nLab model structures on dg-categories

Idea

The simplicial localization of the category of dg-categories at the class of Dwyer-Kan equivalences is the (infinity,1)-category of dg-categories. It is presented by the Dwyer-Kan model structure which we discuss below.

We also discuss two interesting left Bousfield localizations of this model structure which present reflective sub-(infinity,1)-categories.

Definition

(details to be filled in)

Dwyer-Kan model structure

The weak equivalences are the Dwyer-Kan equivalences of dg-categories. The fibrations are the dg-functors that are surjective on all Hom complexes and isofibrations at the level of homotopy categories. The fibrant objects are the locally fibrant dg-categories, i.e. for which all mapping complexes are fibrant objects in the category of chain complexes.

This model structure is cofibrantly generated, see here.

Quasi-equiconic model structure

The fibrant objects are the pretriangulated dg-categories.

Morita model structure

The weak equivalences are the Morita equivalences, i.e. functors $u : A \to B$ inducing equivalences of derived dg-categories

$$D(B) \to D(A).$$

The fibrant objects are the idempotent complete? pretriangulated dg-categories.

References

For a summary of the various model structures on dg-categories, see Section 2 of the paper

• Denis-Charles Cisinski, Goncalo Tabuada, Non-connective K-theory via universal invariants, Compositio Math. 147 (2011), 1281-1320, arXiv:0903.3717, pdf.