Kerodon

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

Theorem 3.5.0.1. The geometric realization functor $| \bullet |: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Top}$ induces an equivalence from the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ to the full subcategory of $\mathrm{h} \mathit{\operatorname{Top}}$ spanned by those topological spaces $X$ which have the homotopy type of a CW complex.

Proof of Theorem 3.5.0.1. Using Construction 3.5.5.1, we see that the geometric realization functor $| \bullet |: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Top}$ induces a functor of homotopy categories $| \bullet |: \mathrm{h} \mathit{\operatorname{Kan}} \rightarrow \mathrm{h} \mathit{\operatorname{Top}}$. It follows from Proposition 3.5.5.2 that this functor is fully faithful, and from Proposition 3.5.5.3 that its essential image consists of those topological spaces $X$ which have the homotopy type of a CW complex. $\square$