Example 2.1.6.8 (00E5): $2$-Cochains as Monoidal Structures

Example 2.1.6.8 ($2$-Cochains as Monoidal Structures). Let $G$ be a group and let $\Gamma $ be an abelian group equipped with an action of $G$. Let $\operatorname{\mathcal{C}}$ be the category introduced in Example 2.1.3.3, whose objects are the elements of $G$ and morphisms are given by

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(g,h) = \begin{cases} \Gamma & \text{ if } g =h \\ \emptyset & \text{otherwise.} \end{cases} \]

Then every $3$-cocycle $\alpha : G \times G \times G \rightarrow \Gamma $ can be regarded as the associativity constraint for a monoidal structure $(\otimes , \alpha )$ on $\operatorname{\mathcal{C}}$. Let us write $\operatorname{\mathcal{C}}(\alpha )$ to indicate the category $\operatorname{\mathcal{C}}$, endowed with the monoidal structure $(\otimes , \alpha )$.

Suppose that we are given a pair of cocycles $\alpha ,\alpha ': G \times G \times G \rightarrow \Gamma $. Unwinding the definitions, we see that monoidal structures on the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}(\alpha ) \rightarrow \operatorname{\mathcal{C}}(\alpha ')$ are given by functions

\[ \mu : G \times G \rightarrow \Gamma \quad \quad (x,y) \mapsto \mu _{x,y} \]

which satisfy the identity

\[ \alpha _{x,y,z} + \mu _{x,yz} + x( \mu _{y,z} ) = \mu _{xy,z} + \mu _{x,y} + \alpha '_{x,y,z} \]

for $x,y,z \in G$. We can rewrite this identity more compactly as an equation $\alpha + d \mu = \alpha '$, where

\[ d: \{ \text{$2$-Cochains $G \times G \rightarrow \Gamma $} \} \rightarrow \{ \text{$3$-Cochains $G \times G \times G \rightarrow \Gamma $} \} \]

is defined by the formula $(d\mu )_{x,y,z} = x(\mu _{y,z}) - \mu _{xy,z} + \mu _{x,yz} - \mu _{x,y}$.

In particular, the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$ can be promoted to a monoidal functor from $\operatorname{\mathcal{C}}(\alpha )$ to $\operatorname{\mathcal{C}}(\alpha ')$ if and only if the cocycles $\alpha $ and $\alpha '$ are cohomologous: that is, they represent the same element of the cohomology group $\mathrm{H}^{3}( G; \Gamma )$.