G-graded finite semisimple 2-vector space

Fix an algebraically closed field $k$ of characteristic zero and a finite group $G$ (Group and abelian group). A finite semisimple category is understood in the sense of Semisimple objects and semisimple abelian categories, with simple objects as in Simple object. A finite semisimple $G$-graded $k$-linear category is a finite semisimple $k$-linear category $\mathcal{A}$ together with a direct-sum decomposition $\mathcal{A}={⨁}_{g\in G}{\mathcal{A}}_g\mathrm{.}$ An object in ${\mathcal{A}}_g$ is homogeneous of degree $g$. A grading-preserving linear functor sends each ${\mathcal{A}}_g$ into the corresponding degree-$g$ summand, and its 2-morphisms are natural transformations. Such categories, functors, and natural transformations form the underlying 2-category denoted $2{\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_G$.

For finite semisimple categories $\mathcal{A},\mathcal{B}$ with simple-label sets $S,T$, their finite semisimple tensor product $\mathcal{A}⊠\mathcal{B}$ has simple labels $S\times T$ and $\mathrm{Hom}(s⊠t,s^{\mathrm{\prime }}⊠t^{\mathrm{\prime }})={\mathrm{Hom}}_{\mathcal{A}}(s,s^{\mathrm{\prime }}){\otimes }_k{\mathrm{Hom}}_{\mathcal{B}}(t,t^{\mathrm{\prime }}),$ extended to direct sums. For a concrete skeleton, choose finite sets $S_g$ of simple labels and replace ${\mathcal{A}}_g$ by the category of finite-dimensional $k$-vector-space families indexed by $S_g$. A linear functor is represented by a finite matrix of finite-dimensional vector spaces within each grade. This skeleton fixes the meaning of “finite” and the homogeneous components; it does not assert any physical realization.