On the integrability of Lie algebroids by diffeological spaces

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.