# nLab free loop space

### Context

#### Topology

topology

algebraic topology

mapping space

# Contents

## Idea

The free loop space of a topological space $X$ (based or not) is the space of all loops in $X$. This is in contrast to the based loop space of a based space $X$ for which the loops are at the fixed base point $x_0\in X$.

## Definition

### Explicit description

For $X$ a topological space, the free loop space $L X$ is the topological space $Map(S^1,X)$ of continuous maps in compact-open topology.

If we work in a category of based spaces, then still the topological space $Map(S^1,X)$ is in the non-based sense bit it itself has a distinguished point which is the constant map $t\mapsto x_0$ where $x_0$ is the base point of $X$.

### General abstract description

If $X$ is a topological space, the free loop space $L X$ of $X$ is defined as the free loop space object of $X$ formed in the (∞,1)-category Top.

Revised on September 18, 2013 01:55:36 by Urs Schreiber (77.251.114.72)