Let $G$ be a Lie group and $p$ a positive integer. A smooth $p$-cochain on $G$ is an element of $$C^p(G):=C^\infty(\underbrace{G\times \ldots \times G}_{p}).$$ We can define coboundary operators $\delta^p:C^p(G)\longrightarrow C^{p+1}(G)$ setting $$(\delta^p f)(g_0, \ldots, g_p):=f(g_1, \ldots, g_p)+\sum_{j=1}^{p} (-1)^j f(g_0, \ldots, g_{p-1} g_p, \ldots, g_p)+(-1)^{p+1} f(g_0, \ldots, g_{p-1}).$$ This will give us $\mathbb R$-linear maps such that $\delta^{p+1}\circ \delta^p=0$ for every $p$. Therefore we have a complex of $\mathbb R$-vector spaces $\{(C^p(G), \delta^p)\}$ and we get a cohomology theory for $G$.
What is the motivation behind the definition of those $\delta^p$? Does anyone know its origins? References are also welcome.
I'm using here cohomology of Lie groups just to illustrate the situation but several others cohomologies are obtained using coboundary operators like the above one.
Thanks.