Example 2.1.3.3 (00CS): Group Cocycles

Example 2.1.3.3 (Group Cocycles). Let $G$ be a group with identity element $1 \in G$, and let $\Gamma $ be an abelian group on which $G$ acts by automorphisms; we denote the action of an element $g \in G$ by $(\gamma \in \Gamma ) \mapsto g(\gamma ) \in \Gamma $. A $3$-cocycle on $G$ with values in $\Gamma $ is a map of sets

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

which satisfies the equations

2.1

\begin{eqnarray} \label{equation:cocycle-identity} w( \alpha _{x,y,z} ) - \alpha _{wx,y,z} + \alpha _{w,xy,z} - \alpha _{w,x,yz} + \alpha _{w,x,y} = 0 \end{eqnarray}

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

Let $\operatorname{\mathcal{C}}$ denote the category whose objects are the elements of $G$, and whose morphisms are given by

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

Using the action of $G$ on $\Gamma $, we can construct a functor

\[ \otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}, \]

given on objects by $(g,h) \mapsto gh$ and on morphisms by

\[ ( (\gamma : g \rightarrow g), (\delta : h \rightarrow h) ) \mapsto (\gamma + g(\delta ): gh \rightarrow gh). \]

Unwinding the definitions, one sees that upgrading the functor $\otimes $ to a nonunital monoidal structure on the category $(\otimes , \alpha )$ on $\operatorname{\mathcal{C}}$ is equivalent to choosing a $3$-cocycle $\alpha : G \times G \times G \rightarrow \Gamma $. More precisely, any map $\alpha : G \times G \times G \rightarrow \Gamma $ can be regarded as a natural transformation of functors

\[ \bullet \otimes ( \bullet \otimes \bullet ) \rightarrow (\bullet \otimes \bullet ) \otimes \bullet , \]

and pentagon identity $(P)$ of Definition 2.1.1.5 translates to the cocycle condition (2.1) above.

For any choice of cocycle $\alpha : G \times G \times G \rightarrow \Gamma $, we can upgrade the associated nonunital monoidal structure $(\otimes , \alpha )$ to a monoidal structure on the category $\operatorname{\mathcal{C}}$, by taking the unit object of $\operatorname{\mathcal{C}}$ to be the identity element $1 \in G$ and the unit constraint $\upsilon : 1 \otimes 1 \simeq 1$ to be the element $0 \in \Gamma $.