  The purpose of this note is to give a concise account of some fundamental
properties of the exponential group and the Maurer-Cartan space associated to a
complete dg Lie algebra. In particular, we give a direct elementary proof that
the Maurer-Cartan space is a delooping of the exponential group. This leads to
a short proof that the Maurer-Cartan space functor is homotopy inverse to
Quillen's functor from simply connected pointed spaces to positively graded dg
Lie algebras.
