Symmetric spectra
Authors:
Mark Hovey, Brooke Shipley and Jeff Smith
Journal:
J. Amer. Math. Soc. 13 (2000), 149208
MSC (2000):
Primary 55P42, 55U10, 55U35
Published electronically:
September 22, 1999
MathSciNet review:
1695653
Fulltext PDF Free Access
Abstract 
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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 wellbehaved 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 modules of Elmendorf, Kriz, Mandell, and May, a completely different symmetric monoidal category of spectra.
 [Ada74]
J.
F. Adams, Stable homotopy and generalised homology, University
of Chicago Press, Chicago, Ill., 1974. Chicago Lectures in Mathematics. MR 0402720
(53 #6534)
 [Bök85]
M. Bökstedt, Topological Hochschild homology, preprint, 1985.
 [Bor94]
Francis
Borceux, Handbook of categorical algebra. 2, Encyclopedia of
Mathematics and its Applications, vol. 51, Cambridge University Press,
Cambridge, 1994. Categories and structures. MR 1313497
(96g:18001b)
 [BF78]
A.
K. Bousfield and E.
M. Friedlander, Homotopy theory of Γspaces, spectra, and
bisimplicial sets, Geometric applications of homotopy theory (Proc.
Conf., Evanston, Ill., 1977), II, Lecture Notes in Math., vol. 658,
Springer, Berlin, 1978, pp. 80–130. MR 513569
(80e:55021)
 [Cur71]
Edward
B. Curtis, Simplicial homotopy theory, Advances in Math.
6 (1971), 107–209 (1971). MR 0279808
(43 #5529)
 [DHK]
W. G. Dwyer, P. S. Hirschhorn, and D. M. Kan, Model categories and general abstract homotopy theory, in preparation.
 [DS]
W. G. Dwyer and Brooke E. Shipley, Hyper symmetric spectra, in preparation.
 [DS95]
W.
G. Dwyer and J.
Spaliński, Homotopy theories and model categories,
Handbook of algebraic topology, NorthHolland, Amsterdam, 1995,
pp. 73–126. MR 1361887
(96h:55014), http://dx.doi.org/10.1016/B9780444817792/500031
 [EKMM97]
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
(97h:55006)
 [GH97]
Thomas Geisser and Lars Hesselholt, Topological cyclic homology of schemes, to appear in Ktheory, Seattle, 1996 volume of Proc. Symp. Pure Math.
 [Hir99]
P. S. Hirschhorn, Localization of model categories, preprint, 1999.
 [Hov98a]
Mark
Hovey, Model categories, Mathematical Surveys and Monographs,
vol. 63, American Mathematical Society, Providence, RI, 1999. MR 1650134
(99h:55031)
 [Hov98b]
Mark Hovey, Stabilization of model categories, preprint, 1998.
 [HPS97]
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
(98a:55017), http://dx.doi.org/10.1090/memo/0610
 [Lew91]
L.
Gaunce Lewis Jr., Is there a convenient category of spectra?,
J. Pure Appl. Algebra 73 (1991), no. 3,
233–246. MR 1124786
(92f:55008), http://dx.doi.org/10.1016/00224049(91)900306
 [Lim59]
Elon
L. Lima, The SpanierWhitehead duality in new homotopy
categories, Summa Brasil. Math. 4 (1959),
91–148 (1959). MR 0116332
(22 #7121)
 [ML71]
Saunders
MacLane, Categories for the working mathematician,
SpringerVerlag, New York, 1971. Graduate Texts in Mathematics, Vol. 5. MR 0354798
(50 #7275)
 [MMSS98a]
M. Mandell, J. P. May, B. Shipley, and S. Schwede, Diagram spaces, diagram spectra and FSPs, preprint, 1998.
 [MMSS98b]
M. Mandell, J. P. May, B. Shipley, and S. Schwede, Model categories of diagram spectra, preprint, 1998.
 [May67]
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
(36 #5942)
 [Qui67]
Daniel
G. Quillen, Homotopical algebra, Lecture Notes in Mathematics,
No. 43, SpringerVerlag, Berlin, 1967. MR 0223432
(36 #6480)
 [Sch98]
Stefan Schwede, Smodules and symmetric spectra, preprint, 1998.
 [SS]
Stefan Schwede and Brooke Shipley, Classification of stable model categories, in preparation.
 [SS97]
Stefan Schwede and Brooke Shipley, Algebras and modules in monoidal model categories, to appear in Proc. London Math. Soc.
