Question about covering spaces extending inverse.

Question

If $p$ is a cover map how would I be able to show that $x\rightarrow p^{-1}(x)$ extends to a functor $p^{-1}$ originating from the Fundamental Group of $X$?

Answer 1

To explain more about my comment, here is the definition of a covering morphism of groupoids $q: H \to G$. First one defines the star at $x \in Ob(H)$ as the set $St_H(x)$ of elements of $H$ which start at $x$. We say $q$ is a covering morphism if each restriction of $q$ to $St_H(x) \to St_G(qx)$ is a bijection. One then proves that if $p: X \to Y$ is a covering map of spaces then the induced morphism $q=\pi_1(p)$ of fundamental groupoids is a covering morphism. This gives the transition from topology to algebra, one aim of algebraic topology. In this algebraic setting you should find it easy to prove that an element $g: px \to py$ in $G$ induces a bijection of sets $q^{-1}(px) \to q^{-1}(py)$. See Topology and Groupoids, Chapter 10. The functoriality is also easy.