Categories in Algebra, Geometry and Mathematical Physics
Conference in honour of Ross Street's sixtieth birthday
July 11-16 2005, Macquarie University, Sydney
Thu, 14 July: 9:00 - 10:00
Lie theory for $L_\infty$-algebras
The Deligne groupoid is a homotopy functor from nilpotent differential graded Lie algebras concentrated in positive degree to groupoids. We generalize this to a functor from nilpotent $L_\infty$-algebras concentrated in degrees $(-n, \infty)$ to $n$-groupoids, or rather, to their nerves. On restriction to abelian differential graded Lie algebras, i.e. chain complexes, our functor is the Dold--Kan functor. The construction uses methods from rational homotopy theory, and gives a generalization of the Campbell-Hausdorff formula.
If time permits, we will discuss the generalization to the non-nilpotent case.