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Spectra and symmetric spectra in general model categories. (English) Zbl 1008.55006
Two general constructions for the passage from unstable to stable homotopy are given. The first construction is based on the standard notion of spectra [R. Vogt, Boardman’s Stable Homotopy Category, Lect. Notes Ser., Åarhus Univ. 21 (1970; Zbl 0224.55014)] and starts from a model category $${\mathcal D}$$ and an endofunctor $$T$$ generalizing the suspension. The second construction is based on symmetric spectra [M. Hovey, B. Shipley and J. Smith, J. Am. Math. Soc. 13, No. 1, 149-208 (2000; Zbl 0931.55006)] and applies to model categories $${\mathcal C}$$ witha compatible monoidal structure and an endofunctor $$T$$ which is given by tensoring with a cofibrant object $${\mathcal K}$$. The constructions apply not only to the homotopy theory for pointed topological spaces but also to the homotopy theory for algebraic varieties and schemes recently developed by F. Morel and V. Voevodsky [Publ. Math., Inst. Hautes Étud. Sci. 90, 45-143 (1999; Zbl 0983.14007); Doc. Math., J. DMV, Extra Vol. ICM Berlin 1998, vol. I, 579-604 (1998; Zbl 0907.19002)] using the affine line $$\mathbb{A}^1$$ instead of the unit interval $$[0,1]$$.
Reviewer: K.H.Kamps (Hagen)

##### MSC:
 55P42 Stable homotopy theory, spectra 18G55 Nonabelian homotopical algebra (MSC2010) 18D99 Categorical structures 55P60 Localization and completion in homotopy theory 55U35 Abstract and axiomatic homotopy theory in algebraic topology
##### Keywords:
unstable homotopy; stable homotopy; localization
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##### References:
 [1] J.F. Adams, Stable Homotopy and Generalised Homology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1974. · Zbl 0309.55016 [2] A.K. Bousfield, E.M. Friedlander, Homotopy theory of γ-spaces, spectra, and bisimplicial sets, in: M. Barratt, M. Mahowald (Eds), Geometric Applications of Homotopy Theory (Proceedings of the Conference, Evanston, IL, 1977), vol. I, Lecture Notes in Mathematics, vol. 657, Springer, Berlin, 1978, pp. 80-130. [3] B. Day, On closed categories of functors, in: S. Mac Lane (Ed.), Reports of the Midwest Category Seminar, IV, Lecture Notes in Mathematics, vol. 137, Springer, Berlin, 1970, pp. 1-38. · Zbl 0203.31402 [4] W.G. Dwyer, P.S. Hirschhorn, D.M. Kan, Model categories and more general abstract homotopy theory, in preparation. [5] W.G. Dwyer, J. Spaliński, Homotopy theories and model categories, in: I.M. James (Ed.), Handbook of Algebraic Topology, North-Holland, Amsterdam, 1995, pp. 73-126. · Zbl 0869.55018 [6] A.D. Elmendorf, I. Kriz, M.A. Mandell, J.P. May, Rings, Modules, and Algebras in Stable Homotopy Theory, American Mathematical Society, Providence, RI, 1997 (with an appendix by M. Cole). · Zbl 0894.55001 [7] Farjoun, E.D., Cellular spaces, null spaces and homotopy localization, Lecture notes in mathematics, vol. 1622, (1996), Springer Berlin [8] Heller, A., Stable homotopy categories, Bull. amer. math. soc., 74, 28-63, (1968) · Zbl 0177.25605 [9] Heller, A., Completions in abstract homotopy theory, Trans. amer. math. soc., 147, 573-602, (1970) · Zbl 0202.22804 [10] Heller, A., Stable homotopy theories and stabilization, J. pure appl. algebra, 115, 2, 113-130, (1997) · Zbl 0868.18001 [11] P.S. Hirschhorn, Localization of model categories, preprint, available at , version dated 4/12/2000. [12] Hovey, M., Model categories, (1999), American Mathematical Society Providence, RI · Zbl 0909.55001 [13] Hovey, M.; Shipley, B.; Smith, J., Symmetric spectra, J. amer. math. soc., 13, 1, 149-208, (2000) · Zbl 0931.55006 [14] J.F. Jardine, Motovic symmetric spectra, Doc. Math. 5 (2000) 445-552. [15] M.W. Johnson, Enriched sheaves as a framework for stable homotopy theory, Ph.D. Thesis, University of Washington, 1999. [16] L.G. Lewis Jr., J.P. May, M. Steinberger, J.E. McClure, Equivariant Stable Homotopy Theory, Springer, Berlin, 1986 (with contributions by J.E. McClure). [17] Mac Lane, S., Categories for the working Mathematician, (1998), Springer New York · Zbl 0906.18001 [18] M. Mandell, J.P. May, S. Schwede, B. Shipley, Model categories of diagram spectra, Proc. London Math. Soc., to appear. · Zbl 1017.55004 [19] F. Morel, V. Voevodsky, $$A\^{}\{1\}$$-homotopy theory of schemes, preprint, 1998. · Zbl 0983.14007 [20] Quillen, D.G., Homotopical algebra, Lecture notes in mathematics, vol. 43, (1967), Springer Berlin · Zbl 0168.20903 [21] Schwede, S., Spectra in model categories and applications to the algebraic cotangent complex, J. pure appl. algebra, 120, 1, 77-104, (1997) · Zbl 0888.55010 [22] S. Schwede, The stable homotopy category has a unique model at the prime 2, preprint, 2000. [23] S. Schwede, B. Shipley, A uniqueness theorem for stable homotopy theory, preprint, 2000. · Zbl 1009.55012 [24] Schwede, S.; Shipley, B.E., Algebras and modules in monoidal model categories, Proc. London math. soc. (3), 80, 2, 491-511, (2000) · Zbl 1026.18004 [25] Shipley, B., Symmetric spectra and topological Hochschild homology, K-theory, 19, 2, 155-183, (2000) · Zbl 0938.55017 [26] B. Stenström, Rings of Quotients, Die Grundlehren der mathematischen Wissenschaften, vol. 217, Springer, Berlin, 1975. [27] V. Voevodsky, A1-homotopy theory, Proceedings of the International Congress of Mathematicians, vol. I, Berlin, Doc. Math. Extra vol. I, pp. 579-604, 1998 (electronic). · Zbl 0907.19002 [28] Vogt, R., Boardman’s stable homotopy category, Lecture notes series, vol. 21, (1970), Matematisk Institut, Aarhus Universitet Aarhus · Zbl 0224.55014
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