nLab
category of covering spaces

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Bundles

Category theory

Contents

Definition

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

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

Properties

Proposition

(fundamental theorem of covering spaces)

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

Fib:Cov(X)Set Π 1(X). Fib \;\colon\; Cov(X) \longrightarrow Set^{\Pi_1(X)} \,.

If XX 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)