dg-localization is the analogue in the world of dg-categories to the notion of simplicial localization. In good cases it is presented by a dg-model category.
The dg-localization of a dg-category at a subset of morphisms is the data of a morphism
in the (infinity,1)-category of dg-categories such that for any dg-category the induced morphism
of mapping spaces is injective on connected components and has image the subset of morphisms in that send morphisms of to equivalences in .
The dg-localization of any dg-category at a set of morphisms exists.
See section 8.2 of
and section 4.3 of
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