Kerodon

Example 6.3.0.8. Let $\operatorname{Set_{\Delta }}$ denote the category of simplicial sets, and let $\mathrm{h} \mathit{\operatorname{Kan}}$ denote the homotopy category of Kan complexes (Construction 3.1.5.10). Then the fibrant replacement functor $\operatorname{Ex}^{\infty }: \operatorname{Set_{\Delta }}\rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ exhibits $\mathrm{h} \mathit{\operatorname{Kan}}$ as a $1$-categorical localization of $\operatorname{Set_{\Delta }}$ with respect to the collection $W$ of weak homotopy equivalences (see Variant 3.1.7.8). However, it does not exhibit $\mathrm{h} \mathit{\operatorname{Kan}}$ as a strict localization of $\operatorname{Set_{\Delta }}$ with respect to $W$ (since it is not bijective on objects).