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 callquasi-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.