Simplicial structures on model categories and functors (Q2731395)

Let \({{\mathcal C}}\) be a closed model category, and let \(s{{\mathcal C}}\) denote the category of simplicial objects in \({{\mathcal C}}\). Then under certain technical, but not unreasonable, hypotheses, there is a model category structure on \(s{{\mathcal C}}\) so that the geometric realization functor \(|-|:s{{\mathcal C}} \to {{\mathcal C}}\) is part of a Quillen equivalence of model categories. In particular, these two model categories yield equivalent homotopy categories and, hence, are both models for the homotopy theory of \({{\mathcal C}}\). More is true. In the standard simplicial structure on \(s{{\mathcal C}}\), the model category structure on \(s{{\mathcal C}}\) becomes a simplicial model category in the sense of Quillen; thus, the authors present conditions under which a model category is Quillen equivalent to a simplicial model category. This is convenient as it is relatively easy to define objects such as mapping spaces and homotopy limits and colimits in a simplicial model category. The technical assumptions on \({{\mathcal C}}\) are that \({{\mathcal C}}\) be left proper, cofibrantly generated, and that geometric realization detect certain level-wise weak equivalences in \(s{{\mathcal C}}\). The last is spelled out in their Realization Axiom 3.4. The main example seems to be that of proper cofibrantly generated stable model categories.