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Monodromy is the name for the action of the homotopy groups of a space $X$ on fibers of covering spaces or locally constant ∞-stacks on $X$.
We discuss monodromy of covering spaces in elementary point-set topology.
(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
to the category Set of sets as follows:
to a point $x \in X$ assign the fiber $p^{-1}(\{x\}) \in Set$;
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 $\hat \gamma(0) = \hat x$
This construction is well defined for a given representative $\gamma$ due to the unique path-lifting property of covering spaces (this lemma) and it is independent of the choice of $\gamma$ in the given homotopy class of paths due to the homotopy-lifting property (this lemma). Similarly, these two lifting properties give that this construction respects composition in $\Pi_1(X)$ and hence is indeed a functor.
Hence this defines a “permutation groupoid representation” of $\Pi_1(X)$.
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
commutes, then the component functions
are compatible with the monodromy $Fib_{E}$ (def. ) along any path $\gamma$ between points $x$ and $y$ from def. in that the following diagrams of sets commute
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 $X$:
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 we have
and hence
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 (this lemma), $f \circ \hat \gamma$ is the unique lift of $\gamma$ with starting point $f(\hat x)$. By def. this means that
This is the equality to be shown.
(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
between the category of covering spaces and the permutation groupoid representations of the fundamental groupoid is known as the fundamental theorem of covering spaces.
(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 equivalently the Grothendieck construction of the monodromy functor $Fib_E \;\colon\; \Pi_1(X) \to Set$
whose
By the uniqueness of the path-lifting (this lemma) and the very definition of the monodromy functor.
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
path-connected precisely if the monodromy $Fib_E$ is a transitive action;
simply connected precisely if the monodromy $Fib_E$ is free action.
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 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 monodromy action of loops based at $x \in X$ on the fiber of the locally constant $\infty$-stack over $x$.