Are generalized cohomology theories a homotopy category of some category of invariants? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T00:14:24Z http://mathoverflow.net/feeds/question/1621 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/1621/are-generalized-cohomology-theories-a-homotopy-category-of-some-category-of-invar Are generalized cohomology theories a homotopy category of some category of invariants? skupers 2009-10-21T10:15:48Z 2009-10-21T16:10:11Z <p>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?</p> <p>If such a thing exists, why do people prefer to use spectra?</p> http://mathoverflow.net/questions/1621/are-generalized-cohomology-theories-a-homotopy-category-of-some-category-of-invar/1642#1642 Answer by Tyler Lawson for Are generalized cohomology theories a homotopy category of some category of invariants? Tyler Lawson 2009-10-21T12:25:03Z 2009-10-21T16:10:11Z <p>Here is a short argument why we don't expect generalized cohomology theories to behave so well.</p> <p>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.</p> <p>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 multiplication-by-2 map is not zero, but the multiplication-by-4 map is.</p> <p>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.)</p>