nLab free infinite loop space

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 $Ho(Spectra)$.

Related concepts

• spectrification

• Omega spectrum – Completion of a suspension spectrum

• stable homotopy group

References

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