  We use the Thom-Whitney construction to show that infinitesimal deformations
of a coherent sheaf F are controlled by the differential graded Lie algebra of
global sections of an acyclic resolution of the sheaf End(E), where E is any
locally free resolution of F. In particular, one recovers the well known fact
that the tangent space to deformations of F is Ext^1(F,F), and obstructions are
contained in Ext^2(F,F).
  The main tool is the identification of the deformation functor associated
with the Thom-Whitney DGLA of a semicosimplicial DGLA whose cohomology is
concentrated in nonnegative degrees with a noncommutative Cech cohomology-type
functor.
