Kerodon

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

Proposition 3.5.5.3. Let $X$ be a topological space. The following conditions are equivalent:

$(1)$

The counit map $| \operatorname{Sing}_{\bullet }(X) | \rightarrow X$ is a homotopy equivalence of topological spaces.

$(2)$

There exists a Kan complex $Y$ and a homotopy equivalence of topological spaces $| Y | \rightarrow X$.

$(3)$

There exists a simplicial set $Y$ and a homotopy equivalence of topological spaces $|Y| \rightarrow X$.

$(4)$

There exists a homotopy equivalence of topological spaces $X' \rightarrow X$, where $X'$ is a CW complex.

Proof. The implication $(1) \Rightarrow (2)$ follows from the observation that $\operatorname{Sing}_{\bullet }(X)$ is a Kan complex (Proposition 1.1.9.8), the implication $(2) \Rightarrow (3)$ is trivial, and the implication $(3) \Rightarrow (4)$ follows from Remark 1.1.8.14. To complete the proof, it will suffice to show that if $X$ has the homotopy type of a CW complex, then the counit map $v: | \operatorname{Sing}_{\bullet }(X) | \rightarrow X$ is a homotopy equivalence. By virtue of Proposition 3.5.3.8, it will suffice to show that $v$ is a weak homotopy equivalence, which follows from Corollary 3.5.4.2. $\square$