TITLE: Are generalized cohomology theories a homotopy category of some category of invariants?
QUESTION [14 upvotes]: I was taught to think of generalized cohomology theories as the homotopy category of (symmetric) spectra. But is there also a category of 'invariants', that is, some category of contravariant functors from a suitable category of topological spaces to a suitable category of algebraic objects, which has a model category structure such that the homotopy category gives the category of generalized cohomology theories, without referring to spectra?
If such a thing exists, why do people prefer to use spectra?

REPLY [12 votes]: An even more compelling reason is that stable natural transformations between generalized cohomology theories on spaces form the wrong category, i.e. are not the same as the morphisms of the stable category. Let $E,F$ be generalized cohomology theories. Let $Z_n$ be the terms of the $\Omega$-spectrum of a CW model of $E$. Then, more or less by definition, stable natural transformations from $E$ to $F$ are the inverse limit of $F^nZ_n=F^0\Sigma^{\infty-n}Z_n$. This is the correct Hom set in the stable homotopy category when $\lim^1=0$, but this may not be the case (for example when $E=S$, $F$ is the wedge of $\Sigma^n HZ/2$ over $n\in \mathbb{Z}$).