Acyclic fibration admits a section?

Question

Is that true that each acyclic fibration admits a section? If so how to see this? If not then under what condition an acyclic fibration admits a section?

Answer 1

Suppose $p:E\to B$ is an acyclic fibration in a general model category. There is then a commutative square $$\require{AMScd} \begin{CD} 0 @>{}>> E\\ @V{}VV @VV{p}V \\ B @>{1}>> B \end{CD}$$ where $0$ is initial. So, if the map $0\to B$ is a cofibration (i.e., if $B$ is cofibrant), there is a map $B\to E$ making the diagram commute, which is exactly a section of $p$. That is, any acyclic fibration over a cofibrant base has a section. In particular, if you're talking about Hurewicz fibrations that are homotopy equivalences, this works for any $B$, since every space is cofibrant in the Strøm model structure.

Alternatively, here's a more direct argument in the latter case. Let $f:B\to E$ be a homotopy inverse of $p$ with homotopy $H:B\times I\to B$ from $pf$ to the identity. Then since $p$ is a Hurewicz fibration, $H$ lifts a homotopy $\tilde{H}:B\times[0,1]\to E$, and then $\tilde{H}(-,1)$ will be a section of $p$.