Four-cocycle twisted monoidal structure on 2Vect_G

Let $G$ be a finite group (Group and abelian group), with identity $e$. A normalized multiplicative 4-cochain is a function $\omega :G^4\to k^{\times }$ equal to $1$ whenever any argument is $e$. It is a group 4-cocycle when, for all $g,h,k,l,m\in G$, $\omega (h,k,l,m)\omega (g,hk,l,m)\omega (g,h,k,lm)=\omega (gh,k,l,m)\omega (g,h,kl,m)\omega (g,h,k,l)\mathrm{.}$

On $2{\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_G$, define the tensor product by convolution: ${(\mathcal{A}\otimes \mathcal{B})}_u={⨁}_{gh=u}{\mathcal{A}}_g⊠{\mathcal{B}}_h,$ where $⊠$ is the finite semisimple tensor product of linear categories. The unit is ${\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_k$ concentrated in degree $e$. The associator and unitors are the canonical rebracketing and unit equivalences. On four homogeneous inputs of degrees $g,h,k,l$, prescribe the pentagonator to be multiplication by the scalar $\omega (g,h,k,l)$ on every simple summand; extend it by direct sums and naturality. Denote this proposed weak monoidal 2-category structure by $2{\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_G^{\omega }$.

The scalar is central, so its naturality is componentwise. The displayed group equation is the exact coherence condition to be checked for five homogeneous inputs. This definition includes no braiding and no claim about classification of all fusion 2-categories.