Kerodon

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

Remark 2.4.2.9. In the situation of Example 2.4.2.8, we will generally abuse notation by identifying the strict $2$-category $\operatorname{\mathcal{C}}$ with the associated simplicial category $\operatorname{\mathcal{C}}_{\bullet }$. Note that the underlying category of $\operatorname{\mathcal{C}}_{\bullet }$ (in the sense of Example 2.4.1.4) agrees with the underlying category of $\operatorname{\mathcal{C}}$ (in the sense of Remark 2.2.0.3). Moreover, since the nerve functor $\operatorname{N}_{\bullet }: \operatorname{Cat}\rightarrow \operatorname{Set_{\Delta }}$ is fully faithful (Proposition 1.2.2.1), the construction of Example 2.4.2.8 supplies a fully faithful embedding

\[ \operatorname{2Cat}_{\operatorname{Str}} \hookrightarrow \operatorname{Cat_{\Delta }}\quad \quad \operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}_{\bullet }, \]

where $\operatorname{2Cat}_{\operatorname{Str}}$ denotes the category of strict $2$-categories (see Definition 2.2.5.5).