Exercise 11.6.0.52. The contents of this tag are now at Remark 1.1.1.7 and Proposition 1.1.1.9. Let $C_{\bullet }$ be a semisimplicial object of a category $\operatorname{\mathcal{C}}$. Show that the face operators of Notation 11.6.0.39 satisfy the following condition:
- $(\ast )$
For $n \geq 2$ and $0 \leq i < j \leq n$, we have $d^{n-1}_{i} \circ d^{n}_{j} = d^{n-1}_{j-1} \circ d^{n}_{i}$ (as a map from $C_{n}$ to $C_{n-2}$).
Conversely, show that any collection of objects $\{ C_ n \} _{n \geq 0}$ and morphisms $\{ d^{n}_ i: C_ n \rightarrow C_{n-1} \} _{0 \leq i \leq n}$, satisfying $(\ast )$ determines a unique semisimplicial object of $\operatorname{\mathcal{C}}$.