Kerodon

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

Example 2.2.0.6 (Ordinary Categories). Every ordinary category can be regarded as a strict $2$-category. More precisely, to each category $\operatorname{\mathcal{C}}$ we can associate a strict $2$-category $\operatorname{\mathcal{C}}'$ as follows:

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

  • For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, objects of the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}'}(X,Y)$ are elements of the set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$, and every morphism in $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}'}(X,Y)$ is an identity morphism.

  • For every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, the composition functor

    \[ \circ : \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}'}( Y, Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}'}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}'}(X,Z) \]

    is given on objects by the composition map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$.

  • For every object $X \in \operatorname{\mathcal{C}}$, the identity object $\operatorname{id}_{X} \in \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}'}(X,X)$ coincides with the identity morphism $\operatorname{id}_ X \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)$.

In this situation, we will generally abuse terminology by identifying the strict $2$-category $\operatorname{\mathcal{C}}'$ with the ordinary category $\operatorname{\mathcal{C}}$ (see Example 2.2.5.7).