nLab Morita morphism

The most useful notion of homomorphism between Lie groupoids is not that of a functor internal to the category SmoothMfd of smooth manifolds (even though Lie groupoids can be regarded as internal groupoids in SmoothMfd), but is that of a morphism of smooth stacks (differentiable stacks) on the site SmoothMfd. To amplify this distinction in the literature a wealth of names and terms for these non-naïve homomorphisms is used, such as Morita morphism . In particular a Morita morphism which is a homotopy equivalence is a Morita equivalence.

Ways to present the correct notion of homomorphism include

• anafunctors

• groupoid principal$\,$ bibundles, Hilsum-Skandalis morphisms