Example 1.1.2.8 (Degenerate $2$-Simplices). Let $S_{\bullet }$ be a simplicial set and let $\sigma $ be a $2$-simplex of $S_{\bullet }$. We say that $\sigma $ is left-degenerate if it has the form $s^{1}_{0}(e)$, for some edge $e: x \rightarrow y$ of $S_{\bullet }$. In this case, the faces of $\sigma $ are depicted in the diagram
\[ \xymatrix { & x \ar [dr]^{e} & \\ x \ar [ur]^{\operatorname{id}_{x}} \ar [rr]^{e} & & y. } \]
We will say that $\sigma $ is right-degenerate if it has the form $s^{1}_{1}(e)$, for some edge $e: x \rightarrow y$ of $S_{\bullet }$; in this case, the faces of $\sigma $ are depicted in the diagram
\[ \xymatrix { & y \ar [dr]^{\operatorname{id}_ y} & \\ x \ar [ur]^{e} \ar [rr]^{e} & & y. } \]
Note that $\sigma $ is degenerate if and only if it is either left-degenerate or right-degenerate.