TITLE: Spectra and localizations of the category of topological spaces
QUESTION [6 upvotes]: 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.]

REPLY [8 votes]: [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 Xn with equivalences Xn → ΩXn+1.