# Lie groupoid with non trivial automorphisms does not come from genuine space

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?

• I added to my answer at the linked question, but this question seems a little broader. – David Roberts Sep 19 '18 at 0:51
• @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? – Praphulla Koushik Sep 19 '18 at 1:49
• 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. – David Roberts Sep 19 '18 at 4:48