Lie groupoids determined by their orbit spaces

he orbit space of a Lie groupoid carries a natural diffeology. More generally, we have a quotient functor from the Hilsum-Skandalis category of Lie groupoids to the category of diffeological spaces. We introduce a class of effective Lie groupoids, called "lift-complete," for which this functor restricts to an equivalence of distinguished sub-categories. In particular, the diffeomorphism class of the orbit space of a lift-complete Lie groupoid determines its Morita class. After explaining this equivalence, we will relate the lift-complete condition to certain PDEs, which justifies both effective quasifolds (hence effective orbifolds) and Riemannian foliations as examples.