It seems to be common to say "no" - but is this true?

Two weeks ago I asked for a counterexample, but received no replies.

To give some background, let's recall that the difference between Lie groupoids and groupoids internal to smooth manifolds is that in the first case we require source and target maps to be submersions, guaranteeing that the domain of the composition map is a smooth manifold, while in the second case we only require that the fibre product that provides the domain of the composition map exists.

The Wikipedia page says that Ehresmann originally considered groupoids internal to smooth manifolds, and that this definition was replaced by Pradines by the current definition of Lie groupoids, because the former "proved to be too weak". I have looked into Pradines' article but could not see anything useful for my question.

The nLab article on Lie groupoids seems to see no difference between the two competing definitions ("A Lie groupoid is a groupoid internal to smooth manifolds"), but does not comment this topic.

Given the fact that Lie groupoids (and higher-categorical generalizations) become more and more important, I think it would be worthwhile to clarify this very basic question...

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    $\begingroup$ As you mentioned, the surjective submersion condition is imposed so that the fiber product exists as a smooth manifold, as in this case one has a transversal intersection. But this is only a sufficient condition for the pullback to exist, in principle one could require the source and target maps to only define a clean intersection (which is strictly weaker than transversal) and the fiber product would still exist. That being said, Mackenzie in his Lie groupoids book comments on this in p.85 saying he is not aware of an example where the source and target maps are not surjective submersions. $\endgroup$ Commented Apr 4 at 19:49
  • $\begingroup$ @AlonsoPerez-Lona: interesting! I am getting more and more convinced that there is no example... $\endgroup$ Commented Apr 4 at 20:43


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