I would probably state it as: for any $x$, the type of $g$ such that $g x = x$ is contractible (or, equivalently, a proposition, since it’s always inhabited by $e$).

]]>One must be careful with transitive action about the empty set, but all is fine here? I guess any group acting on the empty set is a free action.

What’s the HoTT version of freeness? Something like $IsSet(\sum_{\ast: BG} V(\ast))$.

]]>Fixed some hickups in the very first sentence.

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