Terminal object for Prin(X,G) (principal $G$-bundles)

Question

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.

Answer 1

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.

Answer 2

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

1. There is only one isomorphism class of principal bundle, and
2. 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.