Kerodon

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

Example 2.4.1.4 (The Underlying Category of a Simplicial Category). Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category. We let $\operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}_0$ denote the ordinary category obtained by applying Construction 2.4.1.3 in the case $n=0$. We will refer to $\operatorname{\mathcal{C}}$ as the underlying category of the simplicial category $\operatorname{\mathcal{C}}_{\bullet }$. Note that $\operatorname{\mathcal{C}}$ can also be obtained from $\operatorname{\mathcal{C}}_{\bullet }$ by applying the general procedure described in Example 2.1.7.5.

We will sometimes abuse terminology by identifying a simplicial category $\operatorname{\mathcal{C}}_{\bullet }$ with its underlying category $\operatorname{\mathcal{C}}$. In particular, if $X$ and $Y$ are objects of $\operatorname{\mathcal{C}}_{\bullet }$, we will write $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ to denote the set $\operatorname{Hom}_{\operatorname{\mathcal{C}}_0}(X,Y) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{0}$ of morphisms from $X$ to $Y$ in the category $\operatorname{\mathcal{C}}$.