We give a 3-universal property for the Karoubi envelope of 2-category. Using this, we show that 3-categories finite semisimple 2-categories (as introduced in arXiv:1812.11933) and multifusion categories are equivalent.

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.