# zbMATH — the first resource for mathematics

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.

##### MSC:
 55U35 Abstract and axiomatic homotopy theory in algebraic topology 18D20 Enriched categories (over closed or monoidal categories)
##### Keywords:
simplical model categories; Quillen equivalence
Full Text: