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Geometric differentiation of simplicial manifolds

Alejandro Cabrera, Matias del Hoyo

TL;DR

The paper builds a comprehensive geometric framework for differentiating simplicial manifolds to higher Lie algebroids. Central to the construction is the differentiating ideal $\hat{J}$, yielding the Chevalley–Eilenberg algebra $CE(A_G)=C_N(G)/\hat{J}$ and a semi-free higher Lie algebroid $A_G$, with a parallel form theory given by the Weil algebra $W(A_G)$. The authors establish a higher van Est theorem under connectivity hypotheses using a decalage-based interpolating double complex and hypercover techniques, and they show that differentiation is governed by a monoidal adjunction $N'\dashv K$ that generalizes the Dold–Kan correspondence to infinitesimal cosimplicial algebras. They also connect their geometric picture to Severa’s functorial differentiations in the supergeometric setting, providing a concrete, computable framework for higher geometric structures, including shifted symplectic forms and higher stacks. Overall, the work unifies global and infinitesimal viewpoints, offering new tools for integration, deformation, and moduli problems in higher differential geometry.

Abstract

We provide a complete geometric solution to the differentiation problem for simplicial manifolds, extending classical Lie theory and subsuming existing homotopical and formal approaches within a unified 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.

Geometric differentiation of simplicial manifolds

TL;DR

The paper builds a comprehensive geometric framework for differentiating simplicial manifolds to higher Lie algebroids. Central to the construction is the differentiating ideal , yielding the Chevalley–Eilenberg algebra and a semi-free higher Lie algebroid , with a parallel form theory given by the Weil algebra . The authors establish a higher van Est theorem under connectivity hypotheses using a decalage-based interpolating double complex and hypercover techniques, and they show that differentiation is governed by a monoidal adjunction that generalizes the Dold–Kan correspondence to infinitesimal cosimplicial algebras. They also connect their geometric picture to Severa’s functorial differentiations in the supergeometric setting, providing a concrete, computable framework for higher geometric structures, including shifted symplectic forms and higher stacks. Overall, the work unifies global and infinitesimal viewpoints, offering new tools for integration, deformation, and moduli problems in higher differential geometry.

Abstract

We provide a complete geometric solution to the differentiation problem for simplicial manifolds, extending classical Lie theory and subsuming existing homotopical and formal approaches within a unified 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.
Paper Structure (18 sections, 41 theorems, 106 equations)

This paper contains 18 sections, 41 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Frames on simplicial manifolds
  3. From Lie groupoids to simplicial manifolds
  4. Microbundles, tubular neighborhoods and frames
  5. Constructing frames
  6. The differentiation functor
  7. The differentiating ideal
  8. Normal coordinates and monomials
  9. Higher Lie algebroids and the Differentiation Theorem
  10. Differentiation at the level of forms
  11. The higher van Est isomorphism theorem
  12. The decalage and the interpolating double complex
  13. Hypercovers at the infinitesimal level
  14. Connectedness, the van Est theorem and applications
  15. Algebraic aspects of differentiation
  16. ...and 3 more sections

Key Result

Theorem 1.0.1

Every simplicial manifold $(G_n,d_i,s_j)$ admits a frame $\phi=(\phi_n)_{n\geq1}$, namely a family of microbundle tubular neighborhoods $\phi_n:(\nu(G_n,G_0),G_0,e)\rightarrow(G_n,G_0,e)$ compatible with the positive faces and the degeneracies, $d_i\phi_n=\phi_{n-1}d_i$ for $i>0$, and $s_j\phi_n=\ph

Theorems & Definitions (125)

  • Theorem 1.0.1: Frame
  • Theorem 1.0.2: Differentiation
  • Theorem 1.0.3: van Est
  • Theorem 1.0.4: Abstract differentiation
  • Remark 2.1.1
  • Definition 2.1.2
  • Example 2.1.3
  • Remark 2.1.4
  • Definition 2.1.5
  • Example 2.1.6
  • ...and 115 more