nLab model structure on dg-modules

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Contents

Context

Model category theory

model category, model \infty -category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Schanuel's lemma

Homology theories

Theorems

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 𝒜\mathcal{A}, dg-modules are the same as modules over algebras over the associative operad in Ch(𝒜)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 𝒜\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).

See also

Discussion as a model for rational parameterized stable homotopy theory is due to

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