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.
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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.
The fibrant objects are the pretriangulated dg-categories.
The weak equivalences are the Morita equivalences, i.e. functors inducing equivalences of derived dg-categories
The fibrant objects are the idempotent complete? pretriangulated dg-categories.
For a summary of the various model structures on dg-categories, see Section 2 of the paper
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