Higher Lie theory in positive characteristic

The main goal of this article is to develop integration theory for absolute partition $L_\infty$-algebras, which are point-set models for the (spectral) partition Lie algebras of Brantner-Mathew where infinite sums of operations are well-defined by definition. We construct a Quillen adjunction between absolute partition $L_\infty$-algebras and simplicial sets, and show that the right adjoint is a well-behaved integration functor. Points in this simplicial set are given by solutions to a Maurer-Cartan equation, and we give explicit formulas for gauge equivalences between them. We construct the analogue of the Baker-Campbell-Hausdorff formula in this setting and show it produces an isomorphic group to the classical one over a characteristic zero field. We apply these constructions to show that absolute partition $L_\infty$-algebras encode the $p$-adic homotopy types of pointed connected finite nilpotent spaces, up to certain equivalences which we describe by explicit formulas. In particular, these formulas also allow us to give a combinatorial description of the homotopy groups of the $p$-completed spheres as solutions to a certain equation in a given degree, up to an equivalence relation imposed by elements one degree above. Finally, we construct absolute partition $L_\infty$ models for $p$-adic mapping spaces, which combined with the description of the homotopy groups gives an algebraic description of the homotopy type of these $p$-adic mapping spaces parallel to the unstable Adams spectral sequence.