Kerodon

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

Proposition 3.5.6.5. Let $q: X \rightarrow S$ be a continuous function between topological spaces. Then $q$ is a Serre fibration if and only if the induced map of singular simplicial sets $\operatorname{Sing}_{\bullet }(q): \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{Sing}_{\bullet }(S)$ is a Kan fibration.

Proof of Proposition 3.5.6.5. Assume first that the map $\operatorname{Sing}_{\bullet }(q): \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{Sing}_{\bullet }(S)$ is a Kan fibration of simplicial sets. It follows that, for each $n \geq 0$, $\operatorname{Sing}_{\bullet }(q)$ has the right lifting property with respect to the inclusion map $\{ 0\} \times \Delta ^ n \hookrightarrow \Delta ^1 \times \Delta ^ n$ (which is anodyne, by virtue of Proposition 3.1.2.8). It follows that the continuous function $q$ has the right lifting property with respect to the induced map of geometric realizations $| \{ 0\} \times \Delta ^ n | \hookrightarrow | \Delta ^1 \times \Delta ^ n |$, which can be identified with the inclusion $\{ 0\} \times | \Delta ^ n | \hookrightarrow [0,1] \times | \Delta ^ n |$ (see Corollary 3.5.2.2). Allowing $n$ to vary, we deduce that $q$ is a Serre fibration.

We now prove the converse. Suppose that $q$ is a Serre fibration; we wish to show that the induced map of simplicial sets $\operatorname{Sing}_{\bullet }(q): \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{Sing}_{\bullet }(S)$ has the right lifting property with respect to the horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ for every pair of integers $0 \leq i \leq n$ with $n > 0$. Equivalently, we wish to show that $q$ has the right lifting property with respect to the inclusion of geometric realizations $\iota : | \Lambda ^{n}_{i} | \hookrightarrow | \Delta ^ n |$. We proceed by refining the proof of Proposition 1.1.9.8. Define a continuous function $c: | \Delta ^{n} | \rightarrow [0,1]$ by the formula $c( t_0, t_1, \cdots , t_ n) = \min \{ t_0, \ldots , t_{i-1}, t_{i+1}, \ldots , t_ n \} $. Let $h: [0,1] \times | \Delta ^ n | \rightarrow | \Delta ^ n |$ be the continuous function given by the formula

\[ h( s, (t_0, \cdots , t_ n) ) = (t_0 - \lambda , \cdots , t_{i-1} - \lambda , t_{i} + n \lambda , t_{i+1} - \lambda , \cdots , t_ n - \lambda ) \]

\[ \lambda = \max \{ 0, c(t_0, \cdots , t_ n) - s \} \]

By construction, the composition

\[ | \Delta ^{n} | \xrightarrow { (c,\operatorname{id})} [0,1] \times | \Delta ^ n | \xrightarrow {h} | \Delta ^ n | \]

is the identity map. Moreover, the function $(c,\operatorname{id})$ carries the horn $| \Lambda ^{n}_{i} | \subset | \Delta ^ n |$ to the closed subset $\{ 0\} \times | \Delta ^ n | \subseteq | \Delta ^{n} |$, and the function $h$ carries $\{ 0\} \times | \Delta ^ n |$ to the horn $| \Lambda ^{n}_{i} | \subset | \Delta ^ n |$. It follows that $h$ and $(c,\operatorname{id})$ exhibit $\iota $ as a retract of the inclusion map $\iota ': \{ 0\} \times | \Delta ^ n | \hookrightarrow [0,1] \times | \Delta ^ n |$ in the category of topological spaces. Consequently, to show that $q$ has the right lifting property with respect to $\iota $, it will suffice to show that it has the right lifting property with respect to $\iota '$ (Proposition 1.4.4.9), which follows immediately from our assumption that $q$ is a Serre fibration. $\square$