Spectra and symmetric spectra in general model categories

By Mark Hovey. 2000

343 citations recorded by the source corpus.

Abstract We give two general constructions for the passage from unstable to stable homotopy that apply to the known example of topological spaces, but also to new situations, such as the A 1 -homotopy theory of Morel and Voevodsky (preprint, 1998) and Voevodsky (Proceedings of the International Congress of Mathematicians, Vol. I, Berlin, Doc. Math. Extra Vol. I, 1998, pp. 579–604 (electronic)). One is based on the standard notion of spectra originated by Vogt (Boardman's Stable Homotopy Category, Lecture Notes Series, Vol. 21, Matematisk Institut Aarhus Universitet, Aarhus, 1970). Its input is a well-behaved model category D and an endofunctor T, generalizing the suspension. Its output is a model category Sp N ( D ,T) on which T is a Quillen equivalence. The second construction is based on symmetric spectra (Hovey et al., J. Amer. Math. Soc. 13(1) (2000) 149–208) and applies to model categories C with a compatible monoidal structure. In this case, the functor T must be given by tensoring with a cofibrant object K. The output is again a model category Sp Σ ( C ,K) where tensoring with K is a Quillen equivalence, but now Sp Σ ( C ,K) is again a monoidal model category. We study general properties of these stabilizations; most importantly, we give a sufficient condition for these two stabilizations to be equivalent that applies both in the known case of topological spaces and in the case of A 1 -homotopy theory.

Subfields: Geometry and Topology, Mathematical Physics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/202a372e32f335dec3af22e23a4764c4d252424a

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Citations in the corpus

Operads, Algebras and Modules in General Model Categories (2001, background): “ave been considered ([SS], [Hov2]). Also localization techniques for model categories have become important, because they yield many new useful model structures (for example the categoriesofspectra of[Hov3]). The most general statement for the existence of localizations is given in [Hir]. In all the”
Enriched functors and stable homotopy theory (2003, background): “In fact, the work [8] of Hovey shows that G-spectra and symmetric G-spectra are Quillen equivalent via a zig-zag”
Generalized Brown representability in homotopy categories (2005, neutral): “Here, we use the homotopy extension property of cofibrations ([23] 7.”
Homology and cohomology of E∞ ring spectra (2004, background): “The category NB is a topological pointed closed model category with weak equivalences and fibrations the maps that are weak equivalences and fibrations (resp.) of the underlying B-modules.”
Homotopy theory of simplicial sheaves in completely decomposable topologies (2008, methodology): “The standard localization techniques of [2] and [4] apply in the context of this closed model structure and in particular one may consider the class cll(GP) of GP -local equivalences in the sense of [2, Def.”
Motivic symmetric spectra (2000, neutral): “456 1.4 Flasque simplicial presheaves . . . . . . . . . . . . . . . . . . . 460”
K-theory of truncated polynomial algebras (2004, methodology): “As a model for the G-stable category we use symmetric orthogonal G-spectra; see [45] and [ 33, theorem 5.10 ].”
Twists of K-theory and TMF (2010, methodology): “ and a right adjoint f∗, again given by left and right Kan extension, respectively. We can also formally stabilize suitable model categories, using Hovey’s work on spectra in general model categories [Hov01]. Speciﬁcally, given a left proper cellular model category C and an endofunctor of C, Hovey c”
Goodwillie towers and chromatic homotopy: an overview (2004, methodology): “D/ in the ‘usual way’: this has been studied carefully by Schwede [43] (following Bousfield–Friedlander [10]), Hovey [21], and Basterra–Mandell [5].”
Topological equivalences for differential graded algebras (2006, background): “On this category one can form symmetric spectra based on the object Z[1], as in [ Ho2 ]; call the corresponding category Sp (ch+).”
Stable model categories are categories of modules (2003, methodology): “Categories of symmetric spectrum objects over a model category have been considered more generally by Hovey in [23].”
Higher K-theory via universal invariants (2007, methodology): “In particular, we characterize the derivator associated with a left Bousfield localization of a Quillen model category by a universal property, see theorem 5.4.”