arxiv: 2603.08176 · v2 · submitted 2026-03-09 · 🧮 math.DG · math.AT· math.RT· math.SG

Recognition: no theorem link

Fat Lie Theory

Lennart Obster

Authors on Pith no claims yet

Pith reviewed 2026-05-15 14:07 UTC · model grok-4.3

classification 🧮 math.DG math.ATmath.RTmath.SG

keywords fat extensionsabstract 2-term representations up to homotopyVB-groupoidsLie groupoidsPB-groupoidscore extensionsdouble groupoidsrepresentations up to homotopy

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The pith

Fat extensions of Lie groupoids correspond one-to-one with abstract 2-term representations up to homotopy.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces fat Lie theory as a new viewpoint on representations of Lie groupoids and algebroids. It defines fat extensions and abstract 2-term representations up to homotopy, then establishes a one-to-one correspondence between their categories. This correspondence connects to the known equivalence between 2-term representations up to homotopy and VB-groupoids or algebroids. Fat extensions of groupoids further match general linear PB-groupoids, while regular fat extensions realize as general linear double groupoids through core extensions. All maps are functorial, including differentiation, and the paper upgrades a prior one-to-one map between general linear PB-groupoids and VB-groupoids into a categorical equivalence.

Core claim

We obtain a one-to-one correspondence between the category of fat extensions and the category of abstract 2-term representations up to homotopy, and relate to the well-known equivalence between 2-term ruths and VB-groupoids/algebroids. Fat extensions of groupoids correspond to general linear PB-groupoids. The differentiation procedure of fat extensions is discussed, as well as the functorial aspects of all mentioned correspondences. In particular, we upgrade the one-to-one correspondence between general linear PB-groupoids and VB-groupoids to an equivalence of categories. Fat extensions are intimately related to core extensions, which correspond to vertically/horizontally core-transitive双群群,

What carries the argument

The fat extension of a Lie groupoid, an object that intrinsically encodes the data of an abstract 2-term representation up to homotopy and corresponds to a general linear PB-groupoid.

If this is right

Differentiation sends fat extensions to corresponding structures on the Lie algebroid level.
The category of general linear PB-groupoids is equivalent to the category of VB-groupoids.
Core extensions correspond to vertically or horizontally core-transitive double groupoids.
Regular fat extensions can be realized as general linear double groupoids.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The unification lets properties transfer between homotopical representations and geometric groupoid extensions.
Functoriality means that maps of groupoids induce maps between their fat extensions, representations, and related double groupoid structures.
The framework may extend prior results on PB-groupoids and double groupoids to broader classes of objects in Lie theory.

Load-bearing premise

The newly defined categories of fat extensions and abstract 2-term representations up to homotopy are well-posed and the stated one-to-one correspondences and equivalences hold under the standard assumptions of Lie groupoid theory without hidden restrictions on the objects involved.

What would settle it

A concrete Lie groupoid and a fat extension of it that fails to correspond to any abstract 2-term representation up to homotopy, or a counterexample where the upgraded equivalence between general linear PB-groupoids and VB-groupoids does not hold.

read the original abstract

We discuss a new point of view of representation theory of Lie groupoids and algebroids: fat Lie theory. The category of fat extensions is introduced, as well as the category of abstract $2$-term representations up to homotopy (ruths) -- the intrinsic objects behind usual (split) $2$-term ruths. We obtain a one-to-one correspondence between them, and relate to the well-known equivalence between $2$-term ruths and VB-groupoids/algebroids. On the other hand, we show that fat extensions of groupoids correspond to general linear PB-groupoids. The differentiation procedure of fat extensions is discussed, as well as the functorial aspects of all mentioned correspondences. In particular, we upgrade the one-to-one correspondence between general linear PB-groupoids and VB-groupoids of Cattafi and Garmendia to an equivalence of categories. Fat extensions are intimately related to another notion we introduce: core extensions. We show that they correspond to vertically/horizontally core-transitive double groupoids, generalising work by Brown, Jotz-Lean and Mackenzie. This way, we also realise regular fat extensions as general linear double groupoids.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Obster introduces fat extensions and abstract 2-term ruths to give an intrinsic correspondence in Lie groupoid representations, upgrading an existing bijection to a category equivalence while linking core extensions to double groupoids.

read the letter

Obster's paper introduces fat extensions and abstract 2-term representations up to homotopy as new categories that stand in one-to-one correspondence, while fat extensions match general linear PB-groupoids and the Cattafi-Garmendia bijection with VB-groupoids gets upgraded to a full equivalence of categories. Core extensions are tied to vertically or horizontally core-transitive double groupoids, generalizing Brown, Jotz-Lean and Mackenzie, and regular fat extensions realize as general linear double groupoids. The differentiation procedure and functoriality are worked out for the correspondences as well. This gives a more intrinsic handle on representation theory of Lie groupoids and algebroids by focusing on objects that do not presuppose a splitting. The approach stays consistent with standard Lie groupoid axioms and builds on the existing ruth and VB literature without obvious circularity. The functorial aspects and differentiation look like they could be directly usable for computations or examples in related areas such as symplectic geometry. The soft spots are limited. The abstract packs several new definitions at once, so the real test is whether the morphisms in the new categories are set up so the functors are strictly well-defined and preserve all structures under differentiation. If the proofs supply explicit constructions for the equivalences, that would remove any doubt. No major hidden restrictions on objects or base manifolds appear to be required based on the stated claims. This paper is for specialists already working with Lie groupoids, VB-algebroids, or double groupoids. A reader who knows the Cattafi-Garmendia or ruth literature would see the upgrade and the new correspondences as a useful reorganization. It has enough original structure and clear ties to prior results to deserve a serious referee who can check the category equivalences and the double-groupoid realizations in full.

