# Kerodon

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Example 2.2.7.4. 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 2.1.3.3, and let $B\operatorname{\mathcal{C}}$ be the $2$-category obtained by delooping $\operatorname{\mathcal{C}}$ (Example 2.2.2.4). 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 2.2.7.1.