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\begin{document}

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\section*{monodromy}

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

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

[[!include homotopy - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{InPointSetTopology}{In point-set topology}\dotfill \pageref*{InPointSetTopology} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{in_cohesive_toposes}{In cohesive $\infty$-Toposes}\dotfill \pageref*{in_cohesive_toposes} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\emph{Monodromy} is the name for the [[action]] of the [[homotopy group (of an ∞-stack)|homotopy groups]] of a [[space]] $X$ on [[fibers]] of [[covering spaces]] or [[locally constant ∞-stacks]] on $X$.

\hypertarget{InPointSetTopology}{}\subsection*{{In point-set topology}}\label{InPointSetTopology}

We discuss monodromy of [[covering spaces]] in elementary [[point-set topology]].

\hypertarget{definition}{}\subsubsection*{{Definition}}\label{definition}

\begin{defn}
\label{CoveringSpaceMonodromy}\hypertarget{CoveringSpaceMonodromy}{}
\textbf{(monodromy of a [[covering space]])}

Let $X$ be a [[topological space]] and $E \overset{p}{\to} X$ a [[covering space]]. Write $\Pi_1(X)$ for the [[fundamental groupoid]] of $X$.

Define a [[functor]]

\begin{displaymath}
Fib_E
    \;\colon\;
  \Pi_1(X)
    \longrightarrow
  Set
\end{displaymath}
to the [[category]] [[Set]] of [[sets]] as follows:

\begin{enumerate}%
\item to a point $x \in X$ assign the [[fiber]] $p^{-1}(\{x\}) \in Set$;


\item to the [[homotopy class]] of a [[path]] $\gamma$ connecting $x \coloneqq \gamma(0)$ with $y \coloneqq \gamma(1)$ in $X$ assign the function $p^{-1}(\{x\}) \longrightarrow p^{-1}(\{y\})$ which takes $\hat x \in p^{-1}(\{x\})$ to the endpoint of a path $\hat \gamma$ in $E$ which lifts $\gamma$ through $p$ with starting point $\hat \gamma(0) = \hat x$

\begin{displaymath}
\itexarray{
    p^{-1}(x) &\overset{}{\longrightarrow}& p^{-1}(y)
    \\
    (\hat x = \hat \gamma(0)) &\mapsto& \hat \gamma(1)
  }
  \,.
\end{displaymath}


\end{enumerate}
This construction is well defined for a given representative $\gamma$ due to the unique path-lifting property of covering spaces (\href{covering+space#CoveringSpacePathLifting}{this lemma}) and it is independent of the choice of $\gamma$ in the given homotopy class of paths due to the homotopy-lifting property (\href{covering+space#CoveringSpacesHomotopyLifting}{this lemma}). Similarly, these two lifting properties give that this construction respects composition in $\Pi_1(X)$ and hence is indeed a [[functor]].

\end{defn}
Hence this defines a ``[[permutation representation|permutation]] [[groupoid representation]]'' of $\Pi_1(X)$.

\begin{prop}
\label{}\hypertarget{}{}
Given a [[homomorphism]] between two [[covering spaces]] $E_i \overset{p_i}{\to} X$, hence a [[continuous function]] $f \colon E_1 \to E_2$ which respects [[fibers]] in that the [[diagram]]

\begin{displaymath}
\itexarray{
    E_1 && \overset{f}{\longrightarrow} && E_2
    \\
    & {}_{\mathllap{p_1}}\searrow && \swarrow_{\mathrlap{p_2}}
    \\
    && X
  }
\end{displaymath}
[[commuting diagram|commutes]], then the component functions

\begin{displaymath}
f\vert_{\{x\}} \;\colon\; p_1^{-1}(\{x\}) \longrightarrow p_2^{-1}(\{x\})
\end{displaymath}
are compatible with the monodromy $Fib_{E}$ (def. \ref{CoveringSpaceMonodromy}) along any [[path]] $\gamma$ between points $x$ and $y$ from def. \ref{CoveringSpaceMonodromy} in that the following [[diagrams]] of [[sets]] [[commuting diagram|commute]]

\begin{displaymath}
\itexarray{
     p_1^{-1}(x) &\overset{f\vert_{\{x\}}}{\longrightarrow}& p_2^{-1}(x)
     \\
     {}^{\mathllap{Fib_{E_1}([\gamma])}}\downarrow
       &&
     \downarrow^{\mathrlap{ Fib_{E_2}([\gamma]) }}
     \\
     p_1^{-1}(y) &\underset{f\vert_{\{y\}}}{\longrightarrow}& p_2^{-1}(\{y\})
  }
  \,.
\end{displaymath}
This means that $f$ induces a [[natural transformation]] between the monodromy functors of $E_1$ and $E_2$, respectively, and hence that constructing monodromy is itself a functor from the [[category]] of [[covering spaces]] of $X$ to that of [[permutation representations]] of the [[fundamental groupoid]] of $X$:

