My question refers to a folklore statement that I have now seen a couple of times, but never really precise. One avatar is:
"For spectra every cofibration is equivalent to a fibration" (e.g. in the accepted answer here https://mathoverflow.net/a/56575/18744),
Another one is:
"For spectra, fibrations and cofibrations sequences are the same" (which is stronger than the statement above, because it works both ways)
At this point I am already quite happy with a suitable reference. However I do have some follow-up questions:
To what extent can I switch between spaces and spectra? The suspension functor does not preserve fibrations, correct? So what can be said about the relation of the homotopy fibre (cofibre) of spaces and the homotopy fibre (cofibre) of their suspension spectra?
What if the spaces involved are infinite loop-spaces? The resulting $\Omega$-spectra don't carry much different information from the spaces itself, do they?
Concretely:
I have a sequence $X\to Y\to Z$ of group-like H-spaces, and know that they form a homotopy fibration. I would like to make statements about the homotopy cofibre of $X\to Y$. I got the hint to 'work in spectra', where the two are "the same", but don't know what to make of it.
Edit: Thank you all for your answered. Together they cover quie a number of different points of view. Initially I hoped to be able to return back to spaces after doing an excursion through spectra (having fairly explicit $\Omega$-spectra for group like $H$-spaces). It seems that there is no totally generic way to do this and I think I have now enough material to think about the particularities.
Best Answer
One specific statement that people are likely referring to when they say things about fibrations and cofibrations being "the same" in spectra is that a homotopy pushout square of spectra is also a homotopy pullback square (considering squares with one corner trivial gives homotopy fibration and cofibration sequences). A brief explanation of this is given by Goodwillie here: homotopy pullback/pushout.
If you are interested in a formal statement in one of the model categories of spectra, then you need to look at a proof that these model categories are stable, as defined for instance in Hovey's book Model Categories (Chapter 7). As mentioned here Homotopy limit-colimit diagrams in stable model categories, Hovey explains (Remark 7.1.12) that homotopy pullback squares and homotopy pushout squares coincide in any stable model category. Proofs that the standard model categories of spectra are stable can be found in the basic references for these categories: for instance, for the category of symmetric spectra, Hovey-Shipley-Smith prove this in Theorem 3.1.14 of their paper (Symmetric spectra. J. Amer. Math. Soc. 13 (2000), no. 1, 149–208, available here http://www.ams.org/journals/jams/2000-13-01/S0894-0347-99-00320-3/S0894-0347-99-00320-3.pdf).