Remark 10.2.6.4 (Extra Degeneracies). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X_{\bullet }$ be an augmented simplicial object of $\operatorname{\mathcal{C}}$. For every integer $n \geq -1$, the function
\[ \sigma ^{0}_{n+1}: [n+2] \rightarrow [n+1] \quad \quad i \mapsto \begin{cases} 0 & \text{ if $i=0$ } \\ i-1 & \text{ if $i > 0$ } \end{cases} \]
belongs to the subcategory $\operatorname{{\bf \Delta }}_{\min } \subseteq \operatorname{{\bf \Delta }}$. If $\overline{X}$ is a splitting of $X_{\bullet }$, then evaluation on $\sigma ^{0}_{n+1}$ determines a morphism
\[ h_ n: X_{n} = \overline{X}( [n+1] ) \rightarrow \overline{X}( [n+2] ) = X_{n+1}. \]
Heuristically, one can think of the morphisms $\{ h_{n} \} _{n \geq -1}$ as “extra” degeneracy operators on the augmented simplicial object $X_{\bullet }$. In the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, these operators satisfy the identities
10.11
\begin{eqnarray} \label{equation:face-of-extra-degeneracy1} d^{n+1}_{i} \circ h_ n & \sim & \begin{cases} \operatorname{id}_{ X_{n} } & \text{ if $i = 0$ } \\ h_{n-1} \circ d^{n}_{i-1} & \text{ otherwise } \end{cases}\end{eqnarray}
10.14
\begin{eqnarray} \label{equation:face-of-extra-degeneracy2} s^{n+1}_{i} \circ h_{n} & \sim & \begin{cases} h_{n+1} \circ h_{n} & \text{ if $i=0$ } \\ h_{n+1} \circ s^{n}_{i-1} & \text{ otherwise. } \end{cases}\end{eqnarray}