Example 6.2.2.9. Let $\operatorname{Top}$ denote the category whose objects are topological spaces and whose morphisms are continuous functions Let us regard $\operatorname{Top}$ as a simplicial category (Example 2.4.1.5), and let $\operatorname{\mathcal{T}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Top})$ denote its homotopy coherent nerve. Let $\operatorname{\mathcal{T}}_{0} \subseteq \operatorname{\mathcal{T}}$ be the full subcategory spanned by those topological spaces which have the homotopy type of a CW complex. Then:
A continuous function between topological spaces $f: X \rightarrow Y$ is a weak homotopy equivalence (in the sense of Definition 3.6.3.1) if and only if it exhibits $X$ as a $\operatorname{\mathcal{T}}_0$-colocalization of $Y$. This is restatement of Corollary 3.6.5.4.
The full subcategory $\operatorname{\mathcal{T}}_{0} \subseteq \operatorname{\mathcal{T}}$ is coreflective. That is, for every topological space $Y$, there exists a weak homotopy equivalence $f: X \rightarrow Y$, where $X$ has the homotopy type of a CW complex. For example, we can take $f$ to be the counit map $| \operatorname{Sing}_{\bullet }(Y) | \rightarrow Y$ (see Corollary 3.6.4.2).