Differential Graded Modules and Cosimplicial Modules

The ultimate purpose of this part is to explain the definition of models for the rational homotopy of spaces. In our constructions, we use the classical Sullivan model, defined in terms of unitary commutative cochain dg-algebras, and a cosimplicial version of this model, involving cosimplicial algebra structures. The purpose of this preliminary chapter is to provide a survey of constructions on modules which underlie the definition of these model categories. To be more precise, we explain the definition of a model structure on the base category of cochain graded dg-modules, underlying the unitary commutative cochain dg-algebras of the classical Sullivan model, and the definition of a model structure on the category of cosimplicial modules, underlying our cosimplicial algebra model. We also explain the definition of a cosimplicial version of the Dold-Kan equivalence, between the category of cosimplicial modules and the category of cochain graded dg-modules, which we actually use to define our model structure on cosimplicial modules. We provide a reminder of our conventions on dg-modules, simplicial modules and cosimplicial modules in a preliminary section of the chapter (§5.0). We also provide a survey of the classical Dold-Kan equivalence, between the category of simplicial modules and a category of chain graded dg-modules dual to our category of cochain graded dg-modules. We address the construction of our model structure on cochain graded dg-modules afterwards (in §5.1). We then explain the definition of the cosimplicial Dold-Kan equivalence and the definition of the model structure on the category of cosimplicial modules (§5.2). We devote a second part of the chapter to the study of tensor products and enriched category structures on dg-modules, simplicial modules and cosimplicial modules. In a first step, we recall the definition of the symmetric monoidal structure which we associate to these base categories (§5.3). By the way, we also review the definition of the Eilenberg-Zilber equivalence, which we use to formalize the correspondence between the symmetric monoidal structure of the category of chain (respectively, cochain) graded dg-modules and the symmetric monoidal structure of the category of simplicial (respectively, cosimplicial) modules. We use these constructions in the next chapter, in order to define the categories of unitary commutative algebras that give our models for the rational homotopy of spaces. We address the definition of internal hom-objects and enriched category structures in a second step (§5.4). Let us mention that we only use the results of this section §5.4 in §??, where we tackle the applications of our constructions to the computation of function spaces on the category of operads in simplicial sets.