Enriched model categories and the Dold-Kan correspondence

Abstract

The work we present in this thesis is an application of the monoidal properties of the Dold–Kan correspondence and is constituted of two main parts. In the first one, we observe that by a theorem of Christensen and Hovey, the category of nonnegatively graded chain complexes of left R-modules has a model structure, called the Hurewicz model structure, where the weak equivalences are the chain homotopy equivalences. Hence, the Dold–Kan correspondence induces a model structure on the category of simplicial left R-modules and some properties, notably it is monoidal. In the second part, we observe that changing the enrichment of an enriched, tensored and cotensored category along the Dold–Kan correspondence does not preserve the tensoring nor the cotensoring. Thus, we generalize this observation to any weak monoidal Quillen adjunction and we give an insight of which properties are preserved and which are weakened after changing the enrichment of an enriched model category along a right weak monoidal Quillen adjoint.