Kerodon

Example 10.3.1.7. Let $\operatorname{\mathcal{C}}$ be a category and let $S$ be a simplicial subset of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Then $S$ is a sieve on $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (in the sense of Definition 10.3.1.1) if and only if it has the form $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}^0 )$, where $\operatorname{\mathcal{C}}^0$ is sieve on $\operatorname{\mathcal{C}}$ (in the usual category-theoretic sense).