Question

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.]

Answer 1

[Removed a paragraph relating to an earlier version of the question]

You can construct Spectra categorically by adjoining an inverse to the endofunctor Σ of Top as a presentable (∞,1)-category. Inverting an endofunctor is a very different operation than inverting maps! It's like the difference between forming ℤ[1/p] and ℤ/(p).

Here is one way to verify the claim. To invert the endomorphism Σ of Top we should form the colimit, in the (∞,1)-category Pres of presentable categories and colimit-preserving functors, of the sequence Top → Top → ... where all the functors in the diagram are Σ. A basic fact about Pres is that we can compute such a colimit by forming the diagram (on the opposite index category) formed by the right adjoints of these functors, and taking its limit as a diagram of underlying (∞,1)-categories [HTT 5.5.3.18]. The functors in the limit cone will have left adjoints which are the functors to the colimit in Pres. In our case we obtain the sequence Top ← Top ← ... where the functors are Ω, and the limit of this sequence is precisely the classical definition of (Ω-)spectrum: a sequence of spaces X_n with equivalences X_n → ΩX_n+1.

Answer 2

I don't know the answer, but I have a related question. What if we let $f$ be the wedge of the maps $X\to \Omega \Sigma X$ (representing suspension) for all countable CW complexes X, and then apply Bousfield/Farjoun localization $L_f$?

It seems to me that, for the purposes of mapping in finite complexes, we have inverted the suspension operation.