Lie Groupoids as generalized atlases

TL;DR

This paper proposes viewing Lie groupoids as generalized atlases for orbit spaces, capturing isotropy and smoothness, and introduces a unifying theory based on formal properties of embeddings and surmersions.

Contribution

It introduces a novel perspective of Lie groupoids as generalized atlases for orbit spaces, emphasizing smooth Morita equivalence and formal properties for a unifying framework.

Findings

Lie groupoids serve as generalized atlases for orbit spaces.

Smooth Morita equivalence characterizes atlas equivalence.

A unifying theory based on embeddings and surmersions is proposed.

Abstract

Starting with some motivating examples (classical atlases for a manifold, space of leaves of a foliation, group orbits), we propose to view a Lie groupoid as a generalized atlas for the "virtual structure" of its orbit space, the equivalence between atlases being here the smooth Morita equivalence. This "structure" keeps memory of the isotropy groups and of the smoothness as well. To take the smoothness into account, we claim that we can go very far by retaining just a few formal properties of embeddings and surmersions, yielding a very polymorphous unifying theory. We suggest further developments.