Replacing model categories with simplicial ones (Q2750932)

The author shows that a model category \(M\) is often Quillen equivalent to a simplicial model category. It is required that \(M\) should be left proper and either cellular or combinatorial; these are technical conditions which enable \(M\) to be localized. The construction uses hocolim-equivalences of diagrams in \(M\), where a morphism of diagrams is called a hocolim-equivalence if one gets a weak equivalence after replacing the objects with cofibrant objects and passing to homotopy colimits. The simplicial model category equivalent to \(M\) is the category of simplicial objects in \(M\) with hocolim-equivalences as weak equivalences. For a model category \(M\) in the classes considered, and for a small category \({\mathcal C}\) with contractible nerve, it is shown that the diagram category \(M^{\mathcal C}\) also has a model structure with hocolim-equivalences as weak equivalences.