On exponential groups and Maurer–Cartan spaces

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.