The 2-cocycle and 3-cocycle equations for group cohomology with coefficients in U(1) arise naturally from associativity relations. Given a projective representation $U(g_1)U(g_2)=\omega(g_1,g_2)U(g_1g_2)$, the 2-cocycle equation arises from requiring the two decompositions of $U(g_1g_2g_3)$ to equal one another. Similarily, the 3-cocycle equation is a special case of the pentagon equation which is itself an associativity relation. I am wondering if there is a way to similarily understand the $n$-cocycle equations for arbitrary $n$.
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