Pretriangulated 2-representations via dg algebra 1-morphisms

TL;DR

This paper develops a theory of pretriangulated 2-representations of dg 2-categories, characterizing them via dg modules over dg algebra 1-morphisms and exploring Morita theory and equivalences.

Contribution

It introduces a new framework for understanding pretriangulated 2-representations through dg algebra 1-morphisms and investigates their Morita theory and equivalences.

Findings

Characterization of cyclic pretriangulated 2-representations

Development of Morita theory for dg 2-representations

Connections to dg categorifications in literature

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.