Stable Homotopy theory
Among sequential pre-spectra , the -spectra are those for which the structure map is a weak homotopy equivalence (beware that some authors require a homeomorphism instead and say “weak -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 the notation for the loop space construction (whence the name),an -spectrum is a sequence of pointed ∞-groupoids (homotopy types) equipped for each with an equivalence of ∞-groupoids
Remark: In terms of model category presentation one may also consider sequences of topological spaces equipped with homeomorphisms See at spectrum the section Omega-spectra.
The inclusion of -spectra into all sequential pre-spectra has a left adjoint, spectrification. See there for more.
-spectrification of suspension spectra
Given a pointed topological space , its suspension spectrum is far from being an -spectrum. The -spectrum that it induces (its spectrification) is given by free infinite loop space constructions:
for the free infinite loop space functor given as the colimit
over iterated suspension and loop space construction.
Then is the -spectrum corresponding to the suspension spectrum of .
The standard incarnation of the spectrum representing complex and real topological K-theory and is already an -spectrum, due to Bott periodicity
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)