Q-system completion for C⁎ 2-categories

K-theoretic classification of inductive limit actions of fusion categories on AF-algebras

“…Proof. This follows from our general theory of realization [CHPJP22]. The connections between realization and crossed product is elaborated on in Section 3.1).…”

“…For a Q-system P , this crossed product is denoted |P | F and is called the realization of P (see Section 3.1). In previous work, we showed that realization extends to a functor from [QSys(C)] to the category of (unital) C *algebras and isomorphism classes of (right finite) correspondences [CHPJP22]. Composing this with the usual K 0 functor from C * -algebras to the category of (pre)-ordered abelian groups (PAbGrp) yields a functor F : [QSys(C)] → PAbGrp.…”

“…The philosophy espoused in [CHPJP22] is that classifying "finite index" inclusions of algebras is closely related to classifying tensor category actions, and thus our results should have something to say about this problem. An analogue of a hyperfinite subfactor in the C * -setting is an inclusion (A ⊆ B, E A ) of unital AF-algebras with trivial centers and faithful conditional expectation E A : B → A.…”

“…Such an inclusion is called finite depth if B, viewed as an A-A correspondence, tensor generates a unitary fusion category. The standard invariant of the inclusion is the pair (C, Q) where C := A B A ⊆ Bim(A) is the fusion category generated by B, and Q = A B A is the Q-system in C corresponding to the algebra structure on B as in [CHPJP22]. An inclusion (A ⊆ B, E A ) is called strongly AF if the inclusion F : C → Bim(A) is an AF-action.…”

“…For any Q-system P , the realization |P | F should be thought of as a "generalized crossedproduct" [CHPJP22]. It can be defined (as a vector space) as…”