Corollary 10.2.2.27. Let $\operatorname{\mathcal{C}}$ be a category and let $\{ X_ n \} _{n \geq -1}$ be a sequence of objects of $\operatorname{\mathcal{C}}$. Then a system of morphisms $\{ d^{n}_{i}: X_{n} \rightarrow X_{n-1} \} _{0 \leq i \leq n}$ arise as the face operators of an augmented semisimplicial object $X_{\bullet }$ of $\operatorname{\mathcal{C}}$ if and only if they satisfy the following condition:
- $(\ast )$
For all integers $n > 0$ and $0 \leq i < j \leq n$, we have an equality $d^{n-1}_{i} \circ d^{n}_{j} = d^{n-1}_{j-1} \circ d^{n}_{i}$ (as morphisms from $X_{n}$ to $X_{n-2}$).
If this condition is satisfied, then the augmented semisimplicial object $X_{\bullet }$ is uniquely determined.