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.
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
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.
The inclusion of $\Omega$-spectra into all sequential pre-spectra has a left adjoint, spectrification. See there for more.
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
for the free infinite loop space functor given as the colimit
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 $X$.
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
and
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)
Last revised on December 24, 2020 at 14:07:14. See the history of this page for a list of all contributions to it.