$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 10.3.1.12. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. The following conditions are equivalent:
- $(1)$
The morphism $U$ restricts to an isomorphism of $\operatorname{\mathcal{E}}$ with a sieve $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$.
- $(2)$
The morphism $U$ is a right covering map (Definition 4.2.3.8) and, for every vertex $C \in \operatorname{\mathcal{C}}$, the fiber $U^{-1} \{ C\} $ has at most one element.
Proof.
The implication $(1) \Rightarrow (2)$ follows from Example 4.2.3.12. Conversely, suppose that condition $(2)$ is satisfied, and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ denote the homotopy category of $\operatorname{\mathcal{C}}$. Our assumption that $U$ is a right covering map guarantees the existence of a pullback square
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r]^-{U} \ar [d] & \operatorname{N}_{\bullet }( \int ^{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} } \mathscr {F}) \ar [d] \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ) } \]
where $\mathscr {F}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \operatorname{Set}$ is the contravariant homotopy transport representation of $\operatorname{\mathcal{C}}$, given concretely by the formula $\mathscr {F}(C) = U^{-1} \{ C\} $ (see Corollary 5.2.7.4). Our assumption that each of the sets $\mathscr {F}(C)$ has at most one element guarantees that the right vertical map induces an isomorphism from $\int ^{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} } \mathscr {F}$ to the sieve $\operatorname{\mathcal{D}}\subseteq \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ spanned by those objects $C$ for which the fiber $U^{-1} \{ C\} $ is nonempty. Condition $(1)$ now follows from Example 10.3.1.7 and Remark 10.3.1.3.
$\square$