  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.
