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# Homology, Homotopy and Applications

## Volume 23 (2021)

### Number 2

### Gauge equivalence for complete $L_\infty$-algebras

Pages: 283 – 297

DOI: https://dx.doi.org/10.4310/HHA.2021.v23.n2.a15

#### Author

#### Abstract

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

Accepted 23 December 2020