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