Higher differentiation via higher formal groupoids

TL;DR

This paper develops a homotopy-theoretic framework for differentiating Lie infinity-groups, extending classical duality and algebraic techniques to higher geometric structures, resulting in a well-behaved differentiation functor.

Contribution

It introduces a homotopically sound, explicit differentiation functor for Lie infinity-groups to Lie infinity-algebras, expanding the tools for higher geometric and algebraic analysis.

Findings

Differentiation of classifying spaces yields dg Lie algebras of normalized chains.

Constructs a homotopy-theoretic framework for pointed formal infinity-groupoids.

Provides an explicit, coordinate-free differentiation functor for Lie infinity-groups.

Abstract

We solve the differentiation problem for Lie $\infty$-groups. Our approach builds on a classical version of Cartier duality which canonically identifies the Hopf algebra of point distributions supported at the identity of a Lie group with the universal enveloping algebra of its Lie algebra. Hence, for Lie $\infty$-groups, we consider simplicial coalgebras of point distributions. To do this properly, we first develop the homotopy theory of pointed formal $\infty$-groupoids within K. Behrend and E. Getzler's framework for higher geometric stacks. These objects are "higher" but not "derived", which is an important distinction for the geometric applications in mind. The second part of our construction relies on a careful analysis of J. Pridham's variation of the Dold-Kan adjunction for cosimplicial algebras. Our main result is a differentiation functor at the level of 1-categories from finite-dimensional Lie $\infty$-groups to finite-type Lie $\infty$-algebras that is homotopically well-behaved, coordinate-free, and explicit yet tractable. In particular, if $G_\bullet$ is a simplicial Lie group with Lie algebra $\mathfrak{g}_\bullet$, we prove that the differentiation of its classifying space $\overline{\mathscr{W}}_{\! \bullet}G$ is canonically isomorphic to the dg Lie algebra of normalized chains $N_\ast(\mathfrak{g}_\bullet)$.