Kerodon

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Example 2.2.6.12. 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 2.2.2.5). A twisting cochain for the $2$-category $B \operatorname{\mathcal{C}}$ (in the sense of Notation 2.2.6.7) can be identified with a map of sets

\[ \mu : G \times G \rightarrow \Gamma \quad \quad (g,f) \mapsto \mu _{g,f}. \]

Let $(B\operatorname{\mathcal{C}})'$ denote the twist of $B \operatorname{\mathcal{C}}$ with respect to $\mu $. Unwinding the definitions, we see that $(B\operatorname{\mathcal{C}})'$ is obtained by delooping the same category $\operatorname{\mathcal{C}}$ with respect to a different monoidal structure: namely, the monoidal structure supplied by the $3$-cocycle $\alpha ': G \times G \times G \rightarrow \Gamma $ given by the formula

\[ \alpha '_{h,g,f} = \alpha _{h,g,f} + h( \mu _{g,f} ) - \mu _{hg,f} + \mu _{h,gf} - \mu _{h,g}. \]

We can summarize the situation as follows:

  • To every $3$-cocycle $\alpha : G \times G \times G \rightarrow \Gamma $, we can associate a $2$-category $B \operatorname{\mathcal{C}}$ in which the $1$-morphisms are the elements of $G$, the $2$-morphisms are the elements of $\Gamma $, and the associativity constraint is given by $\alpha $.

  • If $\alpha , \alpha ': G \times G \times G \rightarrow \Gamma $ are cohomologous $3$-cocycles on $G$ with values in $\Gamma $, then the associated $2$-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{C}}'$ are isomorphic (though not necessarily strictly isomorphic). More precisely, every choice of $2$-cocycle $\mu : G \times G \rightarrow \Gamma $ satisfying $\alpha ' = \alpha + \partial (\mu )$ determines a strictly unitary isomorphism from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{C}}'$. Here $\partial $ denotes the boundary operator from $2$-cochains to $3$-cocycles, given concretely by the formula

    \[ (\partial \mu )_{h,g,f} =h( \mu _{g,f} ) - \mu _{hg,f} + \mu _{h,gf} - \mu _{h,g}. \]