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...)