Let $\operatorname{Klein}$ denote the category of principal homogeneous bundles. An object in this category is a tuple $\mathbf Q = (Q, P; G, H; q, a, \tilde a)$, where:


* $G$ is a Lie group, and $H$ is a closed subgroup;
* $P$ is a [$G$-torsor](http://math.ucr.edu/home/baez/torsors.html) and the manifold $Q$ is diffeomorphic to the quotient $G/H$;
* The (left) actions $\tilde a : G \to \operatorname{Aut}(P)$ and $a : G \to \operatorname{Aut}(Q)$ are morphisms of topological groups; and
* $q :P \to Q$ is a principal $H$-bundle.


A morphism $\Phi : \mathbf Q \to \mathbf Q'$ is described by a tuple $\Phi = (\varphi, \tilde \varphi, f)$, where $\varphi : Q \to Q'$ and $\tilde \varphi : P \to P'$ are diffeomorphisms which commute with the bundle maps $q : P \to Q$ and $q' : P' \to Q'$, and $f : G \to G'$ is a morphism of Lie groups mapping $H$ to $H'$, and which satisfies the identity $$\tilde a'\_{f(g)} \circ \tilde \varphi = \tilde \varphi \circ \tilde a\_g$$ for all $g \in G$. 




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I think that this is the correct categorical description of principal homogeneous bundles; please correct me if I'm wrong. I selected the name $\operatorname{Klein}$ in homage to Felix Klein and his Erlangen Program.


It seems that such a bundle $\mathbf Q$ contains all the data on its symmetries. Namely, I think that its automorphism group $\operatorname{Aut}(\mathbf Q)$ is isomorphic to its Lie group Lie group $G = G(\mathbf Q)$?


It is easy to see that there is a natural map $K : G \hookrightarrow \operatorname{Aut}(\mathbf Q)$, in that each $u \in G$ corresponds to a unique automorphism $K\_u \in \operatorname{Aut}(\mathbf Q)$. The morphism $K\_u = (k\_u, \tilde k\_u, c\_u)$ is defined by $$k\_u(q) = a\_u(q), \quad \tilde k\_u(p) = \tilde a\_u(p), \quad \mathrm{and} \quad c\_u(g) = ugu^{-1}.$$ That is, the morphism $K\_u$ acts by left-multiplication on both $Q$ and $P$, but by left-conjugation on $G$.


* Is this map $K$ surjective? i.e., is $\operatorname{Aut}(\mathbf Q)$ isomorphic to $G = G(\mathbf Q)$?


If the answer is yes, then I think that this captures the notion of "internal symmetries" of the bundle, since these are the transformations which preserve the bundle structure. 


However, I know that groupoids also show up to describe symmetries in a categorical setting, and I would be interested to hear more on that point of view.


* How can groupoids be used to describe symmetries in this category?



What are the symmetries of a principal homogeneous bundle?
No, in general $G=G(\mathbf{Q})$ can be strictly smaller that ${\rm Aut}(\mathbf{Q})$.


Let $G$ be a Lie group and $H\subset G$ be a Lie subgroup. Set $P=G$, $\ Q=G/H$, and define the maps in the obvious way. We get a principal homogeneous bundle $\mathbf{Q}=(Q,P,\dots)$ in your sense.


Now take $G=SL(n,\mathbb{C})$, $\ H=SO(n,\mathbb{C})$. Consider the automorphism $\tau$ of $G$ given by $\tau(g)= (g^T)^{-1}$, there $g^T$ denotes the transpose of $g$. Since $\tau$ takes $H$ to itself, it induces an automorphism of the principal homogeneous bundle $\mathbf{Q}$. It is well known that for $n\ge 3$, the automorphism $\tau$ of $G$ is *not inner*, i.e. is not of the form $c\_u(g)=ugu^{-1}$, which gives the negative answer to your question. 


