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Let $G$ be a finite group and $\operatorname{Vec}_G$ be the tensor category of $G$-graded vector spaces (or, if you prefer, $\pi_{\le 2}(BG)$).

The completion $\overline{\operatorname{Vec}_G}$ of $\operatorname{Vec}_G$ is a 2-category with

  • 0-morphisms Frobenius algebra objects in $\operatorname{Vec}_G$
  • 1-morphisms bimodule objects in $\operatorname{Vec}_G$
  • 2-morphisms morphisms of bimodule objects

Results of Ostrik [https://arxiv.org/abs/math/0202130] give a constructive way of computing these 2-categories.

Simple example: For $G = \mathbb Z/2$, there is exactly one non-trivial indecomposable algebra object $a$. The tensor category of endomorphisms of $a$ is isomorphic to another copy of $\operatorname{Vec}_{\mathbb Z/2}$. There is a single indecomposable bimodule object connecting $a$ to the trivial algebra object 1. The overall 2-category $\overline{\operatorname{Vec}_{\mathbb Z/2}}$ could be described as "shaded Ising" or "shaded $SU(2)_2$". It has a symmetry which exchanges 1 and $a$, and this symmetry explains/implements the well-known $e$-$m$ symmetry of $\operatorname{Vec}_{\mathbb Z/2}$ (a.k.a. toric code).

Question: For which (if any) $G$ has the 2-category $\overline{\operatorname{Vec}_G}$ been worked out in the literature?

By "worked out" I mean, at minimum, that the fusion rules between all 1-morphisms (bimodule objects) are described explicitly. Using the Ostrik reults, this calculation is straightforward but tedious. (I've done the $G = S_3$ case.) I'm hoping that instead of writing an appendix I can cite another paper.

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