Differential graded cell 2-representations

TL;DR

The paper develops a differential graded cell theory for dg 2-categories by introducing strong and weak left preorders and associated cell 2-representations. It constructs maximal ideal spectra for left cells and proves a bijection between the spectra of strong and weak cells, with strong cell representations embedding into their weak counterparts. The framework is extended to dg idempotent completions and triangulated cell structures, and the authors give a complete classification of both weak and strong cell 2-representations for the dg 2-category CA arising from finite-dimensional dg algebras, including explicit examples such as zigzag algebras. The results situate dg cell theory as a robust tool for triangulated categorification, linking combinatorial cell data to quotient-simple dg 2-representations and providing a bridge to classical finitary theories while enabling explicit computations in more general dg contexts.

Abstract

This article develops a theory of cell combinatorics and cell 2-representations for differential graded 2-categories. We introduce two types of partial preorders, called the strong and weak preorder. We then analyse and compare them. The weak preorder is more easily tractable, while the strong preorder is more closely related to the combinatorics of the associated homotopy 2-representations. To each left cell, we associate a maximal ideal spectrum, and each maximal ideal gives rise to a differential graded cell 2-representation. We prove that any strong cell is contained in a weak cell and that there is a bijection between the corresponding maximal ideal spectra. Finally, we classify weak and strong cell 2-representations for dg 2-categories of projective bimodules over finite-dimensional differential graded algebras.

Theorems & Definitions (81)

• Lemma 2.2.1
• Lemma 2.2.2
• Definition 3.2.1
• Lemma 3.2.2
• Lemma 3.2.3
• Definition 3.3.1
• Definition 3.3.2
• Lemma 3.3.3
• proof
• Lemma 3.3.4