Is it true that homotopy pullbacks and homotopy pushouts coincide in the category of spectra? 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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7$\begingroup$ Yes. This is a consequence of the Blakers-Massey theorem. $\endgroup$– Dylan WilsonCommented May 30, 2013 at 16:19
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14$\begingroup$ 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?) $\endgroup$– Jacob BellCommented May 30, 2013 at 16:21
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10$\begingroup$ 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 $\Omega\Sigma X_n\to \Omega X_{n+1}$. $\endgroup$– Tom GoodwillieCommented May 30, 2013 at 20:32
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Although the answer is sketched in the comments, I wanted to remark that this statement is proved carefully by Cary Malkiewich as Proposition 6.2.11 in Parameterized Spectra, A Low Tech Approach. He credits Model Categories of Diagram Spectra as the first place this was proven, but I couldn't find the result there.
answered Jun 18, 2020 at 0:21
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3$\begingroup$ First of all, that proof is unreadable :), second of all, he only reduces from the case of parametrized spectra to usual spectra and cites a different paper for that. $\endgroup$– user147129Commented Jun 18, 2020 at 4:16
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$\begingroup$ Still, good to have references for the result that one can cite. $\endgroup$ Commented Jun 18, 2020 at 12:56