Gauge equivalence for complete $L_{\mathrm{\infty }}$-algebras

We introduce a notion of left homotopy for Maurer–Cartan elements in $L_{\mathrm{\infty }}$-algebras and $A_{\mathrm{\infty }}$-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 $L_{\mathrm{\infty }}$-algebra.