model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant $\infty$-groupoids
for rational $\infty$-groupoids
for rational equivariant $\infty$-groupoids
for $n$-groupoids
for $\infty$-groups
for $\infty$-algebras
general $\infty$-algebras
specific $\infty$-algebras
for stable/spectrum objects
for $(\infty,1)$-categories
for stable $(\infty,1)$-categories
for $(\infty,1)$-operads
for $(n,r)$-categories
for $(\infty,1)$-sheaves / $\infty$-stacks
(also nonabelian homological algebra)
Context
Basic definitions
Stable homotopy theory notions
Constructions
Lemmas
Homology theories
Theorems
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.
module structure on modules in a monoidal model 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 October 31, 2023 at 16:07:42. See the history of this page for a list of all contributions to it.