Descent of Deligne groupoids

Abstract

To any non-negatively graded dg Lie algebra $g$ over a field $k$ of characteristic zero we assign a functor $\Sigma_g: art/k \to Kan$ from the category of commutative local artinian $k$-algebras with the residue field $k$ to the category of Kan simplicial sets. There is a natural homotopy equivalence between $\Sigma_g$ and the Deligne groupoid corresponding to $g$. The main result of the paper claims that the functor $\Sigma$ commutes up to homotopy with the "total space" functors which assign a dg Lie algebra to a cosimplicial dg Lie algebra and a simplicial set to a cosimplicial simplicial set. This proves a conjecture of Schechtman which implies that if a deformation problem is described ``locally'' by a sheaf of dg Lie algebras $g$ on a topological space $X$ then the global deformation problem is described by the homotopy Lie algebra $R\Gamma(X,g)$.