Extension Of The 2-Representation Theory Of Finitary 2-Categories To Locally (Graded) Finitary 2-Categories

TL;DR

This paper generalizes the 2-representation theory of finitary 2-categories to locally finitary and graded cases, providing classification results for simple transitive 2-representations in these broader contexts.

Contribution

It extends classical 2-representation classification to locally finitary and graded 2-categories, including applications to cyclotomic 2-Kac-Moody algebras.

Findings

Extended 2-representation theory to locally finitary 2-categories.

Classified simple transitive 2-representations in new settings.

Proved coalgebra 1-morphisms have homogeneous structures.

Abstract

We extend the 2-representation theory of finitary 2-categories to certain 2-categories with infinitely many objects, denoted locally finitary 2-categories, and extend the classical classification results of simple transitive 2-representations of weakly fiat 2-categories to this environment. We also consider locally finitary 2-categories and 2-representations with a grading, and prove the associated coalgebra 1-morphisms have a homogeneous structure. We use these results to classify simple transitive 2-representations of certain classes of cyclotomic 2-Kac-Moody algebras.