# Are generalized cohomology theories a homotopy category of some category of invariants?

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?

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