Stable frames in model categories (Q436088)

It was shown in [\textit{W. G. Dwyer} and \textit{D. M. Kan}, Topology 19, 427--440 (1980; Zbl 0438.55011)] that the homotopy category of any model category is naturally enriched over the usual homotopy category of CW-complexes. In [\textit{M. Hovey}, Model categories. Mathematical Surveys and Monographs. 63. Providence, RI: American Mathematical Society. (1999; Zbl 0909.55001)], the question of a stable analogue was raised: Is the homotopy category of any stable model category naturally enriched over the stable homotopy category? Recall that a model category is called stable if it is pointed and the suspension functor induces a self-equivalence of the homotopy category.NEWLINENEWLINEThe question was answered in the affirmative in [\textit{D. Dugger}, Homology Homotopy Appl. 8, No. 1, 1--30 (2006; Zbl 1084.55011)], under the technical assumption that the model category be presentable. The main result (Theorem 6.3) of the paper under review is an affirmative answer without assumptions on the model category, not even functorial factorizations.NEWLINENEWLINEThe author makes use of (cosimplicial) frames, a type of cosimplicial resolution, as developed in [\textit{W. G. Dwyer} and \textit{D. M. Kan}, loc. cit.] and [\textit{M. Hovey}, loc. cit.]. Section 3 contains a good overview of the theory of frames in model categories. The key fact (Theorem 3.18) is a canonical equivalence \(Ho(Fr(C)) \cong Ho(C)\) between the homotopy category of frames in a model category \(C\) and the homotopy category of \(C\).NEWLINENEWLINEOne of the main technical results of the paper (Theorem 5.9) is a stable analogue of the above: a canonical equivalence \(Ho(SF(C)) \cong Ho(C)\) between the homotopy category of stable frames in a stable model category \(C\) and the homotopy category of \(C\).NEWLINENEWLINEUsing this, the author proves the main result: that the homotopy category of any stable model category is naturally enriched over the stable homotopy category \(SHC\) (Theorem 6.3). He also shows that the smash product thus obtained in \(SHC\) is the usual one (Theorem 6.4).NEWLINENEWLINEIn Section 7, the author explicitly relates his construction of the smash product in \(SHC\) to the classical smash product of Boardman (Theorem 7.6).