Remark 1.1.1.7 (Relations Among Face Operators). Let $n \geq 2$ be an integer. For every pair of integers $0 \leq i < j \leq n$, the diagram of linearly ordered sets
\[ \xymatrix@R =50pt@C=50pt{ [n-2] \ar [r]^-{\delta ^{i}_{n-1}} \ar [d]^{ \delta ^{j-1}_{n-1} } & [n-1] \ar [d]^{ \delta ^{j}_{n}} \\[n-1] \ar [r]^-{ \delta ^{i}_{n} } & [n] } \]
is commutative: both the clockwise and counterclockwise compositions can be identified with the unique order-preserving bijection $[n-2] \simeq [n] \setminus \{ i < j \} $. It follows that, if $C_{\bullet }$ is a semisimplicial object of a category $\operatorname{\mathcal{C}}$, then the face operators of $C_{\bullet }$ satisfy the following condition:
- $(\ast )$
For $0 \leq i < j \leq n$, we have $d^{n-1}_{i} \circ d^{n}_{j} = d^{n-1}_{j-1} \circ d^{n}_{i}$ (as morphisms from $C_{n}$ to $C_{n-2}$).