nLab
path space

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

Mapping space

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

In topology the path space of a topological space XX is a topological space of all continuous paths in XX.

In topological homotopy theory the path space constrution serves to exhibit homotopies in the guise of right homotopies. This situation generalizes to many other model categories and one speaks more generally of path space objects in this case.

For exposition in the context of point-set topology see at Introduction to Topology -- 1 around this example.

For exposition in the context of topological homotopy theory see at Introduction to Homotopy Theory around this definition.

Definition

For XX a topological space, then its path space is the mapping space X [0,1]X^{[0,1]}, out of the topological interval into XX, i.e. the set of continuous function γ:[0,1]X\gamma \;\colon\; [0,1] \to X equipped with the compact-open topology.

The two endpoint inclusions *:[0,1]\ast \colon [0,1] and the unique projection [0,1]*[0,1] \to \ast induce continuous functions

XX [0,1]X (const 0,const 1)X×X X \overset{}{\longrightarrow} X^{[0,1]} \overset{X^{(const_0,const_1)}}{\longrightarrow} X \times X

(inclusion of constant paths and endpoint evaluation of paths).

Properties

Relation to loop space

The fiber product of the projection with the diagonal on XX is the free loop space ℒ𝒳\mathcal{L X} of XX:

X X [0,1] (pb) X (const 0,const 1) X Δ X X×X \array{ \mathcal{L}X &\longrightarrow& X^{[0,1]} \\ \downarrow &(pb)& \downarrow^{\mathrlap{X^{(const_0,const_1)}}} \\ X &\underset{\Delta_X}{\longrightarrow}& X \times X }

If XX is equipped with a choice of basepoint x:*Xx \colon \ast \to X (making it a pointed topological space), then the further fiber product with this basepoint inclusion is the based loop space Ω xX\Omega_x X:

Ω xX X X [0,1] (pb) (pb) X (const 0,const 1) * x X Δ X X×X \array{ \Omega_x X &\longrightarrow& \mathcal{L}X &\longrightarrow& X^{[0,1]} \\ \downarrow &(pb)& \downarrow &(pb)& \downarrow^{\mathrlap{X^{(const_0,const_1)}}} \\ \ast &\underset{x}{\longrightarrow}& X &\underset{\Delta_X}{\longrightarrow}& X \times X }

See also

Revised on July 1, 2017 12:43:22 by Matt Earnshaw (87.81.151.195)