Lemma 7.7.6.3 (05W3)

Lemma 7.7.6.3. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix { \operatorname{\mathcal{E}}' \ar [rr]^{\Gamma } \ar [dr]_{U'} & & \operatorname{\mathcal{E}}\ar [dl]^{U} \\ & \operatorname{\mathcal{C}}, & } \]

where $U$ and $U'$ are left fibrations. Then $\Gamma $ is an isofibration if and only if it is a left fibration.

Proof. Assume that $\Gamma $ is an isofibration; we will show that it is a left fibration (the reverse implication follows from Corollary 5.6.7.5). Let $\iota : A \hookrightarrow B$ be a left anodyne map of simplicial sets; we wish to show that every lifting problem

7.90

\begin{equation} \begin{gathered}\label{equation:isofibration-between-left} \xymatrix { A \ar [d]^{\iota } \ar [r] & \operatorname{\mathcal{E}}' \ar [d]^{\Gamma } \\ B \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{E}}} \end{gathered} \end{equation}

admits a solution. Let us regard the upper horizontal map and the underlying morphism $B \rightarrow \operatorname{\mathcal{C}}$ as fixed, and let

\[ V: \operatorname{Fun}_{A / \, / \operatorname{\mathcal{C}}}( B, \operatorname{\mathcal{E}}' ) \rightarrow \operatorname{Fun}_{ A / \, / \operatorname{\mathcal{C}}}( B, \operatorname{\mathcal{E}}) \]

be the morphism given by composition with $\Gamma $. Since $U$ and $U'$ are left fibrations, the source and target of $V$ are contractible Kan complexes (Proposition 4.2.5.4); in particular, $V$ is a categorical equivalence of simplicial sets. Since $\Gamma $ is an isofibration, $V$ is also an isofibration (see Proposition 4.5.5.14), and is therefore a trivial Kan fibration (Proposition 4.5.5.20). In particular, $V$ is surjective on vertices, which guarantees that the lifting problem ( 7.90) admits a solution. $\square$