Fat Lie Theory

This paper introduces the framework of "fat Lie theory" to establish a one-to-one correspondence between fat extensions and abstract 2-term representations up to homotopy, while demonstrating that fat extensions of groupoids correspond to general linear PB-groupoids and core extensions to vertically/horizontally core-transitive double groupoids, thereby upgrading existing correspondences to equivalences of categories.

The Gist

Imagine you are trying to understand a complex machine, like a high-end camera or a spaceship. You can look at it in a few different ways:

1. The Blueprints: The abstract mathematical rules that tell you how it should work.
2. The Physical Object: The actual metal and wires you can touch.
3. The "Fat" Version: A special, slightly larger version of the machine that includes all the possible ways you could tweak or adjust it while it's running.

This paper, titled "Fat Lie Theory" by Lennart Obster, is about finding a new, super-useful way to look at these machines. In the world of mathematics, these "machines" are called Lie Groupoids and Lie Algebroids. They are tools used to describe symmetry, motion, and geometry in physics and advanced math.

Here is the simple breakdown of what the paper does, using everyday analogies.

1. The Problem: Too Many Ways to Say the Same Thing

For a long time, mathematicians have studied these "machines" using two main languages:

• VB-Groupoids: Think of this as looking at the machine as a stack of layers (like a sandwich). It's a group of shapes where every point has a little vector space attached to it.
• Representations up to Homotopy (RUTHs): This is the "blueprint" language. It describes how the machine moves and twists, but it allows for some "wiggle room" (homotopy). It's like a recipe that says, "Mix the ingredients, but it's okay if you stir them a little differently than the exact instructions."

Mathematicians knew these two languages were equivalent (they describe the same thing), but translating between them was often messy and required arbitrary choices (like picking a specific "slice" of the sandwich to start with).

2. The Solution: The "Fat" Extension

The author introduces a new concept called a Fat Extension.

The Analogy: The "Fat" Groupoid
Imagine you have a standard group of people (a Lie Groupoid) holding hands in a circle.

• The Standard View: You just see the circle.
• The "Fat" View: Now, imagine that every person in the circle is holding a small, flexible rubber band that connects them to their neighbors. These rubber bands can stretch and twist, but they must stay attached.

The "Fat Groupoid" is the collection of the people plus all the possible ways those rubber bands can be arranged. It's "fatter" because it contains more information.

Why is this useful?
The paper proves that this "Fat" version is the perfect translator.

• It connects the "Sandwich" view (VB-Groupoids) directly to the "Blueprint" view (RUTHs).
• It does this canonically, meaning you don't have to make arbitrary choices. The translation happens naturally, like water flowing downhill.

3. The "Core" of the Matter

The paper also talks about something called Core Extensions.

The Analogy: The Core of an Apple
If you have a complex fruit salad (a Double Groupoid), the "core" is the very center—the seeds.

• Usually, mathematicians study the whole salad.
• This paper says: "Hey, if you just look at the core (the seeds) and how they are arranged, you can rebuild the whole salad!"

The author shows that "Fat Extensions" are actually just a specific type of "Core Extension." This links the new theory to very old, famous math theories (like those by Brown and Mackenzie), proving that this new "Fat" idea is a natural evolution of existing knowledge, not just a random invention.

4. The "Jet" Connection (The Time Machine)

One of the coolest examples in the paper involves Jet Groupoids.

• Analogy: Imagine you are driving a car. A "Jet" is like a snapshot of your car's speed and acceleration at a specific moment.
• The paper shows that the "Fat Groupoid" of a standard machine is exactly the same as the "Jet Groupoid" (the snapshot of all possible movements).
• This means the "Fat" theory is the perfect tool for studying deformations (how a shape changes over time) and symmetries in physics.

5. The Big Picture: A New Lens

The main goal of the paper is to establish a One-to-One Correspondence.

Think of it like having three different maps of the same city:

1. Map A: Shows the roads (VB-Groupoids).
2. Map B: Shows the traffic rules and flow (RUTHs).
3. Map C: The "Fat" Map (Fat Extensions).

Before this paper, switching between Map A and Map B was like trying to translate a book from English to French using a dictionary that had missing words. You had to guess.
This paper says: "Map C is the master key. If you have Map C, you can instantly and perfectly translate to Map A or Map B without guessing."

