Starting with the trivial model structure on the category of simplicial sets (that is the weak equivalences are exactly the isomorphisms and the cofibrations and fibrations are arbitrary maps), is it possible to get the usual Quillen model structure on simplicial sets by performing a number of explicit left and right Bousfield localizations (e.g. by localizing along the inclusions of horns into simplices)?
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Such sequence of localizations/colocalizations does not exist, since homotopy is not concrete. Suppose, for contradiction, that such sequence is constructed, then we obtain a corresponding sequence of reflections/coreflections on the level of homotopy categories, producing a fully faithful embedding of $\mathrm{Ho}(sSet_{\mathrm{standard}})\rightarrow \mathrm{Ho}(sSet_{\mathrm{trivial}})\cong sSet$. Next, it is possible to construct a faithful functor $sSet\rightarrow Set$, say, by sending every simplicial set into the product of the sets of its simplices, obtaining a contradiction with Freyd's theorem. |
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