Kerodon

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

Exercise 5.4.3.12. Let $(X,x)$ be a pointed topological space. Show that $(X,x)$ belongs to the essential image of the functor $\operatorname{\mathcal{S}}_{\ast } \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Top}_{\ast } )$ if and only if the topological space $X$ has the homotopy type of a CW complex and the inclusion map $\{ x\} \hookrightarrow X$ is a Hurewicz fibration (that is, the union $(\{ 0\} \times X) \cup ( [0,1] \times \{ x\} )$ is a retract of the product space $[0,1] \times X$).