Geometric differentiation of simplicial manifolds

TL;DR

This paper develops a comprehensive geometric framework for differentiating simplicial manifolds, extending classical Lie theory and introducing higher Lie algebroids, with applications to cohomology and the van Est map.

Contribution

It introduces a unifying geometric approach to differentiating simplicial manifolds, including a normal form theorem, higher Lie algebroids, and a higher van Est map.

Findings

Established a normal form theorem with compatible tubular neighborhoods.

Identified a differentiating ideal and constructed higher Lie algebroids.

Proved a van Est isomorphism theorem in cohomology.

Abstract

We provide a complete geometric solution to the problem of differentiating simplicial manifolds, extending classical Lie theory and complementing existing homotopical and formal approaches within a unifying framework. First, we establish a normal form theorem setting a system of compatible tubular neighborhoods. Building on this description, we identify a differentiating ideal in the algebra of cochains, prove that the quotient is semi-free, and interpret it as the Chevalley-Eilenberg algebra of the thus defined higher Lie algebroid. As an application, we introduce a higher version of the van Est map and prove a van Est isomorphism theorem in cohomology, under natural connectivity assumptions. Finally, we identify the algebraic mechanism underlying geometric differentiation as a monoidal refinement of the dual Dold-Kan correspondence, providing a conceptual explanation of the construction and relating it to earlier homotopical and functor-of-points approaches.