Remark 1.1.2.11 (Relations Among Degeneracy Operators). For every triple of integers $0 \leq i \leq j \leq n$, the diagram of linearly ordered sets
\[ \xymatrix@R =50pt@C=50pt{ [n+2] \ar [r]^-{\sigma ^{i}_{n+1}} \ar [d]^{ \sigma ^{j+1}_{n+1} } & [n+1] \ar [d]^{ \sigma ^{j}_{n}} \\[n+1] \ar [r]^-{ \sigma ^{i}_{n} } & [n] } \]
is commutative. It follows that, if $C_{\bullet }$ is a simplicial object of a category $\operatorname{\mathcal{C}}$, then the degeneracy operators of $C_{\bullet }$ satisfy the following condition:
- $(\ast '')$
For $0 \leq i \leq j \leq n$, we have an equality $s^{n+1}_{i} \circ s^{n}_ j = s^{n+1}_{j+1} \circ s^{n}_ i$ (as morphisms from $C_{n}$ to $C_{n+2}$).