$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 10.3.1.5. Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then a simplicial subset $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ is a sieve if and only if it satisfies the following condition:
- $(\ast )$
Let $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ be an $n$-simplex of $\operatorname{\mathcal{C}}$. If the final vertex $\sigma (n)$ is contained in $\operatorname{\mathcal{C}}^{0}$, then $\sigma $ is contained in $\operatorname{\mathcal{C}}^{0}$.
Proof.
For every integer $n \geq 0$, the inclusion map $\{ n\} \hookrightarrow \Delta ^ n$ is right anodyne (Example 4.3.7.11). If the inclusion map $\iota : \operatorname{\mathcal{C}}^{0} \hookrightarrow \operatorname{\mathcal{C}}$ is a right fibration, then condition $(\ast )$ is a special case of Proposition 4.2.4.5. Conversely, suppose that condition $(\ast )$ is satisfied, and let $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ be an $n$-simplex of $\operatorname{\mathcal{C}}$. For every integer $0 < i \leq n$, the horn $\Lambda ^{n}_{i}$ contains the final vertex $\{ n\} \subseteq \Delta ^ n$. Consequently, if the restriction $\sigma |_{ \Lambda ^{n}_{i} }$ factors (uniquely) through $\iota $, then condition $(\ast )$ guarantees that $\sigma $ factors (uniquely) through $\iota $. Allowing $n$ and $i$ to vary, we conclude that $\iota $ is a right fibration.
$\square$