TAILIEUCHUNG - Đề tài " The stable homotopy category is rigid "

The purpose of this paper is to prove that the stable homotopy category of algebraic topology is ‘rigid’ in the sense that it admits essentially only one model: Rigidity Theorem. Let C be a stable model category. If the homotopy category of C and the homotopy category of spectra are equivalent as triangulated categories, then there exists a Quillen equivalence between C and the model category of spectra. Our reference model is the category of spectra in the sense of Bousfield and Friedlander [BF, §2] with the stable model structure. The point of the rigidity theorem is that its. | Annals of Mathematics The stable homotopy category is rigid By Stefan Schwede Annals of Mathematics 166 2007 837 863 The stable homotopy category is rigid By Stefan Schwede The purpose of this paper is to prove that the stable homotopy category of algebraic topology is rigid in the sense that it admits essentially only one model RiGiDity Theorem. Let C be a stable model category. If the homotopy category of C and the homotopy category of spectra are equivalent as triangulated categories then there exists a Quillen equivalence between C and the model category of spectra. Our reference model is the category of spectra in the sense of Bousfield and Friedlander BF 2 with the stable model structure. The point of the rigidity theorem is that its hypotheses only refer to relatively little structure on the stable homotopy category namely the suspension functor and the class of homotopy cofiber sequences. The conclusion is that all higher order structure of stable homotopy theory is determined by these data. Examples of this higher order structure are the homotopy types of function spectra which are not in general preserved by exact functors between triangulated categories or the algebraic K-theory. However the theorem does not claim that a model for the category of spectra can be constructed out of the triangulated homotopy category. Nor does it say that a given triangulated equivalence can be lifted to a Quillen equivalence of model categories. The rigidity theorem completes a line of investigation begun by Brooke Shipley and the author in SS and improved 2-locally in Sch . We refer to those two papers for motivation and for examples of triangulated categories that are not rigid . which admit exotic models. The new ingredients for the odd-primary case are roughly the following. The arguments of Sch reduce the problem at each prime p to a property of the first nonzero p-torsion class in the stable homotopy groups of spheres which is the Hopf map n at the prime 2 and the

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