Foliations determined by their leaf spaces

A regular foliation is a partition of a manifold into connected (weakly-embedded) submanifolds of a fixed dimension, called leaves. The space of leaves should capture the transverse geometry of the foliation, but this is rarely a smooth manifold. We will introduce two approaches to transverse geometry: by viewing the leaf space as a differentiable stack (corresponding to the holonomy groupoid), and by equipping it with its natural quotient diffeology. A priori, the quotient stack is the more refined model, but we show that in some cases, such as for Riemannian foliations, the quotient stack is completely determined by the diffeological leaf space.