Example 2.1.6.15 (00EC): $1$-Cochains as Natural Transformations

Example 2.1.6.15 ($1$-Cochains as Natural Transformations). Let $G$ be a group, let $\Gamma $ be an abelian group equipped with an action of $G$, and choose a pair of $3$-cocycles

\[ \alpha , \alpha ': G \times G \times G \rightarrow \Gamma , \]

which we can regard as associativity constraints for monoidal categories $\operatorname{\mathcal{C}}(\alpha )$ and $\operatorname{\mathcal{C}}(\alpha ')$ having the same underlying category $\operatorname{\mathcal{C}}$ (Example 2.1.6.8). Suppose we are given a pair of monoidal structures $\mu $ and $\mu '$ on the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$, which we can identify with $2$-cochains $\mu , \mu ': G \times G \rightarrow \Gamma $ satisfying

\[ \alpha + d \mu = \alpha ' \quad \quad \alpha + d \mu ' = \alpha '. \]

Then the difference $\nu = \mu - \mu '$ is a $2$-cocycle: that is, it satisfies the identity

\[ x \nu _{y,z} - \nu _{xy,z} + \nu _{x,yz} - \nu _{x,y} = 0 \]

for every triple of elements $x,y,z \in G$.

Note that a natural transformation from the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$ to itself can be identified with a function

\[ \gamma : G \rightarrow \Gamma \quad \quad x \mapsto \gamma _{x}; \]

that is, with a $1$-cochain on $G$ taking values in the group $\Gamma $. Unwinding the definitions, we see that the natural transformation $\gamma $ is monoidal (with respect to the monoidal structures supplied by $\mu $ and $\mu '$, respectively) if and only if it satisfies the identity

\[ \mu '_{x,y} + x \gamma _{y} + \gamma _{x} = \mu _{x,y} + \gamma _{xy} \]

for every pair of elements $x,y \in G$. We can rewrite this identity more conceptually as $\mu ' + d \gamma = \mu $, where

\[ d: \{ \text{$1$-Cochains $G \rightarrow \Gamma $} \} \rightarrow \{ \text{$2$-Cochains $G \times G \rightarrow \Gamma $} \} \]

is defined by the formula $(d\gamma )_{x,y} = x(\gamma _{y}) - \gamma _{xy} + \gamma _{x}$. In particular, the monoidal functors $(\operatorname{id}_{\operatorname{\mathcal{C}}}, \mu )$ to $( \operatorname{id}_{\operatorname{\mathcal{C}}}, \mu ' )$ are isomorphic if and only if the $2$-cocycle $\nu = \mu - \mu '$ is a coboundary: that is, it has vanishing image in the cohomology group $\mathrm{H}^{2}(G; \Gamma )$.