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\begin{document}

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\section*{fundamental theorem of covering spaces}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{bundles}{}\paragraph*{{Bundles}}\label{bundles}

[[!include bundles - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In [[topological homotopy theory]], the \emph{fundamental theorem of covering spaces} says that for a sufficiently well-behaved [[topological space]] $X$, then the [[functor]] which sends a [[covering space]] of $X$ to the [[Set]]-[[action]] ([[permutation representation]]) of the [[fundamental groupoid]] of $X$ on the [[fibers]] of $E$ is an [[equivalence of categories]].

This is a basic instance of the general principle of [[Galois theory]].

It follows in particular that for [[connected topological space|connected]] $X$ then the [[automorphism group]] of the [[universal covering space]] of $X$ coincides with the [[fundamental group]] $\pi_1(X,x)$ itself (for any basepoint $x$). This often yields a convenient means to determine the [[fundamental group]] of $X$ in the first place.

\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

\begin{theorem}
\label{FundamentalTheoremOfCoveringSpaces}\hypertarget{FundamentalTheoremOfCoveringSpaces}{}
\textbf{(fundamental theorem of covering spaces)}

Let $X$ be a [[locally path-connected topological space|locally path-connected]] and [[semi-locally simply-connected topological space]]. Then the operations on

\begin{enumerate}%
\item extracting the [[monodromy]] $Fib_{E}$ of a [[covering space]] $E$ over $X$


\item [[reconstructing a covering space from monodromy]] $Rec(\rho)$



\end{enumerate}
constitute an [[equivalence of categories]]

\begin{displaymath}
Cov(X)
    \underoverset
      {\underset{Fib}{\longrightarrow}}
      {\overset{Rec}{\longleftarrow}}
      {\simeq}
  Set^{\Pi_1(X)}
\end{displaymath}
between the [[category of covering spaces]], and the category of [[permutation representation|permutation]] [[groupoid representations]] of the [[fundamental groupoid]] of $X$.

\end{theorem}
\begin{proof}
With the standard definitions of the two functors, both are in fact inverse [[isomorphisms]] of categories instead of just [[equivalences of categories]] (meaning that the required [[natural isomorphisms]] from the composites of the two functors to the [[identity functor]] are componentwise [[equalities]]), which establishes the claim right away. For definiteness, we make this explicit:

Given $\rho \in Set^{\Pi_1(X)}$ a [[permutation representation]], we need to exhibit a [[natural isomorphism]] of permutation representations.

\begin{displaymath}
\eta_{\rho} 
    \;\colon\; 
  \rho \longrightarrow Fib(Rec(\rho))
\end{displaymath}
First consider what the right hand side is like: By \href{reconstruction+of+covering+spaces+from+monodromy#ElementaryReconstructionCoveringSpace}{this def.} of $Rec$ and \href{monodromy#CoveringSpaceMonodromy}{this def.} of $Fib$ we have for every $x \in X$ an actual equality

\begin{displaymath}
Fib(Rec(\rho))(x) = \rho(x)
  \,.
\end{displaymath}
To similarly understand the value of $Fib(Rec(\rho))$ on morphisms $[\gamma] \in \Pi_1(X)$, let $\gamma \colon [0,1] \to X$ be a representing [[path]] in $X$. As in the proof of the path lifting lemma for covering spaces (\href{covering+space#CoveringSpacePathLifting}{this lemma}) we find a [[finite number]] of paths $\{\gamma_i\}_{i \in \{1,n\}}$ such that

\begin{enumerate}%
\item regarded as morphisms $[\gamma_i]$ in $\Pi_1(X)$ they [[composition|compose]] to $[\gamma]$:

\begin{displaymath}
[\gamma] = [\gamma_n] \circ \cdots \circ [\gamma_2] \circ [\gamma_1]
\end{displaymath}

\item each $\gamma_i$ factors through an open subset $U_i \subset X$ over which $Rec(\rho)$ trivializes.



