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Let $F: Cov_B \to O(\pi_1(B,b))$ be a functor from the category of covering spaces over B to the orbit category of the fundamental group such that $F((q: E \to B)) \to q^{-1}(b)$. I am not sure how to prove that $F$ is fully faithful and essentially surjective.

I was told that $q^{-1}(b) \cong \pi_1(B,b)/q_*(\pi_1(E,e))$ but I do not see why this is true or how it helps.

Yunus Syed
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  • A related question is the more general https://math.stackexchange.com/questions/135457/covering-spaces-and-the-fundamental-groupoid?rq=1 – Ronnie Brown Feb 13 '18 at 16:47
  • @RonnieBrown I have never studied fundamental groupoids so I am trying to minimize the category theory. – Yunus Syed Feb 13 '18 at 20:37

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