Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 1.4.5.3. Let $p: X_{\bullet } \rightarrow Y_{\bullet }$ 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 1.4.5.1).

$(2)$

The map $p$ has the right lifting property with respect to every monomorphism of simplicial sets $i: A_{\bullet } \hookrightarrow B_{\bullet }$. In other words, every lifting problem

\[ \xymatrix@C =40pt@R=40pt{ A_{\bullet } \ar [d]^{i} \ar [r] & X_{\bullet } \ar [d]^{p} \\ B_{\bullet } \ar@ {-->}[ur] \ar [r] & Y_{\bullet } } \]

admits a solution, provided that $i$ is a monomorphism.

Proof of Proposition 1.4.5.3. Let $p: X_{\bullet } \rightarrow Y_{\bullet }$ be a trivial Kan fibration of simplicial sets and let $T$ be the collection of all morphisms in $\operatorname{Set_{\Delta }}$ which have the left lifting property with respect to $p$. Then $T$ 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 1.4.4.16). It follows from Proposition 1.4.5.12 that every monomorphism of simplicial sets $i: A_{\bullet } \hookrightarrow B_{\bullet }$ belongs to $T$ (and therefore has the left lifting property with respect to $p$). $\square$