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A generalization of Ohkawa’s theorem

Part of: Homotopy theory Homology and cohomology theories

Published online by Cambridge University Press:  03 April 2014

Carles Casacuberta ,
Javier J. Gutiérrez  and
Jiří Rosický
Carles Casacuberta
Affiliation:
Institut de Matemàtica, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain email carles.casacuberta@ub.edu
Javier J. Gutiérrez
Affiliation:
Department of Algebra and Topology, Radboud Universiteit Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands email j.gutierrez@math.ru.nl
Jiří Rosický
Affiliation:
Department of Mathematics and Statistics, Masaryk University, Faculty of Science, Kotlářská 2, 61137 Brno, Czech Republic email rosicky@math.muni.cz
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Abstract

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A theorem due to Ohkawa states that the collection of Bousfield equivalence classes of spectra is a set. We extend this result to arbitrary combinatorial model categories.

Keywords

homotopymodel categoryacyclicityBousfield class

MSC classification

Secondary: 55N20: Generalized (extraordinary) homology and cohomology theories 55P42: Stable homotopy theory, spectra
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 5 , May 2014 , pp. 893 - 902
Copyright
© The Author(s) 2014 

References

Adámek, J. and Rosický, J., Locally presentable and accessible categories, London Mathematical Society Lecture Note Series, vol. 189 (Cambridge University Press, Cambridge, 1994).Google Scholar
Bousfield, A. K., The Boolean algebra of spectra, Comment. Math. Helv. 54 (1979), 368–377.CrossRefGoogle Scholar
Dugger, D. and Isaksen, D. C., Motivic cell structures, Algebr. Geom. Topol. 5 (2005), 615–652.Google Scholar
Dwyer, W. G. and Palmieri, J. H., Ohkawa’s theorem: there is a set of Bousfield classes, Proc. Amer. Math. Soc. 129 (2001), 881–886.Google Scholar
Dwyer, W. G. and Palmieri, J. H., The Bousfield lattice for truncated polynomial algebras, Homology, Homotopy Appl. 10 (2008), 413–436.CrossRefGoogle Scholar
Dugger, D., Combinatorial model categories have presentations, Adv. Math. 164 (2001), 177–201.CrossRefGoogle Scholar
Dundas, B. I., Röndigs, O. and Østvær, P. A., Motivic functors, Doc. Math. 8 (2003), 489–525.Google Scholar
Elmendorf, A. D., Kriz, I., Mandell, M. A. and May, J. P., Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs, vol. 47 (American Mathematical Society, Providence, RI, 1997).Google Scholar
Gabriel, P. and Ulmer, F., Lokal präsentierbare Kategorien, Lecture Notes in Mathematics, vol. 221 (Springer, Berlin, 1971).Google Scholar
Hirschhorn, P. S., Model categories and their localizations, Mathematical Surveys and Monographs, vol. 99 (American Mathematical Society, Providence, RI, 2003).Google Scholar
Hovey, M., Model categories, Mathematical Surveys and Monographs, vol. 63 (American Mathematical Society, Providence, RI, 1999).Google Scholar
Hovey, M. and Palmieri, J. H., The structure of the Bousfield lattice, in Homotopy invariant algebraic structures (Baltimore, 1998), Contemporary Mathematics, vol. 239 (American Mathematical Society, Providence. RI, 1999), 175–196.Google Scholar
Hovey, M., Shipley, B. and Smith, J. H., Symmetric spectra, J. Amer. Math. Soc. 13 (2000), 149–208.Google Scholar
Hu, P., S-modules in the category of schemes, Memoirs of the American Mathematical Society, vol. 161, no. 767 (American Mathematical Society, Providence, RI, 2003).Google Scholar
Iyengar, S. B. and Krause, H., The Bousfield lattice of a triangulated category and stratification, Math. Z. 273 (2013), 1215–1241.Google Scholar
Jardine, J. F., Motivic symmetric spectra, Doc. Math. 5 (2000), 445–553.Google Scholar
Johnson, D. C. and Wilson, W. S., B P-operations and Morava’s extraordinary K-theories, Math. Z. 144 (1975), 55–75.Google Scholar
Makkai, M. and Paré, R., Accessible categories: the foundations of categorical model theory, Contemporary Mathematics, vol. 104 (American Mathematical Society, Providence, RI, 1989).Google Scholar
Morel, F. and Voevodsky, V., A1-homotopy theory of schemes, Publ. Math. Inst. Hautes Études Sci. 90 (1999), 45–143.Google Scholar
Neeman, A., The chromatic tower for D (R), Topology 31 (1992), 519–532.Google Scholar
Naumann, N. and Spitzweck, M., Brown representability in A1-homotopy theory, J. K-Theory 7 (2011), 527–539.CrossRefGoogle Scholar
Ohkawa, T., The injective hull of homotopy types with respect to generalized homology functors, Hiroshima Math. J. 19 (1989), 631–639.Google Scholar
Quillen, D. G., Homotopical algebra, Lecture Notes in Mathematics, vol. 43 (Springer, Berlin, 1967).Google Scholar
Ravenel, D. C., Complex cobordism and stable homotopy groups of spheres (Academic Press, New York, 1986).Google Scholar
Röndigs, O. and Østvær, P. A., Modules over motivic cohomology, Adv. Math. 219 (2008), 689–727.Google Scholar
Rosický, J., Generalized Brown representability in homotopy categories, Theory Appl. Categ. 14 (2005), 451–479.Google Scholar
Rosický, J., On combinatorial model categories, Appl. Categ. Structures 17 (2009), 303–316.Google Scholar
Schwede, S. and Shipley, B., Algebras and modules in monoidal model categories, Proc. Lond. Math. Soc. 80 (2000), 491–511.CrossRefGoogle Scholar
Schwede, S. and Shipley, B., Stable model categories are categories of modules, Topology 42 (2003), 103–153.Google Scholar
Stanley, D., Invariants of t-structures and classification of nullity classes, Adv. Math. 224 (2010), 2662–2689.CrossRefGoogle Scholar
Stevenson, G., An extension of Dwyer’s and Palmieri’s proof of Ohkawa’s theorem on Bousfield classes, Preprint (2011), available at http://www.math.uni-bielefeld.de/∼gstevens.Google Scholar
Sullivan, D., Genetics of homotopy theory and the Adams conjecture, Ann. of Math. (2) 100 (1974), 1–79.Google Scholar
Voevodsky, V., $\mathbb{A}^1$-homotopy theory, Proceedings of the International Congress of Mathematicians (ICM, Berlin, 1998), vol. I, Doc. Math., extra volume, 1998, 579–604.Google Scholar
Wolcott, F. L., Variations of the telescope conjecture and Bousfield lattices for localized categories of spectra, Preprint (2013), arXiv:1307.3351.Google Scholar