3-cocycle for trivial group action

Importance
3-cocycles come up in the problem of classifying pointed $$k$$-linear fusion categories. These are essentially monoidal categories whose objects are one-dimensional $$k$$-vector spaces and whose isomorphism classes are indexed by the elements of a group, and where the morphisms are $$k$$-linear maps. and the monoidal operation induces group multiplication on the group elements representing the isomorphism class. The operation is not strictly associative. Rather there is a natural associativity isomorphism that satisfies the pentagon axiom. To each natural associativity isomorphism we can thus associate an element of $$k^*$$, and this gives a function:

$$f: G \times G \times G \to k^*$$

It turns out that the pentagon axiom translates to the condition that this function must be a 3-cocycle for the trivial group action.

Further, it turns out that for the same category, any two different functions differ by a 3-coboundary, so the categories over a particular group are parametrized by elements of the third cohomology group $$H^3(G,k^*)$$ for the trivial action.