Summary of the "Fat Lie Theory"

• What is it? A new way to organize mathematical structures called Lie Groupoids.
• The "Fat" Idea: Instead of looking at the object alone, look at the object plus all its possible "adjustments" (homotopies).
• The Benefit: It creates a clean, natural bridge between different mathematical languages (VB-groupoids and Representations up to Homotopy).
• The Result: It simplifies complex problems in geometry and physics, making it easier to calculate how things move, change, and interact. It's like upgrading from a rusty wrench to a laser-guided tool kit for building mathematical structures.

In short, Lennart Obster has found a "Fat" perspective that makes the messy, abstract world of Lie theory much clearer, more connected, and easier to work with.

Technical Summary

Technical Summary: Fat Lie Theory

Paper: Fat Lie Theory by Lennart Obster
Date: March 10, 2026
Field: Differential Geometry, Lie Groupoids, Representation Theory, Homotopy Theory

1. Problem Statement

The representation theory of Lie groupoids and Lie algebroids has been studied through various lenses, most notably:

1. Representations up to homotopy (ruths): Specifically, 2-term ruths, which generalize the adjoint representation.
2. VB-groupoids/Algebroids: Vector bundle objects in the category of groupoids/algebroids.
3. PB-groupoids: Principal bundle groupoids with structural strict Lie 2-groupoids.

While equivalences between these models are known (e.g., the equivalence between 2-term ruths and VB-groupoids), the literature often treats them as distinct frameworks or relies on "split" versions (requiring choices of cleavages/splittings) which obscure the intrinsic geometric nature of the objects. Furthermore, the relationship between these structures and fat groupoids (often appearing as jet groupoids or first jet algebroids) has been implicit rather than axiomatic.

The paper addresses the need for a unified, intrinsic framework that:

• Axiomatizes the structure of "fat extensions" without relying on arbitrary splittings.
• Establishes rigorous, functorial equivalences between fat extensions, abstract 2-term ruths, VB-groupoids, and general linear PB-groupoids.
• Provides a natural setting for studying multiplicative tensors and deformation theory.

2. Methodology

The author employs a structural approach centered on the concept of Fat Extensions. The methodology involves:

1. Axiomatization of Fat Extensions: Defining a "fat extension" of a Lie groupoid $G$ over a 2-term complex $C_M \to V_M$ as a Lie groupoid $F_G$ fitting into a short exact sequence with a bundle of invertible homotopies $H(V_M, C_M)$, equipped with a compatible cochain complex representation.
2. Intrinsic vs. Split Models: Distinguishing between "abstract" ruths (intrinsic, defined via differential graded modules over the nerve of the groupoid) and "split" ruths (dependent on a cleavage). The paper demonstrates that abstract ruths arise canonically from fat extensions.
3. Category Theory and Functoriality: Constructing explicit functors between the categories of VB-groupoids, fat extensions, abstract 2-term ruths, and general linear PB-groupoids. The author proves these functors are equivalences of categories, upgrading previous one-to-one correspondences.
4. Differentiation (Lie Functor): Applying the Lie differentiation procedure to fat extensions to derive their infinitesimal counterparts (fat algebroids) and establishing the corresponding equivalences in the algebroid setting.
5. Double Groupoid Analysis: Utilizing the theory of double groupoids and core extensions (generalizing crossed modules) to characterize regular fat extensions and general linear PB-groupoids.

3. Key Contributions

A. The Category of Fat Extensions

The paper introduces the category of fat extensions as a central object. A fat extension of $G$ over $C_M \to V_M$ consists of:

• A Lie groupoid $F_G \Rightarrow M$.
• A short exact sequence: $1 \to H(V_M, C_M) \to F_G \to G \to 1$, where$H(V_M, C_M)$ is the bundle of invertible homotopies.
• A cochain complex representation of $F_G$ on $C_M \to V_M$ that restricts to the canonical representation of $H(V_M, C_M)$ and satisfies specific conjugation compatibility conditions.

