$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 1.5.5.5. Let $p: X \rightarrow Y$ be a trivial Kan fibration of simplicial sets. Then:
- $(a)$
The map $p$ admits a section: that is, there is a map of simplicial sets $s: Y \rightarrow X$ such that the composition $p \circ s$ is the identity map $\operatorname{id}_{Y}: Y \rightarrow Y$.
- $(b)$
Let $s$ be any section of $p$. Then the composition $s \circ p: X \rightarrow X$ is fiberwise homotopic to the identity. That is, there exists a map of simplicial sets $h: \Delta ^1 \times X \rightarrow X$, compatible with the projection to $Y$, such that $h|_{ \{ 0\} \times X } = s \circ p$ and $h|_{ \{ 1\} \times X } = \operatorname{id}_{X}$.
Proof.
To prove $(a)$, we observe that a section of $p$ can be described as a solution to the lifting problem
\[ \xymatrix@C =40pt@R=40pt{ \emptyset \ar [d] \ar [r] & X \ar [d]^{p} \\ Y \ar [r]^-{\operatorname{id}} \ar@ {-->}[ur]^{s} & Y, } \]
which exists by virtue of Proposition 1.5.5.4. Given any section $s$, a fiberwise homotopy from $s \circ p$ to the identity can be identified with a solution to the lifting problem
\[ \xymatrix@C =40pt@R=40pt{ \operatorname{\partial \Delta }^1 \times X \ar [d] \ar [r]^-{(s \circ p, \operatorname{id})} & X \ar [d]^{p} \\ \Delta ^1 \times X \ar [r] \ar@ {-->}[ur]^{h} & Y, } \]
which again exists by virtue of Proposition 1.5.5.4.
$\square$