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
For $\mathcal{A}$ an abelian category, a dg-algebra in $\mathcal{A}$ is a monoid in the category of chain complexes $Ch(\mathcal{A})$.
Equivalently this is an algebra over an operad over the associative operad in $Ch(\mathcal{A})$.
For $A$ a fixed dg-algebra, a dg-module is then a module in $Ch(\mathcal{A})$ over $A$: a module over an algebra over an operad. Correspondingly the category $A Mod_{\mathcal{A}}$ of all $A$-modules carries a model structure on modules over an algebra over an operad. This is a model structure on dg-modules
Let $k$ be a field of characteristic 0.
Write $Ch^\bullet(k)$ for the category of chain complexes.
Let $A \in Mon(Ch^\bullet(k)) = dgAlg_k$ be a differential graded algebra.
Write $A Mod$ for the category of dg-modules over $A$: modules in $Ch^\bullet(k)$ over $A$:
If $A$ is commutative, then $A Mod$ is a closed symmetric monoidal category.
This is a standard construction, for instance Bernstein, p. 5.
If $A$ is cofibrant as an object in $Ch^\bullet(k)$ then the transferred model structure along
exists.
This appears as (Fresse, prop. 11.2.6).
(…)
The homotopy cofibers in $A Mod$ are given by the usual mapping cones of dg-modules in the model structure on chain complexes.
This follows from (Bernstein, 10.3.5).
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