A Q-system in a C * 2-category is a unitary version of a separable Frobenius algebra object and can be viewed as a unitary version of a higher idempotent. We define a higher unitary idempotent completion for C * 2-categories called Q-system completion and study its properties. We show that the C * 2-category of right correspondences of unital C * -algebras is Q-system complete by constructing an inverse realization †-2-functor. We use this result to construct induced actions of group theoretical unitary fusion categories on continuous trace C * -algebras with connected spectra.

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Given an arbitrary countably generated rigid C*-tensor category, we construct a fully faithful bi-involutive strong monoidal functor onto a subcategory of finitely generated projective bimodules over a simple, exact, separable, unital C*-algebra with a unique trace. The C*-algebras involved are built from the category using the Guionnet–Jones–Shlyakhtenko construction. Out of this category of Hilbert C*-bimodules, we construct a fully faithful bi-involutive strong monoidal functor into the category of bifinite spherical bimodules over an interpolated free group factor. The composite of these two functors recovers the functor constructed by Brothier, Hartglass, and Penneys.

Generalized Temperley-Lieb-Jones (TLJ) 2-categories associated to weighted bidirected graphs were introduced in unpublished work of Morrison and Walker. We introduce unitary modules for these generalized TLJ 2-categories as strong * -pseudofunctors into the * -2-category of row-finite separable bigraded Hilbert spaces. We classify these modules up to * -equivalence in terms of weighted bi-directed fair and balanced graphs in the spirit of Yamagami's classification of fiber functors on TLJ categories and DeCommer and Yamashita's classification of unitary modules for Rep(SU q (2)).