$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 10.3.1.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a simplicial subset. Then $\operatorname{\mathcal{C}}^{0}$ is a sieve on $\operatorname{\mathcal{C}}$ if and only if it is a full subcategory of $\operatorname{\mathcal{C}}$ which satisfies the following condition:
- $(\ast )$
If $f: X \rightarrow Y$ is a morphism of $\operatorname{\mathcal{C}}$ and $Y$ belongs to the subcategory $\operatorname{\mathcal{C}}^{0}$, then $X$ also belongs to the subcategory $\operatorname{\mathcal{C}}^{0}$.
Proof.
By definition, $\operatorname{\mathcal{C}}^{0}$ is a sieve on $\operatorname{\mathcal{C}}$ if and only if the inclusion map $\iota : \operatorname{\mathcal{C}}^{0} \hookrightarrow \operatorname{\mathcal{C}}$ is a right fibration. In particular, this guarantees that $\iota $ is an inner fibration, so that $\operatorname{\mathcal{C}}^{0}$ is a subcategory of $\operatorname{\mathcal{C}}$. It also guarantees that a morphism $f: X \rightarrow Y$ is contained in $\operatorname{\mathcal{C}}^{0}$ if and only if the object $Y$ is contained in $\operatorname{\mathcal{C}}^{0}$, so that the subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ is full and satisfies $(\ast )$. Conversely, if $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ is a full subcategory satisfying condition $(\ast )$, then $\iota $ satisfies the criterion of Proposition 10.3.1.5 and is therefore a right fibration.
$\square$