Differentiable stratified groupoids and a de Rham theorem for inertia spaces

TL;DR

This paper introduces differentiable stratified groupoids, explores their properties including Morita equivalence, proves a de Rham theorem for them, and applies these concepts to the inertia groupoid of proper Lie groupoids, revealing new stratification structures.

Contribution

It develops the theory of differentiable stratified groupoids, establishes a de Rham theorem for them, and analyzes the inertia groupoid with a new Whitney B stratification.

Findings

De Rham theorem holds for locally contractible differentiable stratified groupoids.

The inertia groupoid of a proper Lie groupoid admits a natural Whitney B stratification.

The inertia groupoid becomes a locally contractible differentiable stratified groupoid with this stratification.

Abstract

We introduce the notions of a differentiable groupoid and a differentiable stratified groupoid, generalizations of Lie groupoids in which the spaces of objects and arrows have the structures of differentiable spaces, respectively differentiable stratified spaces, compatible with the groupoid structure. After studying basic properties of these groupoids including Morita equivalence, we prove a de Rham theorem for locally contractible differentiable stratified groupoids. We then focus on the study of the inertia groupoid associated to a proper Lie groupoid. We show that the loop and the inertia space of a proper Lie groupoid can be endowed with a natural Whitney B stratification, which we call the orbit Cartan type stratification. Endowed with this stratification, the inertia groupoid of a proper Lie groupoid becomes a locally contractible differentiable stratified groupoid.