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Is it true that homotopy pullback and homotopy pushout coincide in the category of spectrum? I had a feeling that this is the case, but don't know where to find a proof or how to prove it. Thanks!

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Yes. This is a consequence of the Blakers-Massey theorem. –  Dylan Wilson May 30 '13 at 16:19
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if you accept the fact that the category of spectra is stable, then it follows from the axioms of being stable (how tautological was this comment?) –  Jacob Bell May 30 '13 at 16:21
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Dylan, in some sense it's more elementary than Blakers-Massey. To get the idea, here's why $X\to \Omega\Sigma X$ is a weak equivalence when $X$ is a spectrum: $X$ consists of spaces $X_n$; the map is given by Freudenthal maps $X_n\to \Omega\Sigma X_n$; an inverse on spectrum homotopy groups is given by the obvious maps $\Omega\Sigma X_n\to \Omega X_{n+1}$. –  Tom Goodwillie May 30 '13 at 20:32

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