Fundamental groupoids is a embedding of category $Cov(B)\to Cov(\Pi(B))$

Question

It is stated in J.P.May's A Concise Course in Algebraic Topology page 29 that the fundamental groupoid functor induces a bijection $$Cov(E,E')\longleftrightarrow Cov(\Pi(E),\Pi(E')).$$ So does that mean that the functor $\Pi:Cov(B)\to Cov(\Pi(B))$ is an embedding of category? It only remains to be checked that if $p:E\to B$ and $p':E'\to B$ are two covering spaces such that $\Pi(p):\Pi(E)\to \Pi(B)$ coincide with $\Pi(p'):\Pi(E')\to \Pi(B)$, then $p = p'$. This seems to be true, but I just want to confirm.

Answer 1

The statement if that if $X$ is locally path connected and semilocally $1$-connected then the fundamental groupoid functor gives an equivalence of categories

$$\pi_1:TopCov(X) \to GpdCov(\pi_1(X)).$$

This is essentially 10.6.1 of Topology and Groupoids (although the Hausdorff assumption there is not needed). The previous section 10.5 gives conditions on $X$ for a covering morphism $p:G \to \pi_1(X)$ to come from a covering map to $X$.

Feb 9: Your comment, mez, is correct. I will assume $X$ locally path connected, since I have difficulty without that. Under that condition, if a covering morphism $p: G \to \pi_1(X)$ comes from a covering map $Y \to X$, then $X$ must be locally $\chi_p$-connected, in the sense that each $x \in X$ has a path connected neighbourhood $U$ such that the image of $\pi_1(U,x)$ in $\pi_1(X,x)$ lies in the intersection of the images of $G(y)$ under $p$ for all $y \in p^{-1}(x)$. Then a topology on $Ob(G)$ is generated by the images of each such $U$ under the lifts of $\pi_1(U)$ to $G$; the $\chi_p$-condition implies that these lifts cover all of $Ob(G)$. So one can then identify $Y$ with $Ob(G)$.

The paper: J. Brazas, "Semicoverings: a generalisation of covering space theory" Homology, Homotopy and Applications, vol. 14(1), 2012, pp.33–63, may also be relevant.