On the category of pointed topological spaces Top there is a pair of adjoint functors (Σ, Ω), where Σ is the suspension functor and Ω is the loop space functor. In particular, for every topological space X we have the unit and counit morphisms of the above adjunction: X→ΩΣX and ΣΩX→X.

Denote by W the (ordinary or perhaps Bousfield?) localization of Top by the set of all morphisms identified above.

The pair (Σ, Ω) becomes a (homotopy) equivalence pair when we apply the suspension spectrum functor Σ^∞: Top→Sp. In particular, there is a canonical functor F: W→Ho(Sp).

What can we say about F?

In other words, what is the exact relationship between the full subcategory of (the homotopy category of?) the category of spectra consisting of all suspension spectra and the localized category defined above?

More generally, can we construct the category of spectra (or maybe just its homotopy category) from the category of pointed topological spaces using some kind of localization combined with other categorical constructions?

Can we construct the category of spectra (or maybe just its homotopy category) from the category of pointed topological spaces using some kind of localization combined with other categorical constructions?

[The first part of the original question was wrong for a trivial reason pointed out by Reid Barton.]