Pre-Lie algebras up to homotopy with divided powers and homotopy of operadic mapping spaces

Abstract

The purpose of this memoir is to study pre-Lie algebras up to homotopy with divided powers, and to use this algebraic structure for the study of mapping spaces in the category of operads. We define a new notion of algebra called $\Gamma\Lambda\mathcal{PL}_\infty$-algebra which characterizes the notion of $\Gamma(\mathcal{P}re\mathcal{L}ie_\infty,-)$-algebra. We also define a notion of a Maurer-Cartan element in complete $\Gamma\Lambda\mathcal{PL}_\infty$-algebras which generalizes the classical definition in Lie algebras. We prove that for every complete brace algebra $A$, and for every $n\geq 0$, the tensor product $A\otimes\Sigma N^*(\Delta^n)$ is endowed with the structure of a complete $\Gamma\Lambda\mathcal{PL}_\infty$-algebra, and define the simplicial Maurer-Cartan set $\mathcal{MC}_\bullet(A)$ associated to $A$ as the Maurer-Cartan set of $ A\otimes\Sigma N^*(\Delta^\bullet)$. We compute the homotopy groups of this simplicial set, and prove that the functor $\mathcal{MC}_\bullet(-)$ satisfies a homotopy invariance result, which extends the Goldman-Millson theorem in dimension $0$. As an application, we give a description of mapping spaces in the category of non-symmetric operads in terms of this simplicial Maurer-Cartan set. We etablish a generalization of the latter result for symmetric operads.