Pre-Lie algebras with divided powers and the Deligne groupoid in positive characteristic

The purpose of this paper is to develop a deformation theory controlled by pre-Lie algebras with divided powers over a ring of positive characteristic. We show that every differential graded pre-Lie algebra with divided powers comes with operations, called weighted braces, which we use to generalize the classical deformation theory controlled by Lie algebras over a field of characteristic $0$. Explicitly, we define the Maurer-Cartan set, as well as the gauge group, and prove that there is an action of the gauge group on the Maurer-Cartan set. This new deformation theory moreover admits a Goldman-Millson theorem which remains valid on the integers. As an application, we give the computation of the ${\pi }_0$ of a mapping space $\text{Map}(B^c(\mathcal{C}),\mathcal{P})$ with $\mathcal{C}$ and $\mathcal{P}$ suitable cooperad and operad respectively.