Kerodon

Remark 4.8.1.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a simplicial subset which is also an $\infty $-category. If $\operatorname{\mathcal{C}}$ is an $(n,1)$-category for some integer $n \geq -1$, then $\operatorname{\mathcal{C}}_0$ is also an $(n,1)$-category. For $n \geq 1$, this follows from Remark 4.8.1.2. The case $n = 0$ follows from Proposition 4.8.1.15 (since any subcategory of a partially ordered set is also a partially ordered set), and the case $n = -1$ is trivial (see Example 4.8.1.9).