On weak Lie 2‐algebras

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 certain “up to homotopy” structures on the complex; for strictly skew‐symmetric Lie 2‐algebras these are $L_{\infty }$‐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.