$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 1.5.5.4. Let $p: X \rightarrow Y$ be a map of simplicial sets. The following conditions are equivalent:
- $(1)$
The map $p$ is a trivial Kan fibration (in the sense of Definition 1.5.5.1).
- $(2)$
The map $p$ is weakly right orthogonal to every monomorphism of simplicial sets $i: A \hookrightarrow B$. In other words, every lifting problem
\[ \xymatrix@C =40pt@R=40pt{ A \ar [d]^{i} \ar [r] & X \ar [d]^{p} \\ B \ar@ {-->}[ur] \ar [r] & Y } \]
admits a solution, provided that $i$ is a monomorphism.
Proof of Proposition 1.5.5.4.
Let $p: X \rightarrow Y$ be a trivial Kan fibration of simplicial sets and let $S$ be the collection of all morphisms in $\operatorname{Set_{\Delta }}$ which are weakly left orthogonal to $p$. Then $S$ contains each of the inclusions $\operatorname{\partial \Delta }^ n \hookrightarrow \Delta ^ n$ (by virtue of our assumption that $p$ is a trivial Kan fibration) and is weakly saturated (Proposition 1.5.4.13). It follows from Proposition 1.5.5.14 that every monomorphism of simplicial sets $i: A \hookrightarrow B$ belongs to $S$ (and is therefore weakly left orthogonal to $p$).
$\square$