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, 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.