related by the Dold-Kan correspondence
The model structure on enriched categories gives in particular a model structure on dg-categories, called the Dwyer-Kan model structure, which is analogous to the usual model structure on sSet-categories which models (infinity,1)-categories.
There are interesting left Bousfield localizations of this model structure, called the quasi-equiconic and Morita model structures. Here the fibrant objects are the pretriangulated dg-categories, resp. idempotent complete pretriangulated dg-categories. In characteristic zero, the Morita model structure is known to present the (infinity,1)-category of linear stable (infinity,1)-categories (Cohn 13).
There is the structure of a cofibrantly generated model category on where a dg-functor is
a weak equivalence if
a fibration if
for all objects the component is a degreewise surjection of chain complexes;
for each isomorphism in there is a lift to an isomorphism in .
This is due to (Tabuada).
There is another model category structure with more weak equivalences, the Morita equivalences (Tabuada 05). This is in fact the left Bousfield localization of the above model structure with respect to the Morita equivalences, i.e. functors whose induced restriction of scalars functor is an equivalence of categories.
The fibrant objects with respect to this model structure are the dg-categories A for which the canonical inclusion has its essential image stable under cones, suspensions, and direct sums. Hence the homotopy category with respect to this model structure is identified with the full subcategory of Ho(DGCat), the homotopy category of the Dwyer-Kan model structure, spanned by dg-categories of this form.
The pretriangulated envelope of Bondal-Kapranov is a fibrant replacement functor for the Morita model structure. The DG quotient? of Drinfeld is a model for the homotopy cofibre with respect to the Morita model structure.
model structure on dg-categories
The model structure on dg-categories is due to
It is reproduced as theorem 4.1 in
The derived internal Hom in the homotopy category of DG-categories is equivalent to the dg-category of A_infty-functors.
There is also
The model structure with Morita equivalences as weak equivalences is discussed in
That the Morita model structure on dg-categories presents the homotopy theory of -linear stable (infinity,1)-categories was shown in