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Differentiable groupoid objects and their abstract Lie algebroids

Lory Aintablian, Christian Blohmann

TL;DR

The paper develops a categorical framework to differentiate groupoid objects beyond smooth manifolds by using Rosický’s abstract tangent structures and a scalar multiplication by a ring object. It defines differentiable groupoid objects, higher vertical tangent bundles, and invariant vector fields, proving that invariant vector fields form a Lie subalgebra and give rise to an abstract Lie algebroid via a differentiation procedure. The main result shows that differentiable groupoids in cartesian tangent categories yield abstract Lie algebroids whose bracket is induced by invariant vector fields, unifying classical Lie groupoid theory with a broad, axiomatic setting. This approach enables diffeological and elastic settings (e.g., diffeomorphism groups, gauge groups, and holonomy groupoids) to be treated uniformly, with applications to general relativity, diffeological stacks, and geometric deformation theory.

Abstract

The infinitesimal counterpart of a Lie groupoid is its Lie algebroid. As a vector bundle, it is given by the source vertical tangent bundle restricted to the identity bisection. Its sections can be identified with the invariant vector fields on the groupoid, which are closed under the Lie bracket. We generalize this differentiation procedure to groupoid objects in any category with an abstract tangent structure in the sense of Rosický and a scalar multiplication by a ring object that plays the role of the real numbers. We identify the categorical conditions that the groupoid object must satisfy to admit a natural notion of invariant vector fields. Then we show that invariant vector fields are closed under the Lie bracket defined by Rosický and satisfy the Leibniz rule with respect to ring-valued morphisms on the base of the groupoid. The result is what we define axiomatically as an abstract Lie algebroid, by generalizing the underlying vector bundle to a module object in the slice category over its base. Examples include diffeomorphism groups, bisection groups of Lie groupoids, the diffeological symmetry groupoids of general relativity (Blohmann/Fernandes/Weinstein), symmetry groupoids in Lagrangian Field Theory, holonomy groupoids of singular foliations, elastic diffeological groupoids, groupoid objects in differentiable stacks, and affine groupoid schemes.

Differentiable groupoid objects and their abstract Lie algebroids

TL;DR

The paper develops a categorical framework to differentiate groupoid objects beyond smooth manifolds by using Rosický’s abstract tangent structures and a scalar multiplication by a ring object. It defines differentiable groupoid objects, higher vertical tangent bundles, and invariant vector fields, proving that invariant vector fields form a Lie subalgebra and give rise to an abstract Lie algebroid via a differentiation procedure. The main result shows that differentiable groupoids in cartesian tangent categories yield abstract Lie algebroids whose bracket is induced by invariant vector fields, unifying classical Lie groupoid theory with a broad, axiomatic setting. This approach enables diffeological and elastic settings (e.g., diffeomorphism groups, gauge groups, and holonomy groupoids) to be treated uniformly, with applications to general relativity, diffeological stacks, and geometric deformation theory.

Abstract

The infinitesimal counterpart of a Lie groupoid is its Lie algebroid. As a vector bundle, it is given by the source vertical tangent bundle restricted to the identity bisection. Its sections can be identified with the invariant vector fields on the groupoid, which are closed under the Lie bracket. We generalize this differentiation procedure to groupoid objects in any category with an abstract tangent structure in the sense of Rosický and a scalar multiplication by a ring object that plays the role of the real numbers. We identify the categorical conditions that the groupoid object must satisfy to admit a natural notion of invariant vector fields. Then we show that invariant vector fields are closed under the Lie bracket defined by Rosický and satisfy the Leibniz rule with respect to ring-valued morphisms on the base of the groupoid. The result is what we define axiomatically as an abstract Lie algebroid, by generalizing the underlying vector bundle to a module object in the slice category over its base. Examples include diffeomorphism groups, bisection groups of Lie groupoids, the diffeological symmetry groupoids of general relativity (Blohmann/Fernandes/Weinstein), symmetry groupoids in Lagrangian Field Theory, holonomy groupoids of singular foliations, elastic diffeological groupoids, groupoid objects in differentiable stacks, and affine groupoid schemes.
Paper Structure (81 sections, 45 theorems, 233 equations)

This paper contains 81 sections, 45 theorems, 233 equations.

Table of Contents

  1. Introduction
  2. Goal of the paper
  3. Lie groupoids and Lie algebroids
  4. Differentiation of Lie groupoids
  5. The Lie bracket of vector fields
  6. Bundles
  7. Submersions and differentiability
  8. Main results
  9. Applications
  10. Relation to previous work
  11. Outline of the paper
  12. Conventions and notation
  13. Abstract tangent structures
  14. Bundles with algebraic structure
  15. Bundles of abelian groups
  16. ...and 66 more sections

Key Result

Theorem 1

Let $G$ be a differentiable groupoid object in a cartesian tangent category $\mathcal{C}$ with scalar $R$-multiplication. Then there is an abstract Lie algebroid $A \to G_0$ with the Lie bracket of invariant vector fields on $\Gamma(G_0, A) \cong \mathcal{X}(G_1)^G$.

Theorems & Definitions (118)

  • Definition : Definition \ref{['def:DiffGroupoid']}
  • Definition : Definition \ref{['def:AbstractLieAlgd']}
  • Theorem : Theorem \ref{['thm:SecAInvVec']}, Theorem \ref{['thm:LieAlgdOfGroupoid']}
  • Definition 2.1
  • Proposition 2.2
  • Example 2.3
  • Proposition 2.5
  • Example 2.7
  • Definition 2.8
  • Proposition 2.9
  • ...and 108 more