Remark 1.3.1.8 (Degeneracy Operators on $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$). Let $\operatorname{\mathcal{C}}$ be a category and suppose we are given an $n$-simplex $\sigma $ of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ which we identify with a diagram
\[ C_0 \xrightarrow {f_1} C_1 \xrightarrow {f_2} C_2 \rightarrow \cdots \xrightarrow {f_ n} C_ n. \]
Then, for $0 \leq i \leq n$, we can identify the degenerate simplex $s^{n}_{i}(\sigma ) \in \operatorname{N}_{n+1}(\operatorname{\mathcal{C}})$ with the diagram
\[ C_0 \xrightarrow {f_1} \cdots \xrightarrow {f_{i-1}} C_{i-1} \xrightarrow {f_ i} C_{i} \xrightarrow { \operatorname{id}_{C_ i} } C_ i \xrightarrow { f_{i+1} } C_{i+1} \rightarrow \cdots \xrightarrow {f_ n} C_ n \]
obtained from $\sigma $ by “inserting” the identity morphism $\operatorname{id}_{ C_{i} }$.