Higher Lie theory in positive characteristic
Victor Roca i Lucio
TL;DR
This work develops a comprehensive integration theory for absolute partition $\mathcal{L}_\infty$-algebras in positive characteristic, providing a Quillen adjunction with simplicial sets and proving the right adjoint yields a well-behaved integration functor that produces Kan complexes from qp-complete algebras.It delivers explicit MC solutions, gauge equivalences, and a characteristic free analogue of the Baker–Campbell–Hausdorff formula via horn-fillers expressed through left-handed and corked planar trees, unifying deformation theory with algebraic models in positive characteristic.Applying these constructions to $p$-adic homotopy theory, the paper shows absolute partition $\mathcal{L}_\infty$-algebras model the $p$-adic homotopy types of pointed connected finite nilpotent spaces, provides a combinatorial description of homotopy groups of $p$-completed spheres, and furnishes algebraic models for mapping spaces, all while connecting to established frameworks in characteristic zero.Overall, the work extends higher Lie theory to positive characteristic, clarifies how to recover classical moduli problems algebraically, and offers a robust toolkit for studying $p$-adic homotopy types through algebraic means.
Abstract
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.