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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. (Edit 2023 - the paper was published in 2012 as tac.mta.ca/tac/volumes/26/29/26-29abs.html) $\endgroup$
    – David Roberts ♦
    Commented Nov 22, 2011 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 ♦
    Commented Nov 22, 2011 at 1:23
  • $\begingroup$ I read this paper several days ago, nice 2 c u! $\endgroup$
    – Ma Ming
    Commented Nov 22, 2011 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
    Commented Nov 22, 2011 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 ♦
    Commented Dec 5, 2011 at 0:44

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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
    Commented Nov 21, 2011 at 16:53
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    $\begingroup$ 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
    Commented Nov 21, 2011 at 17:05
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    $\begingroup$ I believe that these are also the same in 2-category and as ringed etendues. $\endgroup$
    – Benjamin Steinberg
    Commented Nov 21, 2011 at 18:11
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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
    Commented Nov 21, 2011 at 18:21
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    $\begingroup$ @Ma Ming: I'm not sure I understand what you are saying (it's hard to squish this into a comment). Feel free to email me instead. Anyhow, one thing that I didn't mention is that there is also the "data" of the natural isomorphism between the "two different induced maps" $H_{U \cap U'} \to H$ but these two maps actually agree on the nose. $\endgroup$
    – David Carchedi
    Commented Nov 21, 2011 at 18:24

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