Fat Lie Theory

TL;DR

The paper introduces 'fat Lie theory', establishing new categorical correspondences between fat extensions, 2-term representations up to homotopy, and VB-groupoids, enriching the understanding of Lie groupoid structures.

Contribution

It develops a novel framework connecting fat extensions, 2-term ruths, and VB-groupoids, and extends existing equivalences to broader categorical contexts.

Findings

Established a one-to-one correspondence between fat extensions and 2-term ruths.

Related fat extensions to general linear PB-groupoids.

Upgraded known equivalences to an entire category of structures.

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.