Kerodon

Warning 10.3.0.10. Let $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_0 )$ be the nerve of an ordinary category $\operatorname{\mathcal{C}}_0$, and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. Then $f$ is a quotient morphism if and only if it is a regular epimorphism in $\operatorname{\mathcal{C}}_0$, in the sense of Definition 10.3.0.2 (see Corollary 10.3.2.7). In particular, every quotient morphism in $\operatorname{\mathcal{C}}$ is an epimorphism. Beware that, if $\operatorname{\mathcal{C}}$ is not assumed to be the nerve of an ordinary category, then the analogous statement is false: quotient morphisms in $\operatorname{\mathcal{C}}$ are usually not epimorphisms (that is, they are not monomorphisms when viewed as morphisms in the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$). See Warning 10.3.2.10).