Abstract
Let J-I be the homotopy category of all finite spectra, ~the homotopy category of all spectra and Y--[ 89 be the homotopy category of all prime-to-2 spectra. Let H: ~--y ~ (Abelian groups) be a homological functor (i.e. a generalized homology theory). A natural question is whether H can be lifted to a triangulated functor; is there a functor /-t: ~--y~D (Abelian groups) so that /~ composed with the usual homology functor on D(Abelian groups) is H. Here D (Abelian groups) = D(Z) is the derived category of Z-modules. If H is ordinary homology, the answer is clearly yes. After all, the ordinary homology of a space is given by the singular chain complex; almost by definition one has a lifting/~. By the Brown Representability Theorem, the general statement follows from the special case H = H . , the stable homotopy functor. Let us be a little more precise:
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