An arbitrary Lie groupoid gives rise to a groupoid of germs of local diffeomorphisms over its base manifold, known as its effect. The effect of any bundle of Lie groups is trivial. All quotients of a given Lie groupoid determine the same effect. It is natural to regard the effects of any two Morita equivalent Lie groupoids as being ``equivalent''. In this paper we shall describe a systematic way of comparing the effects of different Lie groupoids. In particular, we shall rigorously define what it means for two arbitrary Lie groupoids to give rise to ``equivalent'' effects. For effective 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.
Keywords: Lie groupoids, effective orbifolds, categories of fractions
2010 MSC: Primary 58H05; Secondary 22A22, 18E35
Theory and Applications of Categories, Vol. 30, 2015, No. 31, pp 1032-1066.