Symmetric spectra

Hovey, Mark; Shipley, Brooke; Smith, Jeff

doi:10.1090/S0894-0347-99-00320-3

Symmetric spectra
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by Mark Hovey, Brooke Shipley and Jeff Smith

J. Amer. Math. Soc. 13 (2000), 149-208

DOI: https://doi.org/10.1090/S0894-0347-99-00320-3

Published electronically: September 22, 1999

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Abstract:

The stable homotopy category, much studied by algebraic topologists, is a closed symmetric monoidal category. For many years, however, there has been no well-behaved closed symmetric monoidal category of spectra whose homotopy category is the stable homotopy category. In this paper, we present such a category of spectra; the category of symmetric spectra. Our method can be used more generally to invert a monoidal functor, up to homotopy, in a way that preserves monoidal structure. Symmetric spectra were discovered at about the same time as the category of $S$-modules of Elmendorf, Kriz, Mandell, and May, a completely different symmetric monoidal category of spectra.

References

J. F. Adams, Stable homotopy and generalised homology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1974. MR 0402720
M. Bökstedt, Topological Hochschild homology, preprint, 1985.
Francis Borceux, Handbook of categorical algebra. 1, Encyclopedia of Mathematics and its Applications, vol. 50, Cambridge University Press, Cambridge, 1994. Basic category theory. MR 1291599
A. K. Bousfield and E. M. Friedlander, Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977) Lecture Notes in Math., vol. 658, Springer, Berlin, 1978, pp. 80–130. MR 513569
Edward B. Curtis, Simplicial homotopy theory, Advances in Math. 6 (1971), 107–209 (1971). MR 279808, DOI 10.1016/0001-8708(71)90015-6
W. G. Dwyer, P. S. Hirschhorn, and D. M. Kan, Model categories and general abstract homotopy theory, in preparation.
W. G. Dwyer and Brooke E. Shipley, Hyper symmetric spectra, in preparation.
W. G. Dwyer and J. Spaliński, Homotopy theories and model categories, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 73–126. MR 1361887, DOI 10.1016/B978-044481779-2/50003-1
A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs, vol. 47, American Mathematical Society, Providence, RI, 1997. With an appendix by M. Cole. MR 1417719, DOI 10.1090/surv/047
Thomas Geisser and Lars Hesselholt, Topological cyclic homology of schemes, to appear in K-theory, Seattle, 1996 volume of Proc. Symp. Pure Math.
P. S. Hirschhorn, Localization of model categories, preprint, 1999.
Mark Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999. MR 1650134
Mark Hovey, Stabilization of model categories, preprint, 1998.
Mark Hovey, John H. Palmieri, and Neil P. Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997), no. 610, x+114. MR 1388895, DOI 10.1090/memo/0610
L. Gaunce Lewis Jr., Is there a convenient category of spectra?, J. Pure Appl. Algebra 73 (1991), no. 3, 233–246. MR 1124786, DOI 10.1016/0022-4049(91)90030-6
Elon L. Lima, The Spanier-Whitehead duality in new homotopy categories, Summa Brasil. Math. 4 (1959), 91–148 (1959). MR 116332
Saunders MacLane, Categories for the working mathematician, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York-Berlin, 1971. MR 0354798
M. Mandell, J. P. May, B. Shipley, and S. Schwede, Diagram spaces, diagram spectra and FSPs, preprint, 1998.
M. Mandell, J. P. May, B. Shipley, and S. Schwede, Model categories of diagram spectra, preprint, 1998.
J. Peter May, Simplicial objects in algebraic topology, Van Nostrand Mathematical Studies, No. 11, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0222892
Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432, DOI 10.1007/BFb0097438
Stefan Schwede, S-modules and symmetric spectra, preprint, 1998.
Stefan Schwede and Brooke Shipley, Classification of stable model categories, in preparation.
Stefan Schwede and Brooke Shipley, Algebras and modules in monoidal model categories, to appear in Proc. London Math. Soc.
B. Shipley, Symmetric spectra and topological Hochschild homology, to appear in K-theory.
V. Voevodsky, The Milnor conjecture, preprint, 1997.
Rainer Vogt, Boardman’s stable homotopy category, Lecture Notes Series, No. 21, Aarhus Universitet, Matematisk Institut, Aarhus, 1970. MR 0275431
Friedhelm Waldhausen, Algebraic $K$-theory of spaces, Algebraic and geometric topology (New Brunswick, N.J., 1983) Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 318–419. MR 802796, DOI 10.1007/BFb0074449
George W. Whitehead, Generalized homology theories, Trans. Amer. Math. Soc. 102 (1962), 227–283. MR 137117, DOI 10.1090/S0002-9947-1962-0137117-6