Definition 10.3.4.1 (Universal Quotient Morphisms). 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 say that $f$ is a universal quotient morphism if the sieve $\operatorname{\mathcal{C}}^{0}_{/X}$ is dense (in the sense of Definition 10.3.1.26).
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