K-theoretic Classification of Inductive Limit Actions of Fusion Categories on AF-algebras

We introduce a K-theoretic invariant for actions of unitary fusion categories on unital \({\textrm{C}}^*\) ${\text{C}}^{\ast }$ -algebras. We show that for inductive limits of finite dimensional actions of fusion categories on AF-algebras, this is a complete invariant. In particular, this gives a complete invariant for inductive limit actions of finite groups on unital AF-algebras. We apply our results to obtain a classification of finite depth, strongly AF-inclusions of unital AF-algebras.