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.

Keywords

Maurer–Cartan element, differential graded Lie algebra, homotopy, model category, deformation

2010 Mathematics Subject Classification

17B55, 18G55

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Accepted 23 December 2020

Published 3 November 2021