Classically, cohomologies of Lie teams/algebras parametrize extensions. To be exact, given an linear $G$-action on $M$, there’s an bijection between $H^2(G;M)$ and the set of extension $E$ of $G$ by $M$

$$ 1to Mto Eto Gto 1.$$

Apparently, increased cohomologies correspond to longer extensions reminiscent of

$$ 1to Mto Fto Eto Gto 1.$$

Comparable statements maintain for Lie algebras.

In Greater-Dimensional Algebra VI: Lie 2-Algebras, J. Baez et al confirmed one other which means of upper cohomologies. Given a Lie algebra $frak{g}$, a linear illustration $V$, and a 3-cohomology class $alpha in H^3(frak{g}$$, M)$. In actual fact, they confirmed that all Lie 2-algebras come up this fashion!

A naive however pure query is whether or not this story extends to increased circumstances. Particularly, do $(n+1)$-th cohomology classifies Lie n-algebras? In that case, how does can we join this again to the classical extension idea?