Kerodon

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

Example 1.3.1.3. Let $\operatorname{\mathcal{C}}$ be an ordinary category, and regard the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ as an $\infty $-category. Then:

  • The objects of the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ are the objects of $\operatorname{\mathcal{C}}$.

  • The morphisms of the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ are the morphisms of $\operatorname{\mathcal{C}}$. Moreover, the source and target of a morphism of $\operatorname{\mathcal{C}}$ coincide with the source and target of the corresponding morphism in $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.

  • For every object $X \in \operatorname{\mathcal{C}}$, the identity morphism $\operatorname{id}_{X}$ does not depend on whether we view $X$ as an object of the category $\operatorname{\mathcal{C}}$ or the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.