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The headline of the question is already the question itself: Is a spectrum a colimit of shifted suspension spectra?

By a spectrum I mean a sequence of spaces $E_n$ indexed over the natural numbers and structure maps $\Sigma E_n\to E_{n+1}$. A map of spectra are maps in all degrees commuting with the structure maps.

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I think that this is not a good way to get a feeling for what spectra actually are. The important thing is what the category of spectra looks like. You can build most things out of cofiber sequences inductively.This is the best way to think about it. Also, think about the different categories you do understand and how they "embed" in Spectra. – Sean Tilson Feb 3 '12 at 18:22
For the record I completely agree with Sean here :). Some particular examples of categories "embedding" within are (1) R-mod where R is a ring and (2) R-mod where is a DGA. (So really you get the derived categories, but it's okay.) – Dylan Wilson Feb 4 '12 at 6:59
And spaces! It is quite amazing. – Sean Tilson Feb 5 '12 at 0:22

Yes. Any spectrum should be the colimit of its finite subspectra (which can always be taken in the form you require).

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