Lie groupoids are groupoids with smooth structures. There is a nature 2-category of Lie groupoids: Lie groupoids, smooth functors of Lie groupoids, smooth natural transformations of smooth functors. However this is usually not the correct one, partially because weak equivalence (fully-faithful and essential surjective in the smooth setting) of Lie groupoids cannot be inverted. We need more morphisms between Lie groupoids. There are three (or 2.5) ways to define the more morphisms:

1) Span: Span of Lie groupoid morphism G<<--K-->H such that the left leg is a weak equivalence. Equivalence of such spans: G<<--K1-->H and G<<--K2-->H are equivalent if there exists G<<--K3-->H with K1<<--K3-->>K2 such that all triangles 2-commute.

1') Span:Span of Lie groupoid morphism G<<--K-->H such that the left leg is a weak equivalence and the map between objects are surjective submersion. This is a slightly different version of the 1).

2) Bibundle: right principal bibundle of Lie groupoids. Equivalence of bibundles: equivariant map of bibundles (this is in fact a diffeomorphism).

Modulo equivalent relation in three cases we obtain three categories. A classic result tell us that they are isomorphic.

One could go a bit further, it is possible to define three 2-categories (i.e. bi-categories, in fact (2,1)-categories, 2-morphisms are invertible), and naturally there are functors between them. The construction is given in Hellen Colman http://www.springerlink.com/content/3472617rj6178271/.

One expect also that they are 2-equivalent 2-categories. Recall that a equivalent functor of 2-categories must be locally equivalent and surjecttive-up-to-equivalence on objects (See Leinster "Basic Bicategories"). This means that 2-morphisms must be the same. However it seems that 2-morphisms of (1) and (2) are different. The 2-morphisms induced from (2) to (1) must be and strict isomorphism (namely, K1 and K2 are isomorphic). What's wrong with my reasoning?

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