Let $P:E\to B$ be a cover map for connected grupoids and objects $e\in E$ and $b=P(e)\in B$, prove that the following are the same:
- The subgroup $P(E(e,e)\subset B(b,b)$ is normal.
- For every $e'\in E$ such that $P(e')=b$ we have that $P(E(e,e))=P(E(e',e'))$.
- For every $e'\in E$ such that $P(e')=b$ we have that there exist a grupoid isomorphism $G:E\to E$ such that $PG=P$ and $G(e)=e'$
I'm having trouble with this excercise since I recently started with covering maps and its kind of confusing to be honest.
What I tried: For $1\Rightarrow 2$ let $\gamma\in P(E(e,e))$ that is $\gamma:P(e)\to P(e)$ since $P$ is a functor. Let $\Phi:P(e')\to \Phi(e)$ be any arrow connecting the object (it exist since $B$ is connected), we have that $\Phi\gamma\Phi^{-1}\in P(E(e',e'))$ by definition (the arrows can be composed since $P(e')=b=P(e)$) and it is also an element of $P(E(e,e))$ since it is normal. Here I stopped since I've never used the covering map propertie and thought my thinking has to lead to nowhere.
Edit: Just read the notes and saw a theorem that proved that $P(E(e,e))$ is cojungated to $P(E(e',e'))$ so the equivalency is quite straight forward.
I got stuck in the first part so I didnt really tried the other ones, any help appreciated.
Edit: (Some definitions)
Let $E$ be a grupoid and $e\in E$ an object define the star in $e$ as $St_{E}(e)=\{e\to x\}$ that is all arrows that have $e$ as domain. Given $P:E\to B$ a functor between connected grupoids we say that $P$ is a covering if for every $e\in E$ we have that the functor $St_{E}(e)\xrightarrow{P}St_{B}(P(e))$ given by $(e\xrightarrow{f} x)\mapsto (P(e)\xrightarrow{P(f)} P(x))$ is bijective.