is isomorphic to $G$. A representative set of these classes is $\{{\Bbbk}_g\}_{g \in G}$.

Assume that $G$ is abelian. Any bi-character $\chi{\colon}G \times G \to {\Bbbk}^*={\Bbbk}\setminus \{0\}$ of $G$ such that $\chi(g,h)\chi(h,g)=1$ for all $g,h \in G$ induces a symmetry for $G{\,\mbox{-}\,}\mathrm{Vect}_{\Bbbk}$ defined by $$\tau_{U,V}(u\otimes v)=\chi(|u|,|v|) \, v\otimes u,$$ where $U$ and $V$ are $G$-graded [$\Bbbk$-]{}vector spaces, $u\in U$ and $v\in V$ are homogeneous elements, and $|x|\in G$ is the degree of a homogeneous element $x$. We denote the resulting symmetric monoidal category by $G{\,\mbox{-}\,}\mathrm{Vect}_{\Bbbk}^\chi$. Note that for any $g \in G$, the scalar associated to the invertible object ${\Bbbk}_g$ of $G{\,\mbox{-}\,}\mathrm{Vect}_{\Bbbk}^\chi$ is computed by $$\sigma_{{\Bbbk}_g}=\chi(g,g).$$

Example of super vector spaces {#sect-ex-super-vect}
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Let ${\Bbbk}$ be a field. By Example \[sect-ex-G-vect\], the bi-character $\chi$ of the group ${\mathbb{Z}}/2{\mathbb{Z}}=\{0,1\}$ defined by $\chi(g,h)=(-1)^{gh}$ gives rise to the symmetric monoidal category $$\mathrm{SVect}_{\Bbbk}=({\mathbb{Z}}/2{\mathbb{Z}}){\,\mbox{-}\,}\mathrm{Vect}_{\Bbbk}^\chi$$ known as the category of super vector spaces over ${\Bbbk}$. Its symmetry is given by $$\tau_{U,V}(u\otimes v)=(-1)^{|u||v|}v\otimes u,$$ where $U$ and $V$ are objects of $\mathrm{SVect}_{\Bbbk}$, $u\in U$ and $v\in V$ are homogeneous elements, and $|x|\in \{0,1\}$ is the degree of a homogeneous element $x$. The monoidal unit ${\mathbb{1}}$ of $\mathrm{SVect}_{\Bbbk}$ is defined by ${\Bbbk}$ in degree $0$ and $0$ in 