MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).
show/hide this revision's text 3 clarify the 2 vs 4 thing.

Here is a short argument why we don't expect generalized cohomology theories to behave so well.

In the stable homotopy category, there is a generalized homology/cohomology theory represented by the sphere spectrum S, so that S_*(X) are the stable homotopy groups of X. It has a multiplication-by-2 self-map and we can use the triangulated structure to find an exact triangle S -> S -> M -> S[1]. We think of M as "the sphere mod 2", and it is called the mod-2 Moore spectrum.

In most derived categories coming from algebra, such an exact triangle would have the property that the multiplication-by-2 map on M was zero. However, we know that this is not the case here; the identity self-map actually has order 4multiplication-by-2 map is not zero, but the multiplication-by-4 map is.

One of the problems with using a "functor" language to get at generalized cohomology theories is that given a natural transformation E -> F of generalized cohomology theories, it is not clear what the associated cofiber should be in order to produce a triangulated structure. Spectra have a natural triangulated structure and they rectify this problem. (A map between spectra also includes "phantom" data that isn't easily detected by the natural transformation between associated generalized cohomology theories.)
show/hide this revision's text 2 fixed typo

Here is a short argument why we don't expect generalized cohomology theories to behave so well.

In the stable homotopy category, there is a generalized homology/cohomology theory represented by the sphere spectrum S, so that S_*(X) are the stable homotopy groups of X. It has a multiplication-by-2 self-map and we can use the triangulated structure to find an exact triangle S -> S -> M -> S[1]. We think of M as "the sphere mod 2", at and it is called the mod-2 Moore spectrum.

In most derived categories coming from algebra, such an exact triangle would have the property that the multiplication-by-2 map on M was zero. However, we know that this is not the case here; the identity self-map actually has order 4.

One of the problems with using a "functor" language to get at generalized cohomology theories is that given a natural transformation E -> F of generalized cohomology theories, it is not clear what the associated cofiber should be in order to produce a triangulated structure. Spectra have a natural triangulated structure and they rectify this problem. (A map between spectra also includes "phantom" data that isn't easily detected by the natural transformation between associated generalized cohomology theories.)
show/hide this revision's text 1

Here is a short argument why we don't expect generalized cohomology theories to behave so well.

In the stable homotopy category, there is a generalized homology/cohomology theory represented by the sphere spectrum S, so that S_*(X) are the stable homotopy groups of X. It has a multiplication-by-2 self-map and we can use the triangulated structure to find an exact triangle S -> S -> M -> S[1]. We think of M as "the sphere mod 2", at it is called the mod-2 Moore spectrum.

In most derived categories coming from algebra, such an exact triangle would have the property that the multiplication-by-2 map on M was zero. However, we know that this is not the case here; the identity self-map actually has order 4.

One of the problems with using a "functor" language to get at generalized cohomology theories is that given a natural transformation E -> F of generalized cohomology theories, it is not clear what the associated cofiber should be in order to produce a triangulated structure. Spectra have a natural triangulated structure and they rectify this problem. (A map between spectra also includes "phantom" data that isn't easily detected by the natural transformation between associated generalized cohomology theories.)