nLab Omega-spectrum

Idea

Among sequential pre-spectra $X$, the $\Omega$-spectra are those for which each structure map $X_n \to \Omega X_{n+1}$ (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 $\Omega$-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.

Definition

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

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

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

Properties

$\Omega$-spectrification

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

Examples

$\Omega$-spectrification of suspension spectra

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

write

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

for the free infinite loop space functor given as the colimit

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

over iterated suspension and loop space construction.

Then $(Q \Sigma^\infty X)_n \coloneqq Q(\Sigma^n X)$ is the $\Omega$-spectrum corresponding to the suspension spectrum of $X$.

K-theory spectrum

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

$$\Omega^2(B U \times \mathbb{Z}) \simeq B U \times \mathbb{Z}$$

and

$$\Omega^8 B O \simeq B O \times \mathbb{Z} \,.$$

Related concepts

• coordinate-free spectrum

• excisive functor

References

• Frank Adams, part III, section 2 of Stable homotopy and generalised homology, 1974

• Anthony Elmendorf, Igor Kriz, Peter May, section 1 of Modern foundations for stable homotopy theory, in Ioan Mackenzie James, Handbook of Algebraic Topology, Amsterdam: North-Holland (1995) pp. 213–253, (pdf)

• Stanley Kochmann, section 3.3 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

• Cary Malkiewich, section 2.2 of The stable homotopy category, 2014 (pdf)