfrak{S}_3 \arrow[r, "\mathrm{can}"] & \mathbb{Z}/2\mathbb{Z} \\
\mathbb{D}'_3 \arrow[u] \arrow[r] & \mathbb{Z}/4\mathbb{Z} \arrow[u, "\mathrm{can}"']
\end{tikzcd}

\begin{tikzcd}
\Pi^{\mathbb{D}'}_{0,3} \arrow[r] \arrow[d] & \Pi^{\mathbb{D}}_{0,3} = \{\rho, \sigma \mid \rho^3 = \sigma^2 = 1\} \arrow[d, Rightarrow] \\
\mathbb{D}'_3 \arrow[r] \arrow[d] & \mathbb{D}_3 \arrow[d] \\
\mathbb{Z}/4\mathbb{Z} \arrow[r, no head] & \mathbb{Z}/2\mathbb{Z}
\end{tikzcd}

\begin{tikzcd}
 & R \arrow[dl] \arrow[d] \arrow[dr] & \\
S & A & F
\end{tikzcd}

\begin{tikzcd}
1 & & 1 \\
\mathcal{T}_{11} \arrow[u] \arrow[rr, Rightarrow, "\struck{\ill{}}"] & & \mathbb{Z}/12\mathbb{Z} \arrow[u] \\
S\mathcal{T}_{11} \arrow[u] \arrow[rr] & & \mathbb{Z} \arrow[u] \\
\mathbb{Z} \arrow[u] \arrow[rr, "\mathrm{id}"] & & \mathbb{Z} \arrow[u, "12"'] \\
1 \arrow[u] & & 1 \arrow[u]
\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}]
\pi \arrow[r, hook] & \widetilde{E} \arrow[r] & \mathfrak{S}_n \\
Z\pi \arrow[u] \arrow[r] & \mathbb{Z}^n \arrow[u, hook]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
1 \arrow[r] & S\mathrm{Aut}^{+}(X) \arrow[r] \arrow[d, "\wr"] & \mathrm{Aut}^{+}(X) \arrow[r] & \mathrm{Aut}^{+}(\partial X) \arrow[r] & 1 \\
 & S\mathcal{T} & \mathrm{Aut}^{+}_{!}(\widetilde{X}) \arrow[u] & \mathfrak{S}_n \cdot \mathbb{T}^n \arrow[u, "\wr"'] &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
1 \arrow[r] & S\mathcal{T} \arrow[r] & E \arrow[r] \arrow[d, no head, "\parallel" description] & \mathfrak{S}_n \arrow[r] & 1 \\
 & \mathbb{Z}^n \arrow[u, hook] & \widetilde{E}/\pi & &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
1 \arrow[r] & \widetilde{F} \arrow[r] \arrow[d] & \widetilde{A} \arrow[r] \arrow[d] & \widetilde{G} \arrow[d] & \\
1 \arrow[r] & F \arrow[r] & A \arrow[r] & G \arrow[dr] & \\
 & & & & 1
\end{tikzcd}

\begin{tikzcd}
\mathbb{D}'_3 \arrow[r] & \mathbb{D}_3
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
PM_{11}(\tilde 2) \simeq PU_{03} \arrow[r, Rightarrow] \arrow[d] & PM_{11}(2) \simeq TU_{03} \arrow[d] \\
U_{0,3} \simeq M_{11}(\tilde 2) \arrow[r] & (U_{0,3}, \{1, \omega\}) \simeq M_{11}(2)
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
M_{11}(\tilde 2) \arrow[rr] \arrow[dr] & & M_{11}(2) \arrow[dl] \arrow[d] \\
 & M_{11} \arrow[d] & M_{11}(2)' \arrow[dl] \\
 & M'_{11} &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
U_{03} \arrow[r, "{\{1, \omega\}}"] \arrow[dr, "\mathbb{D}'_3"'] & (U_{0,3}, \{1, \omega\}) \arrow[d, "\mathbb{D}_3"'] \arrow[dr, "\mathrm{gerbe}"] & \\
 & (U_{0,3}, \mathbb{D}'_3) \arrow[d, "\mathrm{gerbe}"'] & (U_{0,3}) \arrow[dl, "\mathbb{D}_3"] \\
 & (U_{0,3}, \mathbb{D}_3) &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
(\mathcal{D}, \Pi_{0,3}) \arrow[r, Rightarrow, "\mathrm{gal}"] \arrow[dr, "\mathrm{gal}"'] & (\mathcal{D}, \Pi_{0,3} \times \{1, \omega\}) \arrow[d, "\mathrm{gal}"'] \arrow[dr, "2\text{-}\mathrm{gerbe}"] & \\
 & (\mathcal{D}, \mathrm{SL}(2, \mathbb{Z})) \arrow[d, "2\text{-}\mathrm{gerbe}"'] & (\mathcal{D}, \Pi_{0,3}) \arrow[dl, "\mathrm{gal}"] \\
 & (\mathcal{D}, \mathrm{SL}(2, \mathbb{Z})') &
\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}]
\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}]
\mathbb{D}'''_3 \arrow[r, "2"] \arrow[d, "3"'] \arrow[dr, "3"] & \mathbb{D}''_3 \arrow[d, "3"] \arrow[dr, "3"] & \\
\mathbb{Z}/12\mathbb{Z} \arrow[dr, "3"'] & \mathbb{D}'_3 \arrow[r, "2"] \arrow[d, "3"] & \mathbb{D}_3 \arrow[d, "3"] \\
 & \mathbb{Z}/4\mathbb{Z} \arrow[r, "2"] & \mathbb{Z}/2\mathbb{Z}
\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, nodes={font=\scriptsize}]
0 \arrow[r] & \overset{\omega}{\mathbb{Z}} \arrow[r, "p"] \arrow[d, "q"] & \overset{\widetilde\Pi_0}{\mathbb{Z}} \arrow[r, "\alpha\varphi_p"] \arrow[d] & \overset{\Pi_0}{\mathbb{Z}/p\mathbb{Z}} \arrow[r] \arrow[d, no head, "\parallel" description] & 0 \\
0 \arrow[r] & \mathbb{Z} \arrow[r] & \widetilde\Pi_0(q) \arrow[r] \arrow[d] & \mathbb{Z}/p\mathbb{Z} \arrow[r] & 0 \\
 & & \mathbb{Z}/q\mathbb{Z} & &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & \overset{\omega}{\mathbb{Z}} \arrow[r] \arrow[dl, Rightarrow] & \widetilde\Pi_0 \arrow[r] \arrow[dl] & (\mathbb{Z}/p\mathbb{Z} \times 0) \arrow[dl] & \\
0 \arrow[r] & \overset{\omega}{\mathbb{Z}} \arrow[r] & \widetilde\Pi_0(q) \arrow[r] & \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/q\mathbb{Z} \arrow[r] & 0 \\
 & \underset{\omega}{\mathbb{Z}} \arrow[r] \arrow[u, Rightarrow, "q"] & \underset{\varpi}{\mathbb{Z}} \arrow[r] \arrow[u] & 0 \times \mathbb{Z}/q\mathbb{Z} \arrow[u] &
\end{tikzcd}

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

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\Pi^{D}_{0,3} \arrow[r, "\alpha'"] & \mathbb{Z}/6\mathbb{Z} \\
\pi_1(PU_{03}, \mathfrak{S}_3) = \widetilde\Pi^{D}_{03} \arrow[u] \arrow[r] & \mathbb{Z} \arrow[u] \\
\varpi' \in \mathbb{Z} \arrow[u] \arrow[r, "="] & \mathbb{Z} \arrow[u, "6"']
\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}