Pyramids and 2-representations

TL;DR

This paper introduces a diagrammatic method to lift monoidal actions to complex categories without direct sums and proves an equivalence for simple transitive 2-representations of certain 2-categories.

Contribution

It presents a new diagrammatic procedure for lifting monoidal actions and establishes an equivalence result for simple transitive 2-representations of projective bimodule categories.

Findings

Diagrammatic procedure for lifting actions without direct sums

Equivalence of simple transitive 2-representations to cell 2-representations

Application to categories of projective bimodules over finite dimensional algebras

Abstract

We describe a diagrammatic procedure which lifts strict monoidal actions from additive categories to categories of complexes avoiding any use of direct sums. As an application, we prove that every simple transitive $2$-representation of the $2$-category of projective bimodules over a finite dimensional algebra is equivalent to a cell $2$-representation.