Lennart Obster | Searcharxiv

arXiv2026

Fat Lie Theory

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.

math.DG↗

arXiv2021

Blow-ups of Lie groupoids and Lie algebroids

In this master's thesis, we will go into the (projective) blow-up construction for Lie groupoids and Lie algebroids. In the literature, there are different methods to be found on how to do this, especially for Lie groupoids. The main goal of the thesis is to explain, in detail, the Lie groupoid and the Lie algebroid blow-up constructions, but also to examine and compare different points of view. More explicitly, we will rigorously explain the blow-up construction for Lie groupoids by Claire Debord and Georges Skandalis. Moreover, we will show that the blow-up construction for Lie groupoids by Songhao Li and Marco Gualtieri, and the construction by Kirsten Wang, fit into this setting. Also, we will show that, analogously, we obtain a general geometric blow-up construction for Lie algebroids. This construction for Lie algebroids coincides with the construction of lower elementary modification in the codimension one case (by e.g. Songhao Li and Marco Gualtieri, Melinda Lanius, or Ralph Klaasse). Examples that are discussed include the blow-up of a pair groupoid (resp. tangent bundle) along a pair groupoid (resp. tangent bundle), the blow-up of a groupoid (resp. algebroid) along the groupoid (resp. algebroid) restricted to a saturated submanifold, and the blow-up of a regularly foliated manifold along a leaf. The blow-up construction by Debord and Skandalis uses the theory of deformation to the normal cones. We will give a broad introduction to this theory and go into some of its known applications. We also discuss the observation made by Debord and Skandalis about Morita invariance of the construction of blow-up and of deformation to the normal cone.

math.DG↗