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Vague question is the following:

Is there a classifcation of Lie groupoids?

Slightly less vague question is the following:

Is there a (short?) list of "types" of Lie groupoids such that any arbitrary Lie groupoid is Morita equivalent to one from the list?

Some Lie groupoids I know are the following:

  1. Lie groupoids coming from manifolds $[M\rightrightarrows M]$,
  2. Lie groupoids coming from Lie group $[G\rightrightarrows *]$,
  3. Lie groupoids coming from action of a Lie group on a manifold $[G\times M\rightrightarrows M]$,
  4. Lie groupoids coming from foliation on a manifold $\mathcal{F}(M)$,
  5. Lie groupoids coming from a submersion, the submersion groupoid $[M\times_NM\rightrightarrows M]$
  6. Lie groupoids coming from a principal $G$-bundle, the Gauge groupoid,
  7. Lie groupoids coming from a vector bundle over $M$, the general linear Lie groupoid $[GL(E)\rightrightarrows M]$.

Do we know any "interesting" Lie groupoids that do not belong (not Morita equivalent) to the above list?

Only classification I know is if a Lie groupoid is proper and étale, then it is locally looks like action Lie groupoid.

I might have forgot one or two types, but these are the only types of Lie groupoids I have come across till now.

Edit: It turns out my question was not conveyed correctly. I do know that there is no hope of classifying Lie groups. I do not want to classify Lie groups. For me, all Lie groupoids that comes from Lie groups are of "same type". So, the classification I am looking for is only based on types. I think I conveyed it correctly now. If this is not clear, please do ask for more clarification.

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    $\begingroup$ Lie groups sit inside Lie groupoids via the delooping construction, and there is no hope of classifying Lie groups. $\endgroup$
    – Dmitri Pavlov
    Commented Sep 20, 2020 at 20:05
  • $\begingroup$ @DmitriPavlov It looks like I have not mentioned it clearly.. Yes, I do know that there is no hope of classifying Lie groups... I do not want to classify Lie groups :D For me Lie groupoids coming from Lie groups are of "same type"... Does it now make it clear what I was trying to understand? $\endgroup$
    – Praphulla Koushik
    Commented Sep 21, 2020 at 2:14
  • $\begingroup$ What is a “type” of a Lie groupoid? Your list of 7 items looks like a fairly ordinary list of examples of Lie groupoids that one would give in a typical expository text after a definition. Clearly, there are many more examples, and I do not understand why one would expect to exhaust all Lie groupoids by adding other examples to this list. $\endgroup$
    – Dmitri Pavlov
    Commented Sep 21, 2020 at 2:44
  • $\begingroup$ @DmitriPavlov "type" of a Lie groupoid is not a standard terminology, I just invented now... I call Lie groupoids coming from Lie groups as one type.. Lie groupoids coming from manifolds as one type.. Lie groupoids coming from foliations as one type and so on... Let me rephrase this in another way. Are there any "interesting" examples of Lie groupoids which are not Morita equivalent to any of the above 7 types? Here, interesting does not have a unique meaning and I leave it to you to interpret in a fair manner :) $\endgroup$
    – Praphulla Koushik
    Commented Sep 21, 2020 at 2:55
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    $\begingroup$ You don't need étaleness to get locally like an action groupoid: proper is enough. $\endgroup$
    – David Roberts ♦
    Commented Sep 21, 2020 at 4:07

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