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Simplicial structures on model categories and functors. (English) Zbl 0979.55013
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.

55U35 Abstract and axiomatic homotopy theory in algebraic topology
18D20 Enriched categories (over closed or monoidal categories)
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