on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
(also 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.
(projective model structure)
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,
the fibrations are object-wise epimorphisms.
Moreover, this is a dg-model structure.
(injective 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 (Toën 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
Discussion as a model for rational parameterized stable homotopy theory is due to
Last revised on May 27, 2018 at 15:56:59. See the history of this page for a list of all contributions to it.