Definition 10.3.1.18. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $Y$ be an object of $\operatorname{\mathcal{C}}$. A sieve on $Y$ is a sieve on the slice $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$; that is, a full subcategory $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ satisfying the following condition:
- $(\ast )$
For every $2$-simplex
\[ \xymatrix@R =50pt@C=50pt{ X' \ar [rr] \ar [dr]_{f'} & & X \ar [dl]^{f} \\ & Y & } \]in the $\infty $-category $\operatorname{\mathcal{C}}$, if $f$ is contained in the subcategory $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$, then $f'$ is also contained in $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$.