  The main goal of this article is to develop integration theory over a
positive characteristic field, and as a by-product, construct Lie-type models
for the $p$-adic homotopy types of spaces. We introduce a new type of algebraic
structure called curved \textit{absolute} partition
$\mathcal{L}_\infty$-algebra, which always admits a well-defined Maurer-Cartan
equation. We construct a Quillen adjunction between curved absolute partition
$\mathcal{L}_\infty$-algebras and simplicial sets, where the right adjoint is
the integration functor. We show that it satisfies the same properties that its
characteristic zero analogues. We also give an explicit combinatorial
description of gauge equivalences. And using our constructions, we show how one
can recover the $\mathbb{E}_\infty$-formal moduli problem associated to a
(spectral) partition $\mathcal{L}_\infty$-algebra, under a homotopy
completeness assumption. Finally, we use the left adjoint of the integration
functor to construct Quillen-like models for the $p$-adic homotopy types of
spaces and of their mapping spaces.
