Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

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.