Kerodon

Remark 6.2.5.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S$ be a class of short morphisms of $\operatorname{\mathcal{C}}$. Then the simplicial set $\operatorname{\mathcal{C}}^{\mathrm{short}}$ is never an $\infty $-category, except in the trivial situation where $S$ is the class of all morphisms of $\operatorname{\mathcal{C}}$ (in which case we have $\operatorname{\mathcal{C}}^{\mathrm{short}} = \operatorname{\mathcal{C}}$). However, for every object $Z \in \operatorname{\mathcal{C}}$, the simplicial set $\operatorname{\mathcal{C}}^{\mathrm{short}}_{/Z} = ( \operatorname{\mathcal{C}}^{\mathrm{short}})_{/Z}$ is an $\infty $-category, since it can be identified with the full subcategory of $\operatorname{\mathcal{C}}_{/Z}$ spanned by those morphisms $s: Y \rightarrow Z$ which belong to $S$.