Differential graded cell 2-representations

TL;DR

This paper develops a theory of cell combinatorics and 2-representations for differential graded 2-categories, introducing new preorder concepts and classifying cell 2-representations in this context.

Contribution

It introduces strong and weak preorders for dg 2-categories and classifies their cell 2-representations, linking combinatorics with homotopy theory.

Findings

Weak preorder is more tractable than strong preorder.

Strong cells are contained within weak cells, with a bijection of maximal ideal spectra.

Complete classification of cell 2-representations for dg categories of projective bimodules.

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.