\end{enumerate}
Hence by [[functor|functoriality]] of $Fib(Rec(\rho))$ it is sufficient to understand its value on these paths $\gamma_i$. But on these we have again by direct unwinding of the definitions that

\begin{displaymath}
Fib(Rec(\rho))([\gamma_i]) = \rho([\gamma_i])
  \,.
\end{displaymath}
This means that if we take

\begin{displaymath}
\eta_\rho(x) \colon \rho(x) \overset{=}{\longrightarrow} Fib(Rec(\rho))
\end{displaymath}
to be the above identification, then this is a [[natural transformation]] and hence in a particular a natural isomorphism, as required.

It remains to see that these morphisms $\eta_\rho$ are themselves natural in $\rho$, hence that for each morphism $\phi \colon \rho \to \rho'$ the diagram

\begin{displaymath}
\itexarray{
    \rho &\overset{\phi}{\longrightarrow}& \rho'
    \\
    {}^{\mathllap{eta_\rho}}\downarrow 
      && 
    \downarrow^{\mathrlap{\eta_{\rho'}}}
    \\
    Fib(Rec(\rho))
    &\underset{Fib(Rec(\phi))}{\longrightarrow}&
    Fib(Rec(\rho'))
  }
\end{displaymath}
commutes as a diagram in $Rep(\Pi_1(X), Set)$. Since these morphisms are themselves groupoid homotopies (natural isomorphisms) this is the case precisely if for all $x \in X$ the corresponding component diagram commutes. But by the above this is

\begin{displaymath}
\itexarray{
    \rho(x) &\overset{\phi(x)}{\longrightarrow}& \rho'(x)
    \\
    {}^{\mathllap{=}}\downarrow && \downarrow^{\mathrlap{=}}
    \\
    Fib(Rec(\rho))(x)
    &\underset{Fib(Rec(\phi))(x) }{\longrightarrow}&
    Fib(Rec(\rho'))(x)
  }
\end{displaymath}
and hence this means that the top and bottom horizontal morphism are in fact equal. Direct unwinding of the definitions shows that this is indeed the case.

Conversely, given $E \in Cov(X)$ a covering space, we need to exhibit a natural isomorphism of covering spaces of the form

\begin{displaymath}
\epsilon_E 
    \;\colon\;
  Rec(Fib(E)) \longrightarrow E
  \,.
\end{displaymath}
Again by \href{reconstruction+of+covering+spaces+from+monodromy#ElementaryReconstructionCoveringSpace}{this def.} of $Rec$ and \href{monodromy#CoveringSpaceMonodromy}{this def.} of $Fib$ the underlying set of $Rec(Fib(E))$ is actually equal to that of $E$, hence it is sufficient to check that this [[identity function]] on underlying sets is a [[homeomorphism]] of [[topological spaces]].

By the assumption that $X$ is [[locally path-connected topological space|locally path-connected]] and [[semi-locally simply connected topological space|semi-locally simply connected]], it is sufficient to check for $U\subset X$ an open path-connected subset and $x \in X$ a point with the property that $\pi_1(U,x) \to \pi_1(X,x)$ lands is constant on the trivial element, that the open subsets of $E$ of the form $U \times \{\hat x\} \subset p^{-1}(U)$ form a basis for the topology of $Rec(Fib(E))$. But this is the case by definition of $Rec$.

It remains to see that $\epsilon_E$ is itself natural in $E$. But as for the converse direction, since the components of $\epsilon_E$ are in fact equalities, this follows by direct unwinding of the definitions.

This establishes an equivalence as required. In fact this is an [[adjoint equivalence]].

\end{proof}
\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\begin{itemize}%
\item [[fundamental group of the circle is the integers]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Lecture notes include

\begin{itemize}%
\item [[Friedhelm Waldhausen]], around p. 122 of \emph{Topologie} (\href{https://www.math.uni-bielefeld.de/~fw/ein.pdf}{pdf})


\item [[Jesper Møller]], \emph{The fundamental group and covering spaces} (2011) (\href{http://www.math.ku.dk/~moller/f03/algtop/notes/covering.pdf}{pdf})



\end{itemize}


\end{document}