Kerodon

Remark 10.2.5.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $C_{\bullet }$ be an augmented simplicial object of $\operatorname{\mathcal{C}}$, which we identify with a (semi)simplicial object $C'_{\bullet }$ of the slice $\infty $-category $\operatorname{\mathcal{C}}_{ / C_{-1} }$ (see Remark 10.2.1.15). Then, for $n \geq -1$, the augmented simplicial object $C_{\bullet }$ is $n$-coskeletal (in the sense of Definition 10.2.5.10) if and only if the simplicial object $C'_{\bullet }$ is $n$-coskeletal (in the sense of Definition 10.2.4.9). Moreover, the analogous statement holds for semisimplicial objects. See Remark 7.3.2.4.