A differentiable n -cochain of G ⇉ ⇉ M with coefficients in some representation E → → M is a smooth section of the pullback vector bundle ( t ( n ) ) ∗ ∗ E → → G ( n ) . One denotes by C n ( G , E ) the space of such n -cochains, and considers the differential d n : C n ( G , E ) → → C n + 1 ( G , E ) , defined as