Kerodon

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

Remark 1.1.1.7. The category $\operatorname{{\bf \Delta }}_{\operatorname{inj}}$ of Variant 1.1.1.6 can be regarded as a (non-full) subcategory of the category $\operatorname{{\bf \Delta }}$ of Definition 1.1.1.2. Consequently, any simplicial object $C_{\bullet }$ of a category $\operatorname{\mathcal{C}}$ determines a semisimplicial object of $\operatorname{\mathcal{C}}$, given by the composition

\[ \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} \hookrightarrow \operatorname{{\bf \Delta }}^{\operatorname{op}} \xrightarrow { C_{\bullet } } \operatorname{\mathcal{C}}. \]

We will often abuse notation by identifying a simplicial object $C_{\bullet }$ of $\operatorname{\mathcal{C}}$ with the underlying semisimplicial object of $\operatorname{\mathcal{C}}$.