\begin{tikzcd}[column sep=small, row sep=normal]
 & \Sigma_{ik} \arrow[dl, "{(F_{ik})}"'] \arrow[rr, hook] & & \Sigma_{k} & \\
\Sigma_{i} & & \Sigma_{ijk} \arrow[ul, hook] \arrow[rr, hook] \arrow[dl] & & \Sigma_{jk} \arrow[ul, hook] \arrow[dl, "{(F_{jk})}"] \\
 & \Sigma_{ij} \arrow[ul, "{(F_{ij})}"] \arrow[rr, hook] & & \Sigma_{j} &
\end{tikzcd}

\begin{tikzcd}
V_{ijk} \arrow[r] \arrow[d] & V_{jk} \arrow[d] & X_k^{*} \arrow[l, hook'] \arrow[d] \\
V_{ij} \arrow[r, hook] \arrow[d] & V(X_j^{*},X_k) & X_k \\
X_i^{*} & &
\end{tikzcd}

\begin{tikzcd}
 & \Sigma_{02} \arrow[dl] \arrow[r, hook] & \Sigma_{2} \\
\Sigma_0 & \Sigma_{012} \arrow[l, "{[l]}"'] \arrow[u] \arrow[r, hook] \arrow[d, "{[l]}"'] & \Sigma_{12} \arrow[u] \arrow[d, "{(l)}"] \\
 & \Sigma_{01} \arrow[ul, "{[l]}"] \arrow[r, hook] & \Sigma_1
\end{tikzcd}

\begin{tikzcd}[column sep=large]
\mathbf{I}\times\dot T \arrow[r, "{(\lambda,x)\mapsto\lambda x}"] & T \\
\lbrace 0\rbrace\times\dot T \arrow[u, hook] \arrow[r, "{(0,x)\mapsto 0.x}"'] & Y \arrow[u, hook, "\mathrm{inc}"']
\end{tikzcd}

\begin{tikzcd}
Y \arrow[r, hook] \arrow[dr, "{y\mapsto(y,0)}"'] & T \arrow[d, "\wr"] \\
 & Y\times\mathbf{I}
\end{tikzcd}

\begin{tikzcd}
\dot T \arrow[r, hook] \arrow[dr, "{t\mapsto(t,0)}"'] & E \arrow[d, "\wr"] \\
 & \dot T\times[0,1]
\end{tikzcd}

\begin{tikzcd}
 & j' \arrow[r, leftrightarrow, "\text{préd}"'] \arrow[d, no head, "\geq" description] & j \arrow[d, no head, "\geq" description] \\
 & k' \arrow[r, leftrightarrow] & k \\
i \arrow[uur] \arrow[urr] & &
\end{tikzcd}

\begin{tikzcd}
X_{j'} \arrow[r] & X_j \\
X_{k'} \arrow[u] \arrow[r] & X_k \arrow[u]
\end{tikzcd}

\begin{tikzcd}
\Sigma_{d''} \arrow[r, hook] \arrow[d] & \Sigma_{d'} \arrow[d] \\
\Sigma_d \arrow[r, hook] & \Sigma_i
\end{tikzcd}

\begin{tikzcd}
\Sigma_{ij} \arrow[r, hook] \arrow[d] & \Sigma_j \\
\Sigma_i &
\end{tikzcd}

\begin{tikzcd}
\Sigma_{ij} \arrow[r, hook] \arrow[d] & \Sigma_j \\
\Sigma_i &
\end{tikzcd}

\begin{tikzcd}
\Sigma_{**} \arrow[r, "\underline b"] \arrow[d, "\underline s"'] & \Sigma_{*} \\
\Sigma_{*} &
\end{tikzcd}

\begin{tikzcd}[column sep=small]
 & d''=d\cup d' & \\
d \arrow[ur, hook] & & d' \arrow[ul, hook'] \\
 & \delta=d\cap d' \arrow[ul, hook] \arrow[ur, hook'] &
\end{tikzcd}

\begin{tikzcd}[column sep=small]
 & \Sigma^{*}(d'') \arrow[dl] \arrow[dr] & \\
\Sigma^{*}(d) \arrow[dr] & & \Sigma^{*}(d') \arrow[dl] \\
 & \Sigma^{*}(\delta) &
\end{tikzcd}

\begin{tikzcd}[column sep=normal, row sep=small]
 & \Sigma^{*}(i_0,i_n) \arrow[dr, "\varphi_0"] & \\
\Sigma^{*}(d) \arrow[ur] \arrow[r] \arrow[dr] & \Sigma^{*}(i_1,i_n) \arrow[r, "\varphi_1"] & \Sigma^{*}(i_n) \\
 & \Sigma^{*}(i_{n-1},i_n) \arrow[ur, "\varphi_{n-1}"'] &
\end{tikzcd}

