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I have a question for a reference regarding classification of covering maps on a topological set $X$ with symmetric group Čech cohomology. In Classifying Covering Spaces using First Cohomology there is already given the result:

There is a bijection between the equivalence class of $i$-sheeted covering spaces of $X$ and the first Čech cohomology with coefficients in $S_i$.

I know the classification theorem for covering spaces in Hatcher, Algebraic Topology but only on the subgroups of the fundamental group of $X$, as also mentioned in the post.

Can someone suggest a reference for the bijection to the Čech cohomology?

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I have found the missing bijection. In Hatcher, Algebraic Topology there is the classification of n-sheeted covering spaces of a connected set $X$ with the homomorphisms of $\pi(X,x_0)\rightarrow S_n$ modulo conjugates, and in William Fulton, Algebraic Topology A First Course Corollary 15.5 there is the bijection from $H^1(U,G)$ to $\operatorname{Hom}(\pi_1(X,x),G)/\text{conjugacy}$. Thanks for the online publication of the Bachelor Thesis from M.P. Noordman, where I have found this!

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