Definition 10.3.2.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ be the sieve generated by $f$ (see Example 10.3.1.19). We will say that $f$ is a quotient morphism if the composite map
\[ (\operatorname{\mathcal{C}}^{0}_{/Y})^{\triangleright } \hookrightarrow (\operatorname{\mathcal{C}}_{/Y})^{\triangleright } \rightarrow \operatorname{\mathcal{C}} \]
is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$.