Pretriangulated 2-representations via dg algebra 1-morphisms

Abstract

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.