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