Remark 6.2.0.1. This tag refers to a construction whose contents is now contained in Example example:monoidal-category-of-cocycle (and uses the language of monoidal categories, rather than $2$-categories).

Let $G$ be a group, let $\Gamma $ be an abelian group equipped with an action of $G$ by automorphisms, and let $\operatorname{\mathcal{C}}$ be the $2$-category obtained by applying the construction of Example 6.3.0.3 to some $3$-cocycle $\alpha : G \times G \times G \rightarrow \Gamma $. We can describe $\operatorname{\mathcal{C}}$ informally as follows: it is a $2$-category whose $1$-morphisms are the elements of the group $G$, whose $2$-morphisms are the elements of the abelian group $\Gamma $, and whose associativity constraint is the $3$-cocycle $\alpha $.