A covariant Stinespring theorem

We prove a finite-dimensional covariant Stinespring theorem for compact quantum groups. Let G be a compact quantum group, and let T := Rep(G) be the rigid C*-tensor category of finite-dimensional continuous unitary representations of G. Let Mod(T) be the rigid C*-2-category of cofinite semisimple finitely decomposable T-module categories. We show that finite-dimensional G-C*-algebras can be identified with equivalence classes of 1-morphisms out of the object T in Mod(T). For 1-morphisms X : T -> M-1, Y : T -> M-2, we show that covariant completely positive maps between the corresponding G-C*-algebras can be "dilated" to isometries tau : X -> Y circle times E, where E : M-2 -> M-1 is some "environment" 1-morphism. Dilations are unique up to partial isometry on the environment; in particular, the dilation minimizing the quantum dimension of the environment is unique up to a unitary. When G is a compact group, this recovers previous covariant Stinespring-type theorems. (C) 2022 Author(s).