Classically, cohomologies of Lie groups/algebras parametrize extensions. To be precise, given an linear $G$-action on $M$, there is an bijection between $H^2(G;M)$ and the set of extension $E$ of $G$ by $M$
$$ 1\to M\to E\to G\to 1.$$
Apparently, higher cohomologies correspond to longer extensions such as
$$ 1\to M\to F\to E\to G\to 1.$$
Similar statements hold for Lie algebras.
In Higher-Dimensional Algebra VI: Lie 2-Algebras, J. Baez et al showed another meaning of higher cohomologies. Given a Lie algebra $\frak{g}$, a linear representation $V$, and a 3-cohomology class $\alpha \in H^3(\frak{g}$$, M)$ we can construct a Lie 2-algebra. In fact, they showed that all Lie 2-algebras arise this way!
A naive but natural question is whether this story extends to higher cases. Namely, do $(n+1)$-th cohomology classifies Lie n-algebras, if any? If so, how does do we connect this back to the classical extension theory?
