Kerodon

Remark 10.3.1.20. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ be a sieve on an object $Y$. Then $\operatorname{\mathcal{C}}^{0}_{/Y}$ coincides with $\operatorname{\mathcal{C}}_{/Y}$ if and only if it contains the identity morphism ${\operatorname{id}}_{Y}: Y \rightarrow Y$. In particular, if $\operatorname{\mathcal{C}}^{0}_{/Y}$ is the sieve generated by a morphism $f: X \rightarrow Y$ (Example 10.3.1.19), then $\operatorname{\mathcal{C}}^{0}_{/Y} = \operatorname{\mathcal{C}}_{/Y}$ if and only if the morphism $f$ admits a right homotopy inverse $s: Y \rightarrow X$.