on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
and
nonabelian homological algebra
The category of dg-modules over a dg-algebra, or more generally a dg-category, admits dg-model structures which present the derived dg-category.
In the case of dg-algebras in an abelian category $\mathcal{A}$, dg-modules are the same as modules over algebras over the associative operad in $Ch(\mathcal{A})$. These admit model structures as described in model structure on modules over an algebra over an operad, transferred from the model structures on chain complexes in $\mathcal{A}$.
More generally there are model structures on dg-modules over a dg-category, analogous to model structures on simplicial presheaves.
There is a cofibrantly generated model structure on the category of dg-modules, called the projective model structure, where the weak equivalences are object-wise quasi-isomorphisms of chain complexes, and the fibrations are object-wise epimorphisms. This is a dg-model structure.
There is a model structure on the category of dg-modules, called the injective model structure, where the weak equivalences are object-wise quasi-isomorphisms of chain complexes, and the cofibrations are object-wise monomorphisms.
See (Toen 04, section 3) and (Keller 06, theorem 3.2). These model structures present the derived dg-category.
Section 3 of
Paragraph 3.2 of
For the case of dg-algebras, see the references below.
A general account is around section 11.2.5 of
and in section 3 of
The homotopy category and triangulated category of dg-modules is discussed for instance also in
See also