\begin{displaymath}
Fib
   \;\colon\;
  Cov(X)
    \longrightarrow
  Set^{\Pi_1(X)}
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
For any $\hat x \in p_1^{-1}(x)$ let $\hat \gamma$ be the unique path in $E$ with $\hat \gamma(0) = \hat x$ and $p \circ \hat \gamma = \gamma$. By definition \ref{CoveringSpaceMonodromy} we have

\begin{displaymath}
Fib_{E_1}([\gamma(f)])(\hat x) = \hat \gamma(1)
\end{displaymath}
and hence

\begin{displaymath}
f(Fib_{E_1}([\gamma(f)])(\hat x)) = f(\hat \gamma(1))
\end{displaymath}
Now $f \circ \hat \gamma$ satisfies $f \circ \hat \gamma(0) = f(\hat x)$ and $p \circ f \circ \hat \gamma = \gamma$ by the fact that $f$ preserves fibers. Hence by uniqueness of path lifting (\href{covering+space#CoveringSpacePathLifting}{this lemma}), $f \circ \hat \gamma$ is the unique lift of $\gamma$ with starting point $f(\hat x)$. By def. \ref{CoveringSpaceMonodromy} this means that

\begin{displaymath}
Fib_{E_2}([\gamma])(f(\hat x))
  = 
  f (\hat \gamma(1))
  \,.
\end{displaymath}
This is the equality to be shown.

\end{proof}
\hypertarget{properties}{}\subsubsection*{{Properties}}\label{properties}

\begin{remark}
\label{}\hypertarget{}{}
\textbf{([[fundamental theorem of covering spaces]])}

The [[reconstruction of covering spaces from monodromy]] is an [[inverse functor]] to the monodromy functor. The resulting [[equivalence of categories]]

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

\end{remark}
\begin{example}
\label{CoveringSpaceFundamentalGroupoid}\hypertarget{CoveringSpaceFundamentalGroupoid}{}
\textbf{([[fundamental groupoid]] of covering space)}

Let $E \overset{p}{\longrightarrow} X$ be a covering space.

Then the [[fundamental groupoid]] $\Pi_1(E)$ of the total space $E$ is [[equivalence of categories|equivalently]] the [[Grothendieck construction]] of the [[monodromy]] functor $Fib_E \;\colon\; \Pi_1(X) \to Set$

\begin{displaymath}
\Pi_1(E)
   \;\simeq\;
  \int_{\Pi_1(X)} Fib_E
\end{displaymath}
whose

\begin{itemize}%
\item [[objects]] are pairs $(x,\hat x)$ consisting of a point $x \in X$ and en element $\hat x \in Fib_E(x)$;


\item [[morphisms]] $[\hat \gamma] \colon (x,\hat x) \to (x', \hat x')$ are morphisms $[\gamma] \colon x \to x'$ in $\Pi_1(X)$ such that $Fib_E([\gamma])(\hat x) = \hat x'$.



\end{itemize}
\end{example}
\begin{proof}
By the uniqueness of the path-lifting (\href{covering+space#CoveringSpacePathLifting}{this lemma}) and the very definition of the [[monodromy]] functor.

\end{proof}
\begin{prop}
\label{MonodromyConnectednessOfCoveringSpace}\hypertarget{MonodromyConnectednessOfCoveringSpace}{}
Let $X$ be a [[path-connected topological space]] and let $E \overset{p}{\to} X$ be a [[covering space]]. Then the total space $E$ is

\begin{enumerate}%
\item [[path-connected topological space|path-connected]] precisely if the [[monodromy]] $Fib_E$ is a [[transitive action]];


\item [[simply connected topological space|simply connected]] precisely if the [[monodromy]] $Fib_E$ is [[free action]].



\end{enumerate}
\end{prop}
\begin{proof}
By example \ref{CoveringSpaceFundamentalGroupoid}.

\end{proof}
\hypertarget{in_cohesive_toposes}{}\subsection*{{In cohesive $\infty$-Toposes}}\label{in_cohesive_toposes}

Let $\mathbf{H}$ be a [[cohesive (∞,1)-topos]] and $X \in \mathbf{H}$ any object. Then the [[locally constant ∞-stacks]] on $X$ are represented by morphisms $X \to LConst Core(\infty Grpd)$. By adjunction such morphisms are equivalent to [[(∞,1)-functors]] $\Pi(X) \to Core(\infty Grpd)$ This morphism exhibits the \textbf{monodromy} of the [[locally constant ∞-stack]].

Specifically, the restriction $\mathbf{B}\Omega_x \Pi(X) \hookrightarrow \Pi(X) \to \infty Grpd$ to the [[delooping]] $\mathbf{B}\Omega_x \Pi(X)$ of the [[loop space object]] $\Omega_x \Pi(X)$ at a chosen baspoint $x : {*} \to X$ is the \textbf{monodromy [[action]]} of loops based at $x \in X$ on the fiber of the locally constant $\infty$-stack over $x$.

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

\begin{itemize}%
\item [[eom]], \emph{\href{http://eom.springer.de/m/m064700.htm}{Monodromy transformation}}

\end{itemize}
[[!redirects monodromies]]



\end{document}