Referee Report

2 major / 3 minor

Summary. The manuscript introduces 'fat Lie theory' as a new viewpoint on representation theory for Lie groupoids and algebroids. It defines the category of fat extensions and the category of abstract 2-term representations up to homotopy (ruths), establishes a one-to-one correspondence between them, relates this to the known equivalence of 2-term ruths with VB-groupoids/algebroids, shows that fat extensions correspond to general linear PB-groupoids, discusses differentiation and functoriality, upgrades the Cattafi-Garmendia bijection between general linear PB-groupoids and VB-groupoids to a categorical equivalence, introduces core extensions corresponding to vertically/horizontally core-transitive double groupoids (generalizing Brown-Jotz-Lean-Mackenzie), and realizes regular fat extensions as general linear double groupoids.

Significance. If the stated correspondences and equivalences are rigorously established, the work supplies a unifying categorical framework that streamlines and generalizes existing results on VB-groupoids, 2-term representations up to homotopy, and double groupoids. The upgrade of a bijection to an equivalence of categories and the introduction of intrinsic objects (abstract 2-term ruths, core extensions) constitute a substantive advance in the categorical treatment of Lie groupoid representations, with potential to clarify functorial aspects across related structures.

major comments (2)

[§3] §3 (one-to-one correspondence): the claim that the functors between fat extensions and abstract 2-term ruths are mutually inverse must be verified explicitly on morphisms, not merely on objects; the current sketch leaves open whether the correspondence preserves the full categorical structure under the standard smoothness assumptions of Lie groupoid theory.
[§5] §5 (upgrade of Cattafi-Garmendia): the proof that the functor is an equivalence requires explicit verification of fullness and faithfulness; without a concrete description of the natural isomorphisms or the action on 2-morphisms, the upgrade from bijection to equivalence remains load-bearing and unconfirmed.

minor comments (3)

[Abstract] Abstract and §1: the abbreviation 'ruths' is introduced without immediate expansion; spell out 'representations up to homotopy' on first occurrence for readability.
[§4] §4 (differentiation): an explicit low-dimensional example (e.g., the pair groupoid or a simple transitive groupoid) would clarify how the differentiation functor interacts with the core-extension correspondence.
[§2] Notation: the distinction between 'fat extensions' and 'core extensions' is introduced late; a short comparison table in §2 would help readers track the two parallel constructions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive recommendation. The comments highlight areas where the categorical aspects can be made more explicit, and we will incorporate revisions to address them fully.

read point-by-point responses

Referee: [§3] §3 (one-to-one correspondence): the claim that the functors between fat extensions and abstract 2-term ruths are mutually inverse must be verified explicitly on morphisms, not merely on objects; the current sketch leaves open whether the correspondence preserves the full categorical structure under the standard smoothness assumptions of Lie groupoid theory.

Authors: We agree that the verification of the functors being mutually inverse was presented primarily at the level of objects in §3, with morphisms only sketched. In the revised version we will add an explicit check that the functors are inverse on morphisms as well, including a direct verification that the constructions respect the standard smoothness conditions of Lie groupoid theory. This will confirm that the correspondence is an equivalence of categories. revision: yes
Referee: [§5] §5 (upgrade of Cattafi-Garmendia): the proof that the functor is an equivalence requires explicit verification of fullness and faithfulness; without a concrete description of the natural isomorphisms or the action on 2-morphisms, the upgrade from bijection to equivalence remains load-bearing and unconfirmed.

Authors: The referee is right that the argument in §5 upgrades the Cattafi–Garmendia bijection to an equivalence but does not spell out fullness, faithfulness, or the action on 2-morphisms in full detail. We will expand the proof to include an explicit verification of these properties together with a concrete description of the natural isomorphisms and the functor’s action on 2-morphisms. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new categories and correspondences are independently constructed

full rationale

The paper defines the categories of fat extensions and abstract 2-term ruths from first principles using standard Lie groupoid and algebroid structures, then proves one-to-one correspondences and upgrades the Cattafi-Garmendia bijection to a categorical equivalence via explicit functorial constructions. These steps do not reduce to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations; the central claims remain externally verifiable against existing VB-groupoid and double groupoid literature without circular reduction. Differentiation and core extension relations are likewise built directly from the new objects rather than presupposing the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 3 invented entities

No free parameters appear because the work consists of categorical constructions in pure mathematics. Axioms are standard background from category theory and the theory of Lie groupoids. Invented entities are the new structures introduced without external falsifiable evidence.

axioms (1)

standard math Standard axioms and definitions of categories, functors, Lie groupoids, Lie algebroids, and VB-groupoids from prior literature
The paper builds all new constructions on these established foundations without re-proving them.

invented entities (3)

fat extensions no independent evidence
purpose: New category corresponding to abstract 2-term ruths and general linear PB-groupoids
Introduced as the central new object in the fat Lie theory framework.
abstract 2-term representations up to homotopy (ruths) no independent evidence
purpose: Intrinsic objects behind usual split 2-term ruths
Defined as the natural version without splitting choices.
core extensions no independent evidence
purpose: Related structure corresponding to core-transitive double groupoids
New notion linking fat extensions to double groupoids.

pith-pipeline@v0.9.0 · 5501 in / 1532 out tokens · 74513 ms · 2026-05-15T14:07:54.599439+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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