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

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 $\mathrm{\Gamma }\mathrm{\Lambda }{\mathcal{P}\mathcal{L}}_{\mathrm{\infty }}$-algebra which characterizes the notion of $\mathrm{\Gamma }(\mathcal{P}re\mathcal{L}i{e}_{\mathrm{\infty }},-)$-algebra. We also define a notion of a Maurer-Cartan element in complete $\mathrm{\Gamma }\mathrm{\Lambda }{\mathcal{P}\mathcal{L}}_{\mathrm{\infty }}$-algebras which generalizes the classical definition in Lie algebras. We prove that for every complete brace algebra $A$, and for every $n\ge 0$, the tensor product $A\otimes \mathrm{\Sigma }{N}^{\ast }({\mathrm{\Delta }}^n)$ is endowed with the structure of a complete $\mathrm{\Gamma }\mathrm{\Lambda }{\mathcal{P}\mathcal{L}}_{\mathrm{\infty }}$-algebra, and define the simplicial Maurer-Cartan set ${\mathcal{M}\mathcal{C}}_{\bullet }(A)$ associated to $A$ as the Maurer-Cartan set of $A\otimes \mathrm{\Sigma }{N}^{\ast }({\mathrm{\Delta }}^{\bullet })$. We compute the homotopy groups of this simplicial set, and prove that the functor ${\mathcal{M}\mathcal{C}}_{\bullet }(-)$ 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.