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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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    And spaces! It is quite amazing.2012-02-05

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