Coverings and Actions of Structured Lie Groupoids I

TL;DR

This paper explores the theory of coverings and actions of structured Lie groupoids, establishing categorical equivalences and methods to construct coverings, advancing the understanding of their geometric and algebraic properties.

Contribution

It introduces the concepts of actions and coverings of Lie group-groupoids, proving their categorical equivalence and providing methods to construct coverings from Lie group-groupoids.

Findings

Categorical equivalence between smooth actions and coverings of Lie group-groupoids

Methods to obtain covering Lie group-groupoids from given structures

Theoretical framework for structured Lie groupoid coverings

Abstract

In this work we deal with coverings and actions of Lie group- groupoids being a sort of the structured Lie groupoids. Firstly, we define an action of a Lie group-groupoid on some Lie group and the smooth coverings of Lie group-groupoids. Later, we show the equivalence of the category of smooth actions of Lie group-groupoids on Lie groups and the category of smooth cov- erings of Lie group-groupoids. Further, we prove a theorem which denotes how is obtained a covering Lie group-groupoid and a smooth covering morphism of Lie group-groupoids from a Lie group-groupoid.