Definition 1.1.1.2. Let $\operatorname{\mathcal{C}}$ be a category. A semisimplicial object of $\operatorname{\mathcal{C}}$ is a functor $\operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$. We typically use the notation $C_{\bullet }$ to indicate a semisimplicial object of $\operatorname{\mathcal{C}}$, whose value on an object $[n] \in \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}}$ we denote by $C_{n}$. A semisimplicial set is a semisimplicial object of the category of sets.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$