Groupoids in Diffeology

TL;DR

This paper reviews the development and applications of diffeological groupoids, highlighting their role in unifying concepts across geometry, orbifold stratification, noncommutative geometry, and geometric quantization.

Contribution

It provides a comprehensive overview of diffeological groupoids, emphasizing their historical evolution and diverse applications in modern mathematical and physical theories.

Findings

Diffeological groupoids serve as a unifying framework in geometry.

They facilitate analysis of orbifold stratification.

They connect to noncommutative geometry and geometric quantization.

Abstract

This expository paper recounts the development and application of the concept of the diffeological groupoid, from its introduction in 1985 to its use in current research. We demonstrate how this single concept has served as a powerful and unifying tool for defining fundamental structures, analyzing the stratification of complex spaces like orbifolds, building a bridge to noncommutative geometry, and, most recently, forging new approaches to geometric quantization. The paper aims to provide a cohesive narrative of this journey, making explicit certain concepts like the "Klein groupoid" and showcasing the enduring vitality of the diffeological groupoid in modern geometry and physics.