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This paper develops a theory of pretriangulated 2-representations of dg2-categories. We characterize cyclic pretriangulated 2-representations, undercertain compactness assumptions, in terms of modules over dg algebra1-morphisms internal to associated dg 2-categories of compact dg modules.Further, we investigate the Morita theory and quasi-equivalences for such dg2-representations. We relate this theory to various classes of examples of dgcategorifications from the literature.

Pretriangulated 2-representations via dg algebra 1-morphisms