Realizations of algebra objects and discrete subfactors

Abstract

We give a characterization of extremal irreducible discrete subfactors $(N\subseteq M, E)$ where $N$ is type ${\rm II}_1$ in terms of connected W*-algebra objects in rigid C*-tensor categories. We prove an equivalence of categories where the morphisms for discrete inclusions are normal $N-N$ bilinear ucp maps which preserve the state $\tau \circ E$, and the morphisms for W*-algebra objects are categorical ucp morphisms. As an application, we get a well-behaved notion of the standard invariant of an extremal irreducible discrete subfactor, together with a subfactor reconstruction theorem. Thus our equivalence provides many new examples of discrete inclusions $(N\subseteq M, E)$, in particular, examples where $M$ is type ${\rm III}$ coming from non Kac-type discrete quantum groups and associated module W*-categories. Finally, we obtain a Galois correspondence between intermediate subfactors of an extremal irreducible discrete inclusion and intermediate W*-algebra objects.