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Remark 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.