# nLab free infinite loop space

### Context

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

## Idea

The free infinite loop space on a given pointed topological space.

## Definition

Write

$Q\;\colon\; Top^{\ast/} \longrightarrow Top^{*/}$

for the endofunctor on pointed topological spaces given as the colimit

$Q X \coloneqq \underset{\longrightarrow}{\lim}_k \Omega^k \Sigma^k X$

over iterated suspension and loop space construction.

This is the degree-0 part of spectrification of suspension spectra.

Equivalently this is

$Q X \simeq \Omega^\infty \Sigma^\infty X$

for

$(\Sigma^\infty \dashv \Omega^\infty) \;\colon\; Ho(Spectra) \underoverset{\underset{\Omega^\infty}{\longrightarrow}}{\overset{\Sigma^\infty}{\longleftarrow}}{} Ho(Top)$

the stabilization adjunction between the homotopy category Ho(Top) and the stable homotopy category $Ho(Spectra)$.

## References

• Cary Malkiewich, Some facts about $Q X$, 2011 (pdf)

Last revised on February 9, 2016 at 03:23:19. See the history of this page for a list of all contributions to it.