Kerodon

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

Example 2.4.2.7 (Strict $2$-Categories as Simplicial Categories). Let $\operatorname{\mathcal{C}}$ be strict $2$-category (Definition 2.2.0.1). Then we can associate to $\operatorname{\mathcal{C}}$ a simplicial category $\operatorname{\mathcal{C}}_{\bullet }$ as follows:

  • The objects of $\operatorname{\mathcal{C}}_{\bullet }$ are the objects of $\operatorname{\mathcal{C}}$.

  • For every pair of objects $X,Y \in \operatorname{Ob}( \operatorname{\mathcal{C}}_{\bullet } ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, the simplicial set $\operatorname{Hom}_{ \operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ is the nerve of the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$.

  • For every triple of objects $X,Y,Z \in \operatorname{Ob}( \operatorname{\mathcal{C}}_{\bullet } ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, the composition law

    \[ \operatorname{Hom}_{ \operatorname{\mathcal{C}}}( Y,Z )_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X,Z)_{\bullet } \]

    of $\operatorname{\mathcal{C}}_{\bullet }$ is given by the nerve of the composition functor $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z)$.