nLab
free infinite loop space

Contents

Idea

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

Definition

Write

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

for the endofunctor on pointed topological spaces given as the colimit

QXlim kΩ kΣ kX 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

QXΩ Σ X Q X \simeq \Omega^\infty \Sigma^\infty X

for

(Σ Ω ):Ho(Spectra)Ω Σ Ho(Top) (\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)Ho(Spectra).

References

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