  This paper develops a theory of pretriangulated 2-representations of dg
2-categories. We characterize cyclic pretriangulated 2-representations, under
certain compactness assumptions, in terms of modules over dg algebra
1-morphisms internal to associated dg 2-categories of compact dg modules.
Further, we investigate the Morita theory and quasi-equivalences for such dg
2-representations. We relate this theory to various classes of examples of dg
categorifications from the literature.
