  A Lie 2-algebra is a linear category equipped with a functorial bilinear
operation satisfying skew-symmetry and Jacobi identity up to natural
transformations which themselves obey coherence laws of their own. Functors and
natural transformations between Lie 2-algebras can also be defined, yielding a
2-category. Passing to the normalized chain complex gives an equivalence of
2-categories between Lie 2-algebras and 2-term "homotopy everything" Lie
algebras; for strictly skew-symmetric Lie 2-algebras, these reduce to
$L_\infty$-algebras, by a result of Baez and Crans. Lie 2-algebras appear
naturally as infinitesimal symmetries of solutions of the Maurer--Cartan
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.
