nLab Omega-spectrum




Among sequential pre-spectra XX, the Ω\Omega-spectra are those for which each structure map X nΩX n+1X_n \to \Omega X_{n+1} (from each component space to the based loop space of the next component space) is a weak homotopy equivalence.

(Beware that some authors require a homeomorphism instead and say “weak Ω\Omega-spectrum”, for the more general case).

Omega-spectra are particularly good representatives among pre-spectra of the objects of the stable (∞,1)-category of spectra, hence of the stable homotopy category. For instance they are (after geometric realization) the fibrant objects of the Bousfield-Friedlander model structure.


With Ω\Omega the notation for the loop space construction (whence the name), an Ω\Omega-spectrum is a sequence E =(E n) nE_\bullet = (E_n)_{n \in \mathbb{N}} of pointed ∞-groupoids (homotopy types) equipped for each nn \in \mathbb{N} with an equivalence of ∞-groupoids

E nΩE n+1. E_n \stackrel{\simeq}{\longrightarrow} \Omega E_{n+1} \,.

Remark: In terms of model category presentation one may also consider sequences of topological spaces equipped with homeomorphisms E nΩE n+1E_n \longrightarrow \Omega E_{n+1} See at spectrum the section Omega-spectra.



The inclusion of Ω\Omega-spectra into all sequential pre-spectra has a left adjoint, spectrification. See there for more.


Ω\Omega-spectrification of suspension spectra

Given a pointed topological space XX, its suspension spectrum Σ X\Sigma^\infty X is far from being an Ω\Omega-spectrum. The Ω\Omega-spectrum that it induces (its spectrification) is given by free infinite loop space constructions:


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

for the free infinite loop space functor 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.

Then (QΣ X) nQ(Σ nX)(Q \Sigma^\infty X)_n \coloneqq Q(\Sigma^n X) is the Ω\Omega-spectrum corresponding to the suspension spectrum of XX.

K-theory spectrum

The standard incarnation of the spectrum representing complex and real topological K-theory KUK U and KOK O is already an Ω\Omega-spectrum, due to Bott periodicity

Ω 2(BU×)BU× \Omega^2(B U \times \mathbb{Z}) \simeq B U \times \mathbb{Z}


Ω 8BOBO×. \Omega^8 B O \simeq B O \times \mathbb{Z} \,.


Last revised on December 24, 2020 at 19:07:14. See the history of this page for a list of all contributions to it.