Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 9.1.5.2. Let us informally display the standard simplex $\Delta ^3$ as a diagram

9.7
\begin{equation} \begin{gathered}\label{equation:square-in-simplex} \xymatrix@R =50pt@C=50pt{ \bullet \ar [r] \ar [d] \ar [dr] & \bullet \ar [d] \\ \bullet \ar [r] \ar@ {-->}[ur] & \bullet . } \end{gathered} \end{equation}

The morphism $\alpha : \Delta ^1 \times \Delta ^1 \rightarrow \Delta ^3$ appearing in Definition 9.1.5.1 is a monomorphism of simplicial sets, whose image is the simplicial subset $Q \subseteq \Delta ^3$ consisting of those simplices which do not contain the “inner” edge $\operatorname{N}_{\bullet }( \{ 1 < 2 \} )$ which is indicated by the dotted arrow in the diagram (9.7). Stated more informally, $Q$ is the subset of $\Delta ^3$ which is “visible from the top” in the diagram (9.7); in particular, $Q$ contains the inner faces $\operatorname{N}_{\bullet }( \{ 0 < 1 < 3 \} )$ and $\operatorname{N}_{\bullet }( \{ 0 < 2 < 3 \} )$, but not the outer faces $\operatorname{N}_{\bullet }( \{ 0 < 1 < 2 \} )$ and $\operatorname{N}_{\bullet }( \{ 1 < 2 < 3 \} )$.