To any non-negatively graded dg Lie algebra g over a field k of
characteristic zero we assign a functor Σg​:art/k→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 Σg​ and the Deligne groupoid corresponding to g.
The main result of the paper claims that the functor Σ 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 RΓ(X,g).Comment: Minor corrections made AMSLaTeX v 1.2 (Compatibility mode