Automorphisms of which structure form a Lie groupoid

Question

Given a manifold $M$, the collection of all automorphisms of $M$, denoted by $\text{Aut}(M)$ forms a Lie group.

Do we have similar setting in case of Lie groupoid?

Is there "a structure" whose "automorphisms" forms a Lie groupoid?

Answer 1

Here is a construction due to Ehresmann, and covered in detail by Mackenzie in either of his Lie groupoids books.

Take a principal $G$-bundle, $\pi\colon P\to M$, everything here in smooth manifolds. Then there is a Lie groupoid with object manifold $M$ and and a morphism from $m_1$ to $m_2$ is a triple $(m_1,m_2,\phi)$ where $\phi$ is an isomorphism $\phi\colon P_{m_1} \stackrel{\simeq}{\to} P_{m_2}$. This collection of morphisms has the structure of a smooth manifold, and the functions $(m_1,m_2,\phi) \mapsto m_i$ are surjective submersions. Composition is smooth, and inversion is $(m_1,m_2,\phi)\mapsto (m_2,m_1,\phi^{-1})$. This gives a Lie groupoid.

In slightly more detail, the collection of all isomorphisms $P_{m_1} \stackrel{\simeq}{\to} P_{m_2}$ forms a manifold diffeomorphic to the underlying manifold of $G$. The only hard part is to see how these are collected up into a single manifold. One can look locally around $m_1$ and $m_2$, where $P$ is trivial, and then you are looking at isomorphisms of trivial bundles, which are essentially parametrised collections of automorphisms of $G$ as a $G$-space (so, not as a group). The arrow manifold of this Lie groupoid is isomorphic to the quotient of $P\times P$ by the diagonal action of $G$, namely $(p_1,p_2) \mapsto (p_1g,p_2g)$. The source and target maps are given by $[p_1,p_2]\mapsto \pi(p_i)$. (Edited to confirm and correct this definition).

---

The automorphisms $X\stackrel{\simeq}{\to} X$ of a single object $X$ will never give rise to a Lie groupoid that has more than one object, since you somehow need to break being able to always compose automorphisms.

Answer 2

Just some quick addenda to David's comments:

1. You can see that the automorphisms of a Lie groupoid do not form a richer (groupoid) structure in some more specialised examples. It is well known that proper etale Lie groupoids are a Lie groupoid formulation of orbifolds (they are often called orbifold groupoids). Now their automorphisms corresponds to the group of orbifold diffeomorphisms and these can be turned into an infinite dimensional Lie group (this was the whole point of my PhD thesis, see The diffeomorphism group of a non-compact orbifold). The point is that these objects really form a group and not some higher structure like a groupoid or a $2$-Lie group.

2. In the literature people have considered groups of smooth maps on the arrow space of a Lie groupoid. Basically you look at diffeomorphisms $\operatorname{Diff}(G)$ which preserve source and target fibres. Again this yields groups not groupoids and has the advantage that the bisection group of the Lie groupoid identifies a subgroup of the automorphism group, called the inner automorphisms (by sending a bisection to the associated left translation). To my knowledge no smooth structure has yet been constructed on this automorphism group in general (though it seems to be an interesting question if there is one). For more references let me sneak in my recent paper The Lie group of vertical bisections of a regular Lie groupoid. The references can be found in the introduction together with an explanation of how this is related to the automorphism group.