Terminal object for Prin(X,G) (principal $G$-bundles) Consider the category $Prin(X,G)$ of principal $G$-bundles over a topological space $X$ ($G$ is a group). My question is: does this category have a terminal object?
I would say that the category of $G$-bundles over $X$ actually has one, namely $X$ with the trivial action, but this is not a principal bundle, since the action is not free on fibres.
Thank you in advance.
 A: The category of principal bundles is a groupoid, so the only way it could have a terminal object is if it were contractible, meaning both that


*

*There is only one isomorphism class of principal bundle, and

*Its automorphism group is trivial.


However, you can verify that the automorphism group of the trivial $G$-bundle is the group of continuous maps $X \to G$, which is nontrivial as long as $G$ is nontrivial and $X$ is not empty. So the only way this automorphism group could be trivial is if either $X$ is empty or $G$ is trivial. 
A: Suppose that such terminal object $e$ exists, you have a morphism of bundles $X\times G\rightarrow e$, implies that $e$ is isomorphic to $X\times G$, for every $G$-principal bundle $P$, you have a morphism $P\rightarrow X\times G$. This implies that $P$ is trivial. The terminal object exists if every $G$-principal bundle is trivial.
We have used the fact that a morphism between two principal $G$-bundles over $X$ is an isomorphism.
A morphism of principal bundles is an isomorphism.
