Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Exercise 1.1.1.11. Let $C_{\bullet }$ be a simplicial object of a category $\operatorname{\mathcal{C}}$. Show that the face and degeneracy maps of Notations 1.1.1.8 and 1.1.1.9 satisfy the simplicial identities

$(1)$

For $n \geq 2$ and $0 \leq i < j \leq n$, we have $d_{i} \circ d_{j} = d_{j-1} \circ d_{i}$ (as a map from $C_{n}$ to $C_{n-2}$).

$(2)$

For $0 \leq i \leq j \leq n$, we have $s_{i} \circ s_ j = s_{j+1} \circ s_ i$ (as a map from $C_{n}$ to $C_{n+2}$).

$(3)$

For $0 \leq i,j \leq n$, we have

\[ d_{i} \circ s_ j = \begin{cases} s_{j-1} \circ d_ i & \text{ if } i < j \\ \operatorname{id}_{ C_ n } & \text{ if } i = j \text{ or } i = j + 1 \\ s_{j} \circ d_{i-1} & \text{ if } i > j+1. \end{cases} \]

Conversely, show that any collection of objects $\{ C_ n \} _{n \geq 0}$ and morphisms $\{ d_ i: C_ n \rightarrow C_{n-1} \} _{0 \leq i \leq n}$, $\{ s_ i: C_ n \rightarrow C_{n+1} \} _{0 \leq i \leq n}$ satisfying $(1)$, $(2)$, and $(3)$ determines a (unique) simplicial object of $\operatorname{\mathcal{C}}$.