B. Equivalence of Categories

The main result is the establishment of a web of equivalences of categories:
$$\{ \text{VB-groupoids} \} \simeq \{ \text{Fat Extensions} \} \simeq \{ \text{Abstract 2-term Ruths} \} \simeq \{ \text{General Linear PB-groupoids} \}$$

• VB-groupoids $\leftrightarrow$ Fat Extensions: The functor maps a VB-groupoid $V_G$ to its fat groupoid $\widehat{V_G}$. The inverse constructs a VB-groupoid from a fat extension via an associated bundle construction.
• Fat Extensions $\leftrightarrow$ Abstract 2-term Ruths: The paper defines the invariant complex $C_{inv}(F_G; C_M)$ of a fat extension. This complex is shown to be isomorphic to the abstract 2-term ruth associated with the VB-groupoid. This provides a canonical model for ruths that does not require a choice of splitting (cleavage).
• Fat Extensions $\leftrightarrow$ General Linear PB-groupoids: The paper refines the correspondence between VB-groupoids and general linear PB-groupoids (introduced in [GC26]). It shows that a general linear PB-groupoid is essentially a fat extension "twisted" by the general linear groupoid $GL(C_M, V_M)$.

C. Abstract Representations up to Homotopy

The author formalizes abstract ruths as finitely generated projective modules over the differential graded algebra of the groupoid, equipped with two differentials ($\delta$ of degree +1 and $\nu$ of degree -1).

• It is shown that the VB-complex of a VB-groupoid is a valid example of a 2-term abstract ruth.
• The paper clarifies that "split" ruths are merely a specific realization of abstract ruths via a choice of splitting, and the equivalence between VB-groupoids and abstract ruths is canonical.

D. Infinitesimal Theory (Fat Lie Theory)

The paper extends these results to Lie algebroids:

• Fat Algebroids: Defined as the Lie algebroid of a fat groupoid.
• Differentiation: The differentiation of a fat extension yields a fat algebroid. The Van Est map is reformulated in terms of invariant cochains on fat extensions, providing a bridge between the cohomology of the groupoid and the algebroid.
• Deformation Cohomology: The deformation complex of a Lie groupoid is identified with the invariant complex of its jet groupoid (viewed as a fat extension).

E. Core Extensions and Double Groupoids

The paper introduces core extensions as a generalization of crossed modules and core diagrams.

• It proves that vertically core-transitive double groupoids are equivalent to core extensions.
• For regular fat extensions, the associated gauge double groupoid is shown to be a general linear double groupoid, and the fat extension itself is recovered as the core extension of this double groupoid.

4. Key Results

1. Theorem 2.13: The functor from VB-groupoids to abstract 2-term ruths (via the VB-complex) is an equivalence of categories.
2. Proposition 4.5 & Theorem 7.13: There is a one-to-one correspondence (equivalence of categories) between VB-groupoids and fat extensions.
3. Proposition 5.20 & Theorem 7.24: There is an equivalence of categories between fat extensions and general linear PB-groupoids.
4. Proposition 8.26 & Theorem 8.32: The infinitesimal counterparts (VB-algebroids, fat algebroids, abstract ruths) also form equivalent categories.
5. Theorem 9.10 (Van Est Theorem): The Van Est map restricts to an isomorphism on the cohomology of invariant cochains for fat extensions with sufficiently connected source fibers.
6. Proposition 10.3 & 10.5: The deformation cohomology of a Lie groupoid can be naturally described using the invariant complex of its jet groupoid (fat extension), unifying previous results on deformation theory.

5. Significance

• Unification: The paper provides a single, intrinsic framework ("Fat Lie Theory") that unifies VB-groupoids, representations up to homotopy, and PB-groupoids. It removes the reliance on "split" structures (cleavages) for defining morphisms and equivalences, offering a more canonical perspective.
• New Tools for Multiplicative Tensors: By framing multiplicative tensors as objects defined on fat extensions (specifically as invariant cochains or morphisms of fat extensions), the theory offers a natural language for tensor products and deformation theory, which is further developed in the author's upcoming work [MOV].
• Clarification of Higher Structures: The work clarifies the role of "normal weakly flat cleavages" in higher vector bundles (VB-$n$-groupoids), suggesting they arise from the choice of splitting in abstract ruths. This sets the stage for formulating higher-dimensional analogues of these equivalences.
• Deformation Theory: The reformulation of the deformation complex of Lie groupoids in terms of the jet groupoid's invariant cochains provides a powerful new tool for studying deformations, particularly in the context of proper groupoids and linearization problems.
• Generalization of Classical Results: The results generalize the classical correspondence between crossed modules and strict Lie 2-groupoids (via core extensions) to the broader context of fat extensions and double groupoids.

In summary, "Fat Lie Theory" establishes that the "fat" structure (the jet groupoid/algebroid with its canonical homotopy bundle) is not just a technical tool but the fundamental geometric object underlying the representation theory of Lie groupoids and algebroids.