Pretriangulated 2-representations via dg algebra 1-morphisms

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 dg modules over dg algebra 1 -morphisms internal to associated dg 2 -categories of compact objects. 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.