Variant 10.2.2.28. Let $\operatorname{\mathcal{C}}$ be a category and let $\{ X_ n \} _{n \geq -1}$ be a sequence of objects of $\operatorname{\mathcal{C}}$. Then morphisms
are the face and degeneracy operators for an augmented simplicial object $X_{\bullet }$ of $\operatorname{\mathcal{C}}$ if and only if they satisfy the following conditions:
- $(1)$
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 $X_{n-2}$).
- $(2)$
For all integers $0 \leq i \leq j \leq n$, we have an equality $s^{n+1}_{i} \circ s^{n}_ j = s^{n+1}_{j+1} \circ s^{n}_ i$ (as morphisms from $X_{n}$ to $X_{n+2}$).
- $(3)$
For all integers $0 \leq i,j \leq n$, we have an equality
\[ d^{n+1}_{i} \circ s^{n}_ j = \begin{cases} s^{n-1}_{j-1} \circ d^{n}_ i & \text{ if } i < j \\ \operatorname{id}_{ X_ n } & \text{ if } i = j \text{ or } i = j + 1 \\ s^{n-1}_{j} \circ d^{n}_{i-1} & \text{ if } i > j+1 \end{cases} \](as morphisms from $X_{n}$ to $X_{n}$).
If these conditions are satisfied, then the augmented simplicial object $X_{\bullet }$ is uniquely determined.