The Chern-Weil-Lecomte characteristic map for $L_{\infty }$-algebras

Abstract

In this paper we extend the Chern-Weil-Lecomte characteristic map to the setting of $L_{\infty }$-algebras. In this general framework, characteristic classes of $L_{\infty }$-algebra extensions are defined by means of the Chern-Weil-Lecomte map which takes values in the cohomology of an $L_{\infty }$-algebra with coefficients in a representation up to homotopy. This general set up allows us to recover several known cohomology classes in a unified manner, including: the characteristic class of a Lie 2-algebra, the Ševera class of an exact Courant algebroid and the curvature 3-form of a gerbe with connective structure. We conclude by introducing a Chern-Weil map for principal 2-bundles over Lie groupoids.