  We introduce a notion of left homotopy for Maurer--Cartan elements in
$L_{\infty}$-algebras and $A_{\infty}$-algebras, and show that it corresponds
to gauge equivalence in the differential graded case. From this we deduce a
short formula for gauge equivalence, and provide an entirely homotopical proof
to Schlessinger--Stasheff's theorem. As an application, we answer a question of
T. Voronov, proving a non-abelian Poincar\'e lemma for differential forms
taking values in an $L_{\infty}$-algebra.
