0
$\begingroup$

This question came from comments of this question.

Suppose $\mathcal{G}$ is a Lie groupoid, we consider the category $B\mathcal{G}$ whose objects are principal $\mathcal{G}$ bundles and morphisms are $\mathcal{G}$ equivariant maps between principal $\mathcal{G}$ bundles.

Let $\mathcal{D}$ is a stack over manifolds. Suppose further that this is a geometric stack i.e., there is an atlas for $\mathcal{D}$. Given an atlas $p:\underline{X}\rightarrow \mathcal{D}$ we can associate a Lie groupoid with $\mathcal{D}$.

As $p:\underline{X}\rightarrow \mathcal{D}$ is an atlas, the $2$-fibered product $\underline{X}\times_{\mathcal{D}}\underline{X}$ is a manifold, say $P$.

The projections $pr_1:\underline{X}\times_{\mathcal{D}}\underline{X}\rightarrow \underline{X}$ and $pr_2:\underline{X}\times_{\mathcal{D}}\underline{X}\rightarrow \underline{X}$ gives maps $P\rightrightarrows X$ giving a Lie groupoid $P\rightrightarrows X$.

I was trying to understand when a stack it comes from a manifold. David Roberts said

[...] there is an object of the groupoid with a non-trivial automorphism, which is the very hallmark of a non-trivial stack, and the entire motivation for the theory. It also means that that groupoid gives rise to a stack where one of the fibres is a groupoid not equivalent to a set, hence the stack cannot come from a manifold.

I am trying to understand more about this. In other setup also having a nontrivial automorphism group is considered to be not an easy problem to handle.

I was searching with keywords nontrivial automorphism group in google and found this which says in page no $10$

moduli problems involving objects with nontrivial automorphisms will almost never be representable by genuine spaces.

I do not understand so much about moduli problem but I think this is the common point to what I sad before. A stack gives a Lie groupoid which has non trivial automorphisms and it can not possibly come from a manifold. Here also they are saying something similar "will almost never be representable by genuine spaces"

So, can some one help me to see what is going on here?

$\endgroup$
  • $\begingroup$ I added to my answer at the linked question, but this question seems a little broader. $\endgroup$ – David Roberts Sep 19 '18 at 0:51
  • $\begingroup$ @DavidRoberts Yes. I saw that edit.. I am trying to understand what you said... Do you think this is too broad to get a good answer? $\endgroup$ – Praphulla Koushik Sep 19 '18 at 1:49
  • $\begingroup$ I'm not sure that I can add too much to what I said already at the other question. The addition to my answer is not specific to the case of differentiable stacks, but stacks in any setting, with the minor that if a stack is not 'geometric' (that is: comes from some structured groupoid, like a groupoid scheme, or a topological groupoid etc) then it has no hope at all of coming from an object in the base site, so that is a prerequisite also. $\endgroup$ – David Roberts Sep 19 '18 at 4:48

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.