Reduced Smooth Stacks?

An arbitrary Lie groupoid gives rise to a groupoid of germs of local diffeomorphisms over its base manifold, known as its e ect. The e ect of any bundle of Lie groups is trivial. All quotients of a given Lie groupoid determine the same e ect. It is natural to regard the e ects of any two Morita equivalent Lie groupoids as being equivalent . In this paper we shall describe a systematic way of comparing the e ects of di erent Lie groupoids. In particular, we shall rigorously de ne what it means for two arbitrary Lie groupoids to give rise to equivalent e ects. For e ective orbifold groupoids, the new notion of equivalence turns out to coincide with the traditional notion of Morita equivalence. Our analysis is relevant to the presentation theory of proper smooth stacks.