# Classiﬁcation of covering spaces

I just found this document http://www.math.toronto.edu/~drorbn/classes/0405/Topology/CoveringSpaces/CoveringSpaces.pdf in which it is said that we can use the classification of covering spaces theorem :

"Theorem: If $B$ is connected and locally connected with base point $b$ and fundamental group $G=\pi_1(B; b)$, then the map which assigns to every covering $p:X\rightarrow B$ its ﬁber $p^{-1}(b)$ over the basepoint $b$ induces a functor $\mathcal F$ from the category $C(B)$ of coverings of $B$ to the category $S(G)$ of $G$-sets — sets with a right $G$-action and set maps that respect the $G$ action. If in addition if $B$ is semi-locally simply connected then the functor $\mathcal F$ is an equivalence of categories. (In fact, this is iﬀ)"

to show that "Corollary 8: If $B$ is semi-locally simply connected, then for every subgroup $H$ of $\pi_1(B)$ we can found a connected covering $p:X\rightarrow B$ such as $p_*(\pi_1(X))=H$."

But I don't know how to use this theorem (the equivalence of categories) to get the corollary. Could you give me some advices ? Thanks

The covering you are looking for is the one corresponding to the $G$-set $H\backslash G$ by the theorem. This is the quotient of the universal covering of $B$ (which corresponds to the $G$-set $G$) by the action of $H$.

• Thanks, I was writting this answer when you posted yours: The functor $\mathcal F$ associates a covering $p:X\rightarrow B$ to $p^{-1}(b)$ for the monodromy action ($\forall[\gamma]\in\pi_1(B,b),\,\forall x\in p^{-1}(b),\,[\gamma]\cdot x=\tilde\gamma(1)$ where $\tilde\gamma$ is the only lift of $\gamma$ such as $\tilde\gamma(0)=x$). Now we can consider $H$ a subgroup of $\pi_1(B,b)$, then $\pi_1(B,b)$ acts on $\pi_1(B,b)/H$ and the stabilizer of $H$ is the subgroup $H$. But $\mathcal F$ is an equivalence of categories, so it exists a covering whose monodromy action is the one we just defined. Sep 4 '11 at 11:44

These results are illuminated by the notion of covering morphism of groupoids and the result that:

1. If $G$ is a groupoid, then there is an equivalence of categories between covering morphisms of $G$ and actions of $G$ on sets.

2. If $X$ is a suitably nice space, then the fundamental groupoid functor gives an equivalence of categories between covering maps of $X$ and covering groupoids of $\pi_1(X)$.

Note that because this models a map of space by a morphism of groupoids, this model is somewhat nearer to intuition, and is easier to apply, than the corresponding action on sets. These ideas are applied in a recent paper

Jeremy Brazas, Semicoverings: a generalization of covering space theory, Homology, Homotopy and Applications, 14(1), (2012) 33-63. (downloadable)