Compositio Mathematica

Research Article

Integrating $L_\infty $-algebras

André Henriquesa1

a1 Mathematisch Instituut, Universiteit Utrecht, PO Box 80.010 NL-3508, TA Utrecht, The Netherlands (email:


Given a Lie n-algebra, we provide an explicit construction of its integrating Lie n-group. This extends work done by Getzler in the case of nilpotent $L_\infty $-algebras. When applied to an ordinary Lie algebra, our construction yields the simplicial classifying space of the corresponding simply connected Lie group. In the case of the string Lie 2-algebra of Baez and Crans, we obtain the simplicial nerve of their model of the string group.

(Received March 26 2006)

(Accepted November 24 2007)

2000 Mathematics subject classification

  • 53C05;
  • 55U10;
  • 58B10


  • L_\infty-algebra;
  • simplicial manifold;
  • string group;
  • Lie 2-group;
  • Lie 2-algebra;
  • Kan condition