Kerodon

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

Remark 5.3.5.2. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category. Then every degenerate $2$-simplex of $\operatorname{\mathcal{C}}$ is thin. Consequently, to check that a simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ belongs to the pith $\operatorname{Pith}(\operatorname{\mathcal{C}})$, it suffices to check that $\sigma $ carries every nondegenerate $2$-simplex of $\Delta ^ n$ to a thin $2$-simplex of $\operatorname{\mathcal{C}}$. In particular:

  • Every object of $\operatorname{\mathcal{C}}$ belongs to $\operatorname{Pith}(\operatorname{\mathcal{C}})$.

  • Every morphism of $\operatorname{\mathcal{C}}$ belongs to $\operatorname{Pith}(\operatorname{\mathcal{C}})$.

  • A $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ belongs to $\operatorname{Pith}(\operatorname{\mathcal{C}})$ if and only if it is thin.