# nLab model structure on dg-modules

Contents

### Context

#### Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of $(\infty,1)$-categories

Model structures

for $\infty$-groupoids

for ∞-groupoids

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

# Contents

## Idea

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.

## Definitions

###### Theorem

(projective model structure)

There is a cofibrantly generated model structure on the category of dg-modules, called the projective model structure, where

1. the weak equivalences are object-wise quasi-isomorphisms of chain complexes,

2. the fibrations are object-wise epimorphisms.

Moreover, this is a dg-model structure.

###### Theorem

(injective model structure)

There is a model structure on the category of dg-modules, called the injective model structure, where

1. the weak equivalences are object-wise quasi-isomorphisms of chain complexes,

2. 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.

## References

Section 3 of

Paragraph 3.2 of

### For dg-algebras

For the case of dg-algebras, see the references below.

A general account is around section 11.2.5 of

• Benoit Fresse, Modules over operads and functors Springer Lecture Notes in Mathematics, (2009) (pdf)

and in section 3 of

The homotopy category and triangulated category of dg-modules is discussed for instance also in

• Joseph Bernstein, DG-modules and equivariant cohomology (pdf).