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Example Let $G$ be a group with identity element $1 \in G$, let $\Gamma $ be an abelian group on which $G$ acts by automorphisms, let $\alpha : G \times G \times G \rightarrow \Gamma $ be a $3$-cocycle, let $\operatorname{\mathcal{C}}$ be the monoidal category of Example, and let $B\operatorname{\mathcal{C}}$ be the $2$-category obtained by delooping $\operatorname{\mathcal{C}}$ (Example The following conditions are equivalent:

  • The $3$-cocycle $\alpha $ is normalized: that is, it satisfies the equations

    \[ \alpha _{x,y,1} = \alpha _{x,1,y} = \alpha _{1,x,y} = 0 \]

    for every pair of elements $x,y \in G$.

  • The $2$-category $B\operatorname{\mathcal{C}}$ is strictly unitary, in the sense of Definition