Kerodon

Remark 10.3.1.14. Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then the collection of sieves on $\operatorname{\mathcal{C}}$ is closed under the formation of intersections. In particular, for every vertex $X \in \operatorname{\mathcal{C}}$, there is a smallest sieve $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ containing $X$. We will refer to $\operatorname{\mathcal{C}}^{0}$ as the sieve generated by $X$. If $\operatorname{\mathcal{C}}$ is an $\infty $-category, then $\operatorname{\mathcal{C}}^{0}$ admits a more explicit description: it is the full subcategory of $\operatorname{\mathcal{C}}$ spanned by those objects $C$ for which there exists a morphism $f: C \rightarrow X$.