\begin{tikzcd}[column sep=normal, row sep=small]
 & \Sigma^{*}(d\cup j_1) \arrow[dr, "\text{bordant}"] & \\
\Sigma^{*}(d') \arrow[ur] \arrow[dr] & & \Sigma^{*}(d) \\
 & \Sigma^{*}(d\cup j_p) \arrow[ur, "\text{bordant}"'] &
\end{tikzcd}

\begin{tikzcd}[row sep=small]
\vdots \\
\Sigma^{*}(2) \arrow[d, "{s',\ b'}"] \\
\Sigma^{*}(1) \arrow[d, "b"] \\
\Sigma=\Sigma^{*}(0)
\end{tikzcd}

\begin{tikzcd}
\Sigma^{*}(2) \arrow[r, hook, "\underline b'"] \arrow[d, "p_2"'] & \Sigma^{*}(1) \arrow[d, "p_1=\underline s"] \\
\Sigma^{*}(1) \arrow[r, hook, "\underline b"'] & \Sigma
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & \Sigma^{*}(1) \arrow[dl, "p_1"'] & \\
\Sigma & & \Sigma^{*}(2) \arrow[ul, "\underline s'"'] \arrow[dl, "p_2"] \\
 & \Sigma^{*}(1) \arrow[ul, "p_1"] &
\end{tikzcd}

\begin{tikzcd}
\Sigma^{*}(2) \arrow[r, hook, "\underline b'"] \arrow[d, "p_2"'] & \Sigma^{*}(1) \arrow[d, "p_1"] \\
\Sigma^{*}(1) \arrow[r, hook, "\underline b"] & \Sigma
\end{tikzcd}

\begin{tikzcd}[column sep=small]
 & \Sigma^{*}(1) \arrow[r, hook, "\underline b"] \arrow[dl, "\underline s"'] & \Sigma & \\
\Sigma & \Sigma^{*}(2) \arrow[u, hook, "\underline s'"] \arrow[rr, hook, "\underline b'"] \arrow[d, "p_2"'] & & \Sigma^{*}(1) \arrow[ul, hook, "\underline b"'] \arrow[dl, "\underline s"] \\
 & \Sigma^{*}(1) \arrow[r, hook, "\underline b"] \arrow[ul, "\underline s"] & \Sigma &
\end{tikzcd}

\begin{tikzcd}[column sep=small]
 & \Sigma_{ik} \arrow[r, hook, "b_{ik}"] \arrow[dl, "s_{ik}"'] & \Sigma_k & \\
\Sigma_i & \Sigma_{ijk} \arrow[u, hook] \arrow[rr, hook] \arrow[d] & & \Sigma_{jk} \arrow[ul, hook, "b_{jk}"'] \arrow[dl, "s_{jk}"] \\
 & \Sigma_{ij} \arrow[r, hook, "b_{ij}"] \arrow[ul, "s_{ij}"] & \Sigma_j &
\end{tikzcd}

\begin{tikzcd}
X \arrow[d, "f"'] & X' \arrow[l, "f'"'] \arrow[d, "g'"] \\
Y & Y' \arrow[l, "g"]
\end{tikzcd}

\begin{tikzcd}
\mathcal V_{ij} \arrow[r, "b_{ij}"] \arrow[d, hook, "s_{ij}"'] & \mathcal V_j \\
\mathcal V_i &
\end{tikzcd}

\begin{tikzcd}[column sep=small]
X_J=X^{*}_{\overline J} \arrow[r, hook] & X_{J'}=X^{*}_{\overline{J'}} \arrow[r, hook] & X_{I'}=X^{*}_{\overline{I'}} \\
X_{J_0}=X^{*}_{J_0} \arrow[u, hook] \arrow[r, no head] & X_{J'_0}=X^{*}_{J'_0} \arrow[u, hook] \arrow[r, hook] & X_{I'_0}=X^{*}_{I'_0} \arrow[u, hook]
\end{tikzcd}

\begin{tikzcd}[column sep=small]
\overline J \arrow[r, hook] & \overline{J'} \arrow[r, hook] & \overline{I'} \\
J_0 \arrow[u, hook] \arrow[r, hook] & J'_0 \arrow[u, hook] \arrow[r, hook] & I'_0 \arrow[u, hook]
\end{tikzcd}