Let $G$ be a finite abelian group.
We can think of the monoidal category $1Vec_G=Vec_G$ of $G$-graded $k$-vector spaces as a categorification of the group algebra $0Vec_G=kG$. How does this categorification pattern continue? Is there a nice recursive definition of the $n$-category $nVec_G$?
My guess is that we could use the equivalence $Vec_G\cong kG-mod$ in order to define $nVec_G$ as the $n$-category of module $(n-1)$-categories $(n-1)Vec_G-mod$. Is there a description in terms of the grading instead of modules?
Aha, I think its simply $nVec_G= \boxplus_{g \in G}\, nVec$, where $nVec$ is defined as above.