# nLab category of covering spaces

### Context

#### Bundles

bundles

fiber bundles in physics

category theory

# Contents

## Definition

For $X$ a topological space, the category of covering spaces $Cov(X)$ is the category whose

• objects are covering spaces $E \overset{p}{\to} X$ over $X$;

• morphisms are homomorphisms of covering spaces, hence continuous functions $f \colon E_1 \longrightarrow E_2$ such that we have a commuting diagram

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

This is equivalently the full subcategory of the slice category $Top_{/X}$ of the category of topological spaces over $X$ on those objects which are covering spaces.

## Properties

###### Proposition

(fundamental theorem of covering spaces)

For $X$ a topological space then forming monodromy is a functor from the category of covering spaces over $X$ to that of permutation groupoid representations of the fundamental groupoid of $X$:

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

If $X$ is locally path connected and semi-locally simply connected, then this is an equivalence of categories.

See at fundamental theorem of covering spaces for details.

Created on July 10, 2017 13:50:27 by Urs Schreiber (188.100.200.251)