Gauge equivalence for complete $L_\infty$-algebras

Abstract

We introduce a notion of left homotopy for Maurer-Cartan elements in L∞-algebras and A∞-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∞-algebra.