$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 3.6.6.8 (The Homotopy Extension Lifting Property). Let $q: X \rightarrow S$ be a continuous function between topological spaces. The following conditions are equivalent:
- $(1)$
The morphism $q$ is a Serre fibration.
- $(2)$
For every simplicial set $B$, every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \{ 0\} \times |B| \ar [r] \ar@ {^{(}->}[d] & X \ar [d]^{q} \\ \empty [0,1] \times |B| \ar [r] \ar@ {-->}[ur] & S } \]
admits a solution.
- $(3)$
For every monomorphism of simplicial sets $A \hookrightarrow B$, every lifting problem
3.84
\begin{equation} \begin{gathered}\label{equation:Serre-fibration-HELP} \xymatrix@R =50pt@C=50pt{ ( [0,1] \times |A| ) \coprod _{ (\{ 0\} \times |A|) } ( \{ 0\} \times |B|) \ar [r] \ar@ {^{(}->}[d] & X \ar [d]^{q} \\ \empty [0,1] \times | B | \ar [r] \ar@ {-->}[ur] & S } \end{gathered} \end{equation}
admits a solution.
Proof.
The implication $(3) \Rightarrow (2) \Rightarrow (1)$ are immediate from the definition. We will complete the proof by showing that $(1)$ implies $(3)$. Using Corollary 3.6.2.2, we observe that every lifting problem of the form (3.84) can be rewritten as a lifting problem
\[ \xymatrix@R =50pt@C=50pt{ (\Delta ^1 \times _{A} ) \coprod _{ ( \{ 0\} \times A) } ( \{ 0 \} \times B) \ar [r] \ar [d]^{\iota } & \operatorname{Sing}_{\bullet }(X) \ar [d]^{ \operatorname{Sing}_{\bullet }(q) } \\ \Delta ^1 \times B \ar [r] \ar@ {-->}[ur] & \operatorname{Sing}_{\bullet }(S) } \]
in the category of simplicial sets. If $q$ is Serre fibration, then $\operatorname{Sing}_{\bullet }(q)$ is a Kan fibration (Proposition 3.6.6.5), so the existence of the desired lifting follows from the observation that $\iota $ is an anodyne morphism (Proposition 3.1.2.9).
$\square$