In Ehresmann's footsteps: from Group Geometries to Groupoid Geometries

TL;DR

This paper generalizes the classical theory of principal bundles using groupoid actions, introducing a diagrammatic 'butterfly' framework that unifies various geometric and algebraic structures.

Contribution

It develops a symmetric 'butterfly diagram' approach to principal Lie groupoid actions, extending Ehresmann's concepts to non-locally trivial bundles and categorical contexts.

Findings

Unified framework for principal groupoid actions

General theorem of 'universal activation' for various geometric structures

Application to globalization, cocycles, and homogeneous spaces

Abstract

For a smooth (locally trivial) principal bundle in Ehresmann's sense, the relation between the commuting vertical and horizontal actions of the structural Lie group and the structural Lie groupoid (isomorphisms between vertical fibers) is regarded as a special case of a symmetrical concept of conjugation between "principal" Lie groupoid actions, allowing possibly non-locally trivial bundles. A diagrammatic description of this concept via a symmetric "butterfly diagram" allows its "internalization" in a wide class of categories (used by "working mathematicians") whenever they are endowed with two distinguished classes of monomorphisms and epimorphisms mimicking the properties of embeddings and surjective submersions. As an application, a general theorem of "universal activation" encompasses in a unified way such various situations as Palais' theory of globalization for partial action laws, the realization of non-abelian cocycles (including Haefliger cocycles for foliations) or the description of the "homogeneous space" attached to an embedding of Lie groups (still valid for Lie groupoids).