# What is the natural Lie groupoid structure on the Atiyah Lie groupoid of a principal $G$-bundle?

$$\DeclareMathOperator\At{At}\DeclareMathOperator\Obj{Obj}\DeclareMathOperator\Mor{Mor}$$According to https://ncatlab.org/nlab/show/Atiyah+Lie+groupoid#idea the Atiyah Lie groupoid $$\At(P)$$ of a principal $$G$$ bundle $$\pi:P \rightarrow X$$ is a category for which $$\Obj(\At(P))=\lbrace \pi^{-1}(x): x \in X \rbrace$$ and $$\Mor(\At(P))=\big\lbrace f:\pi^{-1}(x)\rightarrow \pi^{-1}(y): \text{f is a G equivariant morphism}\big\rbrace.$$ Structure maps of this category are easy to guess. Now it is easy to see that $$\At(P)$$ is indeed a groupoid.

Although it is mentioned in https://ncatlab.org/nlab/show/Atiyah+Lie+groupoid#idea that the Atiyah Lie groupoid is indeed a Lie groupoid, I am not able to guess appropriate smooth structures on $$\Obj(\At(P))$$ and $$\Mor(\At(P))$$ such that the source and the target maps are surjective submersions and other structure maps are smooth.

Is there any natural choice of such smooth structures on both $$\Obj(\At(P))$$ and $$\Mor(\At(P))$$ such that $$At(P)$$ is a Lie groupoid so that if someone talks about the Atiyah Lie groupoid of a principal $$G$$ bundle then he/she is precisely assuming those natural choice of smooth structures on $$\Obj(\At(P))$$ and $$\Mor(\At(P))$$?

I would also be very grateful if someone point me to any literature in this direction.

• @LSpice I was asking about the definition given in the section idea ncatlab.org/nlab/show/Atiyah+Lie+groupoid#idea not ncatlab.org/nlab/show/…. So are you saying both are actually same notion and the definition given in the idea section is just an informal notion? Jul 27, 2020 at 22:26
• Yes, that is what I am saying. Jul 27, 2020 at 22:27
• There is no definition in the 'idea' section; it is just an idea, and, as not rigorously defined, cannot be checked against a rigorous definition. Jul 27, 2020 at 22:38
• @LSpice Ok I got your point. Thanks. Jul 27, 2020 at 22:47
• The objects as defined on the nLab and on Wikipedia give isomorphic sets. That the morphism sets are the same is less obvious, but it boils down to knowing that a map between principal homogeneous $G$-spaces is determined entirely by its value at a single point. It's worth thinking about the case of a trivialisable bundle first. Jul 27, 2020 at 23:42

Let's see how these smooth structures are constructed. Recall that the set of objects is $$\{π^{−1}(x)\mid x∈X\}$$, i.e., the set of fibers of $$P$$. Fibers are in a bijective correspondence with points in the base $$X$$, and the latter is a smooth manifold.
The set of morphisms is $$\{f\colon π^{−1}(x)→π^{−1}(y)\mid \text{f is a G-equivariant morphism}\}$$. A morphism between two $$G$$-torsors $$U→V$$ is uniquely determined by its value $$v∈V$$ at some point $$u∈U$$. That is, for any pair $$(u,v)∈U⨯V$$ there is exactly one morphism that sends $$u↦v$$. The pair $$(gu,gv)$$ gives rise to the same morphism $$U→V$$ as $$(u,v)$$. It is also easy to see that the converse is true: $$(u,v)$$ and $$(u',v')$$ yield the same morphism if there is $$g∈G$$ such that $$(u',v')=(gu,gv)$$. Thus, the set of morphisms $$U→V$$ is $$(U⨯V)/G$$, where $$G$$ acts on $$U⨯V$$ via $$g(u,v)=(gu,gv)$$. The action of $$G$$ on $$U⨯V$$ is a smooth free proper action, so the quotient $$(U⨯V)/G$$ is a smooth manifold and the quotient map $$U⨯V→(U⨯V)/G$$ is a submersion.
From here, we see that the set of all morphisms is $$(P⨯P)/G$$ and therefore possesses a canonical smooth structure. The source and target maps are surjective submersions by the 2-out-of-3 property.
• Calling a fibre $X$ if the base is already called $X$ is a bit awkward. I suggest to pick different letters in the middle paragraph, maybe $F_x$ and $F_y$. Jul 28, 2020 at 15:02