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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.]

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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. –  Jeff Strom Mar 1 '10 at 0:46

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[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.

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Minor quible. Don't you only get connective spectra by starting with Top and inverting the suspension functor? Also a question: If you start with Top and invert the loops functor do you also get the category of (connective) spectra? –  Chris Schommer-Pries Feb 24 '10 at 4:39
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It's even worse than that, you only get the category of suspension spectra. Also, I had some other argument written here about inverting the loop functor which suffered from me accidentally getting an adjunction on the wrong side. Remember, kids: no Math Overflow late at night. –  Tyler Lawson Feb 24 '10 at 5:23
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I should have mentioned that I'm working in the world of presentable (∞,1)-categories. I'm pretty sure my new statement is correct. –  Reid Barton Feb 24 '10 at 5:28
    
Is the operation of adjoining an inverse to an endofunctor explained somewhere in Lurie's papers or anywhere else? –  Dmitri Pavlov Feb 24 '10 at 5:42
    
I don't know of a specific place where it is written down, but invertibility of an endofunctor is an (∞,1)-categorical (as opposed to (∞,2)-categorical) notion, so it's directly analogous to the situation in classical algebra. –  Reid Barton Feb 24 '10 at 6:06

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