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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    $\begingroup$ May I advertise my article ncatlab.org/davidroberts/files/DRoberts_anafunctors.pdf (an update of arxiv.org/abs/1101.2363, although it is still more improved in the latest version, not yet public) which provides another model (actually a family of such), equivalent to the above? Proposition 6.5 in my paper (proved by Pronk) is what you can use to see that all these are equivalent. $\endgroup$ – David Roberts Nov 22 '11 at 1:20
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    $\begingroup$ I should say that all of these are all models for 'the' bicategorical localisation of LieGpd at the weak equivalences. Concerning the equivalence between the different sort of 2-morphisms, Makkai in his paper on anafunctors essentially proves that the more general 2-arrows in (1'), when restricted to between the 1-arrows of (2) (which are secretly a special case of (1')), are in bijective correspondence with the 2-arrows of (2). $\endgroup$ – David Roberts Nov 22 '11 at 1:23
  • $\begingroup$ I read this paper several days ago, nice 2 c u! $\endgroup$ – Ma Ming Nov 22 '11 at 8:58
  • $\begingroup$ @David Roberts May I ask a question, What is the result if one use weak pullbacks to define 2-morphisms of anafunctors? $\endgroup$ – Ma Ming Nov 22 '11 at 16:28
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    $\begingroup$ Sorry for not replying earlier! In regards to your 22Nov question, you really don't want to use 'weak pullbacks' (really they are isocomma objects), that is the approach used in Pronk's paper, and one reason why the 2-cells are so complicated. The advantage of anafunctors is that using the strict pullback (using the pretopology that one has) means you get representatives of 2-cells instead of using equivalence classes. $\endgroup$ – David Roberts Dec 5 '11 at 0:44

These three bicategories ARE equivalent. To go from (2) to (1) it's helpful to introduce a (2)':

Let (2)' have the same objects, and morphisms as (2) except each principal bundle $P$ for $G$ over $H$, is equipped with a choice of local sections of the map $P \to H_0$ (so this is equivalent to (2)). If $U$ is the cover of $H_0$ over which these local sections are defined, this can easily be seen to the same data as a map $H_U \to G$, where $H_U$ is $H$ pulled back along the map $\coprod U_i \to H_0$. But now if $\alpha:P \to P'$ is an isomorphism and we have two different choices of covers of $H_0$, $U$ and $U'$ over which they admit local sections, then this induces a $2$-cell in (1) between the maps $H_U \to G$ and $H_U' \to G$ in the bicategory of fractions. These $2$-cells are a bit complicated, see the work of Pronk. This two cell is represented by the two (canonically equivalent) induced smooth functors $H_{U \cap U'} \to G$.

You should note that there is ANOTHER bicategory to which these are all equivalent, namely the bicategory of differentiable stacks, which are stacks over the category of smooth manifolds, which admit a surjective representable submersion from a manifold.

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    $\begingroup$ What is a good reference for the equivalence of these four bicategories? $\endgroup$ – Konrad Waldorf Nov 21 '11 at 16:53
  • $\begingroup$ If my understand of biequivalent of bicategories are correct: locally equivalent and surjecttive-up-to-equivalence on objects, then I have a simple counterexample illustrated that 2-cells in (1) is much more than (2). Let G and H be the most trivial groupoid , morphism and objects spaces are both one point manifold. Let M and N also be *. Then the induced spans of Lie groupoids are *<<---->* and <<---->*. However any pair Lie groupoid will give a 2-cell between such spans. $\endgroup$ – Ma Ming Nov 21 '11 at 17:05
  • $\begingroup$ I believe that these are also the same in 2-category and as ringed etendues. $\endgroup$ – Benjamin Steinberg Nov 21 '11 at 18:11
  • $\begingroup$ @Benjamin: Not quite. First, you need these etendues to be "smooth", i.e. be locally isomorphic to $R^n$ with its ring of smooth functors (just like arbitrary locally ringed spaces need nor be manifolds). Secondly, you have to restrict to Lie groupoids which are (Morita equivalent to) etale Lie groupoids. This breaks down when stuff isn't etale. $\endgroup$ – David Carchedi Nov 21 '11 at 18:14
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    $\begingroup$ @Konrad: Dorette Pronk's thesis is a good reference for comparing the bicategory of fractions approach to the one in terms of stacks. She only proves it for etale stacks, but it works without this assumption if you remove her use of topoi and rephrase things in terms of stacks of torsors. If I may be so bold, I would suggest my thesis for the proof that (2) is equivalent to differentiable stacks, as I work this out in some detail: igitur-archive.library.uu.nl/dissertations/2011-0830-200501/… $\endgroup$ – David Carchedi Nov 21 '11 at 18:21

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