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 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 $g \in G$.

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?