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

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 Poincaré lemma for differential forms taking values in an $L_{\infty}$-algebra.

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