Pretriangulated 2-representations via dg algebra 1-morphisms

TL;DR

The paper develops a robust framework for pretriangulated $2$-representations of dg $2$-categories, anchoring cyclic representations to modules over internal dg algebra $1$-morphisms internal to dg theories of compact objects. It builds Morita theory and quasi-equivalences in this setting and connects these abstract constructions to concrete dg categorifications, including finitary and braid-group contexts, through pretriangulated hulls and dg enhancements. A central technical device is the internal hom $[X,X]$ yielding a dg algebra $1$-morphism $A_{X}$ and linking $\\mathbf{M}$ with $\\mathbf{M}_{A_{X}}$ via equivalences when $X$ generates. The formalism clarifies how to classify, compare, and transport $2$-representations across dg functors, and it provides explicit computations in key examples (e.g., categorifications of $\mathbb{Z}[i]$ and braid groups) that illuminate the Morita theory and quotient-simple phenomena in this higher-categorical setting.

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.

Theorems & Definitions (114)

• Lemma 2.1: Or2
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Remark 2.5
• Lemma 2.6
• proof
• Lemma 2.7