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.

中文翻译：

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将有限 $2$-范畴的 $2$-表示理论扩展到局部（分级）有限 $2$-范畴

我们将有限 $2$-categories 的 $2$-representation 理论扩展到具有无限多个对象的特定 $2$-categories，称为局部有限 $2$-categories，并扩展了弱法定 $2 的简单传递 $2$-representations 的经典分类结果$-类别到这个环境。我们还考虑了局部有限的 $2$-categories 和带有分级的 $2$-representations，并证明了相关的余代数 1-morphisms 具有齐次结构。我们使用这些结果对某些类分圆 $2$-Kac–Moody 代数的简单传递 $2$-表示进行分类（分级）。