The model category of right modules

In this part, we aim to prove that right R-module structures give convenient models for the homotopy of associated functors. First of all, we check in this chapter that the category of right R-modules inherits a convenient model structure. This verification is carried out in §14.1. In §14.2, we make explicit the structure of cell complexes of right R-modules in the case where C = dgkMod is the category of dg-modules over a ring k. In §14.3, we apply the general results of §11.1 to the symmetric monoidal model category of right R-modules. To summarize, we obtain that the category of P-algebras in right R-modules, where P a Σ∗-cofibrant operad, is equipped with a natural semi model structure, inherited from right Rmodules. In the case of a non-unitary operad R, we obtain that the category of P-algebras in connected right R-modules is equipped with a semi model structure as long as the operad P is C-cofibrant.