On weak Lie 3-algebras

Abstract

In this article, we introduce a category of weak Lie 3-algebras with suitable weak morphisms. The definition is based on the construction of a partial resolution over $\mathbb{Z}$ of the Koszul dual cooperad of the $\textrm{Lie}$ operad, with free symmetric group action. Weak Lie 3-algebras and their morphisms are then defined via the usual operadic approach---as solutions to Maurer--Cartan equations. As 2-term truncations we recover Roytenberg's category of weak Lie 2-algebras. We prove a version of the homotopy transfer theorem for weak Lie 3-algebras. A right homotopy inverse to the resolution is constructed and leads to a skew-symmetrization construction from weak Lie 3-algebras to 3-term $\textrm{L}_\infty$-algebras. Finally, we give two applications: the first is an extension of a result of Rogers comparing algebraic structures related to $n$-plectic manifolds; the second is the construction of a weak Lie 3-algebra associated to an CLWX 2-algebroid leading to a new proof of a result of Liu--Sheng.