Contents

# Contents

## Idea

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 $X$.

## In point-set topology

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

### Definition

###### Definition

(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

$Fib_E \;\colon\; \Pi_1(X) \longrightarrow Set$

to the category Set of sets as follows:

1. to a point $x \in X$ assign the fiber $p^{-1}(\{x\}) \in Set$;

2. 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 $\hat \gamma(0) = \hat x$

$\array{ p^{-1}(x) &\overset{}{\longrightarrow}& p^{-1}(y) \\ (\hat x = \hat \gamma(0)) &\mapsto& \hat \gamma(1) } \,.$

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 $\Pi_1(X)$.

###### Proposition

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

$\array{ E_1 && \overset{f}{\longrightarrow} && E_2 \\ & {}_{\mathllap{p_1}}\searrow && \swarrow_{\mathrlap{p_2}} \\ && X }$

commutes, then the component functions

$f\vert_{\{x\}} \;\colon\; p_1^{-1}(\{x\}) \longrightarrow p_2^{-1}(\{x\})$

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

$\array{ p_1^{-1}(x) &\overset{f\vert_{\{x\}}}{\longrightarrow}& p_2^{-1}(x) \\ {}^{\mathllap{Fib_{E_1}([\gamma])}}\downarrow && \downarrow^{\mathrlap{ Fib_{E_2}([\gamma]) }} \\ p_1^{-1}(y) &\underset{f\vert_{\{y\}}}{\longrightarrow}& p_2^{-1}(\{y\}) } \,.$

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 $X$:

$Fib \;\colon\; Cov(X) \longrightarrow Set^{\Pi_1(X)} \,.$
###### 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 we have

$Fib_{E_1}([\gamma(f)])(\hat x) = \hat \gamma(1)$

and hence

$f(Fib_{E_1}([\gamma(f)])(\hat x)) = f(\hat \gamma(1))$

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

$Fib_{E_2}([\gamma])(f(\hat x)) = f (\hat \gamma(1)) \,.$

This is the equality to be shown.

### Properties

###### Remark

(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

$Cov(X) \underoverset {\underset{Fib}{\longrightarrow}} {\overset{Rec}{\longleftarrow}} {} Set^{\Pi_1(X)}$

between the category of covering spaces and the permutation groupoid representations of the fundamental groupoid is known as the fundamental theorem of covering spaces.

###### Example

(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 $Fib_E \;\colon\; \Pi_1(X) \to Set$

$\Pi_1(E) \;\simeq\; \int_{\Pi_1(X)} Fib_E$

whose

• objects are pairs $(x,\hat x)$ consisting of a point $x \in X$ and en element $\hat x \in Fib_E(x)$;

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

###### Proof

By the uniqueness of the path-lifting (this lemma) and the very definition of the monodromy functor.

###### Proposition

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

1. path-connected precisely if the monodromy $Fib_E$ is a transitive action;

2. simply connected precisely if the monodromy $Fib_E$ is free action.

By example .

## In cohesive $\infty$-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 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 $x$.