On the integrability of Lie algebroids by diffeological spaces

TL;DR

This paper extends Lie's third theorem to a special class of diffeological groupoids called singular Lie groupoids, connecting them to Lie algebroids and demonstrating the theorem's validity in this broader setting.

Contribution

It introduces quasi-etale diffeological spaces and shows that singular Lie groupoids within this category satisfy Lie's third theorem, extending classical results.

Findings

The ;evera-Weinstein groupoid of an algebroid is a singular Lie groupoid.

Lie's third theorem holds for singular Lie groupoids with manifold units.

A functor from singular Lie groupoids to Lie algebroids is constructed and extended.

Abstract

Lie's third theorem does not hold for Lie groupoids and Lie algebroids. In this article, we show that Lie's third theorem is valid within a specific class of diffeological groupoids that we call `singular Lie groupoids.' To achieve this, we introduce a subcategory of diffeological spaces which we call `quasi-etale.' Singular Lie groupoids are precisely the groupoid objects within this category, where the unit space is a manifold. Our approach involves the construction of a functor that maps singular Lie groupoids to Lie algebroids, extending the classical functor from Lie groupoids to Lie algebroids. We prove that the \v{S}evera-Weinstein groupoid of an algebroid is an example of a singular Lie groupoid, thereby establishing Lie's third theorem in this context.