Dmitry Roytenberg
Weak Lie 2algebras
A Lie 2algebra is a linear category equipped with a functorial bilinear operation satisfying skewsymmetry and Jacobi identity up to natural transformations which themselves obey coherence laws of their own. Functors and natural transformations between Lie 2algebras can also be defined, yielding a 2category. Passing to the normalized chain complex gives an equivalence of 2categories between Lie 2algebras and certain "up to homotopy" structures on the complex; for strictly skewsymmetric Lie 2algebras these are $L_\infty$algebras, by a result of Baez and Crans.
Lie 2algebras appear naturally as infinitesimal symmetries of solutions of the MaurerCartan equation in some differential graded Lie algebras and $L_\infty$algebras. In particular, (quasi) Poisson manifolds, (quasi) Lie bialgebroids and Courant algebroids provide large classes of examples
