Kerodon

Warning 9.3.4.35. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a subcategory. Beware that, if we do not assume that $\operatorname{\mathcal{C}}_0$ is replete (or full), then the inclusion functor $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ need not be a monomorphism in $\operatorname{\mathcal{QC}}$. For example, suppose that $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }( \operatorname{\mathcal{D}})$ is the nerve of a category $\operatorname{\mathcal{D}}$. Then the $0$-skeleton $\operatorname{\mathcal{C}}_0 = \operatorname{sk}_{0}( \operatorname{\mathcal{C}})$ is always subcategory of $\operatorname{\mathcal{C}}$ (namely, the subcategory spanned by the identity morphisms of $\operatorname{\mathcal{C}}$). However, the inclusion $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ is a monomorphism in $\operatorname{\mathcal{QC}}$ if and only if every isomorphism in $\operatorname{\mathcal{D}}$ is an identity morphism.