Extension of the $2$-representation theory of finitary $2$-categories to locally (graded) finitary $2$-categories

We extend the $2$-representation theory of finitary $2$-categories to certain $2$-categories with infinitely many objects, called 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 that the associated coalgebra 1-morphisms have a homogeneous structure. We use these results to classify (graded) simple transitive $2$-representations of certain classes of cyclotomic $2$-Kac–Moody algebras.

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Received 18 May 2021

Received revised 6 December 2021

Accepted 16 December 2021

Published 16 May 2022