Group laws and commutativity conditions on $n$-categories

As the integer $n$ grows, so does the number of distinct natural commutativity conditions which may be satisfied by a given higher group law on an $n$-category (or more generally an $n$-stack). I will describe these conditions for small values of $n$, and discuss their relation with the cohomology of Eilenberg–Mac Lane spaces. This relation also points to some interesting multilinear invariants which may be attached to these categories and stacks.