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I am trying to understand why it generally suffices to work with CW-spectra when working in stable homotopy/generalised (co)homology.

Indeed, it is true (an exercise in Adams' Book) that:

Any spectrum Y is weakly equivalent to a CW-Spectrum.

The hint is to consider the functor $[X,Y]_0$. (I guess X needs to be a CW spectrum for this to work)

Some brief terminology in Adams' book (I include this because I undersand they differ from more modern terminology such as May's)

  • A spectrum is a family of spaces $E_n$ with base points, provided with structure maps $\epsilon_n: \sum E_n \to E_{n+1}$ (or the equivalent adjoint map)

  • A spectrum is a CW spectrum if each $E_n$ are CW-complexes with base-point and each structure map maps isomorphically into a subcomplex of $E_{n+1}$

  • If $E$ and $F$ are spectra, with $E$ a CW spectrum we write $[E,F]_r$ for the set of homotopy maps of degree $r$ from $E$ to $F$

My understand of (classical) Brown Representability is that if I have a contravariant functor $H$ from pointed CW spaces to pointed sets satisfying the axioms, then there is a unique (up to homotopy at least) CW complex $Y$ such that the functor $F: [\quad,Y]$ and $H$ are naturally equivalent. It is then true if we replace CW space with CW spectrum.

In the stable CW category the functor $[X, \quad]_0$ satisfes the required axioms of Brown representability so does Brown representability just say that there is a natural equivalence from the functor $[X,Y]_0$ to the functor $[X,Z]_0$ where $Z$ is a CW spectrum? (Indeed that is what Adams' book defines as a weak equivalence...)

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up vote 6 down vote accepted

When working only ``up to homotopy'', that is, in the stable homotopy category, one can get away with old-fashioned CW spectra as Adams defined them and use them just as you say, and for the reason you say. But, on a foundational level, Adams' CW spectra were regarded as obsolescent when he was writing, and they are certainly long obsolete now. There is much to love in his book, but not in the foundational part on CW spectra. (Historically, the stable homotopy category was first defined by Boardman in his 1964 thesis, which he never published. Boardman knew and rejected Adams' definitions on aesthetic grounds even before Adams' book was written). It is work to absorb any definition, and I would advise learning more recent foundations. There are many variants from the 1970's and 1980 (Lewis-May-Steinberger and Bousfield-Friedlander for examples) but the 1990's saw a revolution that allows really good structure before passage to homotopy, allowing rings, modules, and algebras of spectra (Elmendorf-Kriz-Mandell-May, Hovey-Shipley-Smith, and others).

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Would you recommend skipping all the "classical" theory (and even the symmetric spectra theory) altogether and just reading, say, Lurie's "Higher Algebra" to learn about the $\infty$-category of spectra? –  Akhil Mathew Jul 26 '11 at 2:58
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Well, there are options between reading what was obsolescent at the time Adams was writing and what was written last week :) –  Mariano Suárez-Alvarez Jul 26 '11 at 3:02
    
I think I recall reading something similar in Neil Strickland's 'Bestiary' - that the construction of the smash product in Adams' book should be ignored, because of the construction by Elmendorf-Kriz-Mandell-May that has better properties before passing to homotopy. In fact he recommends Chapter XII of 'Equivariant Homotopy and Cohomology Theory' as a more gentle read than "Rings, Modules, and Algebras in Stable Homotopy Theory" (which I have had a glance at - I'm a while off understanding that I think) –  Juan S Jul 26 '11 at 9:37
    
there are many models of the smash product, and all of them are equivalent in that they induce the same thing in the homotopy category. Each version has its benefit. I think the main advantage Adams construction is in getting you to look at the modern approach. –  Sean Tilson Aug 8 '11 at 5:02

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