 [Shi]
B. Shipley, Symmetric spectra and topological Hochschild homology, to appear in Ktheory.
 [Voe97]
V. Voevodsky, The Milnor conjecture, preprint, 1997.
 [Vog70]
Rainer
Vogt, Boardman’s stable homotopy category, Lecture Notes
Series, No. 21, Matematisk Institut, Aarhus Universitet, Aarhus, 1970. MR 0275431
(43 #1187)
 [Wal85]
Friedhelm
Waldhausen, Algebraic 𝐾theory 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
(86m:18011), http://dx.doi.org/10.1007/BFb0074449
 [Whi62]
George
W. Whitehead, Generalized homology
theories, Trans. Amer. Math. Soc. 102 (1962), 227–283. MR 0137117
(25 #573), http://dx.doi.org/10.1090/S00029947196201371176
 [Ada74]
 J. F. Adams, Stable homotopy and generalised homology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.London, 1974, x+373 pp. MR 53:6534
 [Bök85]
 M. Bökstedt, Topological Hochschild homology, preprint, 1985.
 [Bor94]
 Francis Borceux, Handbook of categorical algebra. 2. Categories and structures, Encyclopedia of Mathematics and its Applications, vol. 51, Cambridge University Press, Cambridge, 1994. MR 96g:18001b
 [BF78]
 A. K. Bousfield and E. M. Friedlander, Homotopy theory of spaces, spectra, and bisimplicial sets, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II (M. G. Barratt and M. E. Mahowald, eds.), Lecture Notes in Math., vol. 658, Springer, Berlin, 1978, pp. 80130. MR 80e:55021
 [Cur71]
 Edward B. Curtis, Simplicial homotopy theory, Advances in Math. 6 (1971), 107209. MR 43:5529
 [DHK]
 W. G. Dwyer, P. S. Hirschhorn, and D. M. Kan, Model categories and general abstract homotopy theory, in preparation.
 [DS]
 W. G. Dwyer and Brooke E. Shipley, Hyper symmetric spectra, in preparation.
 [DS95]
 W. G. Dwyer and J. Spalinski, Homotopy theories and model categories, Handbook of algebraic topology (Amsterdam), NorthHolland, Amsterdam, 1995, pp. 73126. MR 96h:55014
 [EKMM97]
 A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules, and algebras in stable homotopy theory. With an appendix by M. Cole, Mathematical Surveys and Monographs, vol. 47, American Mathematical Society, Providence, RI, 1997, xii+249 pp. MR 97h:55006
 [GH97]
 Thomas Geisser and Lars Hesselholt, Topological cyclic homology of schemes, to appear in Ktheory, Seattle, 1996 volume of Proc. Symp. Pure Math.
 [Hir99]
 P. S. Hirschhorn, Localization of model categories, preprint, 1999.
 [Hov98a]
 Mark Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1998. MR 99h:55031
 [Hov98b]
 Mark Hovey, Stabilization of model categories, preprint, 1998.
 [HPS97]
 Mark Hovey, John H. Palmieri, and Neil P. Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997), no. 610. MR 98a:55017
 [Lew91]
 L. Gaunce Lewis, Jr., Is there a convenient category of spectra?, J. Pure Appl. Algebra 73 (1991), 233246. MR 92f:55008
 [Lim59]
 Elon L. Lima, The SpanierWhitehead duality in new homotopy categories, Summa Brasil. Math. 4 (1959), 91148. MR 22:7121
 [ML71]
 S. Mac Lane, Categories for the working mathematician, Graduate Texts in Mathematics, vol. 5, SpringerVerlag, 1971. MR 50:7275
 [MMSS98a]
 M. Mandell, J. P. May, B. Shipley, and S. Schwede, Diagram spaces, diagram spectra and FSPs, preprint, 1998.
 [MMSS98b]
 M. Mandell, J. P. May, B. Shipley, and S. Schwede, Model categories of diagram spectra, preprint, 1998.
 [May67]
 J. P. May, Simplicial objects in algebraic topology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill., 1967, viii+161pp. MR 36:5942
 [Qui67]
 Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, vol. 43, SpringerVerlag, 1967. MR 36:6480
 [Sch98]
 Stefan Schwede, Smodules and symmetric spectra, preprint, 1998.
 [SS]
 Stefan Schwede and Brooke Shipley, Classification of stable model categories, in preparation.
 [SS97]
 Stefan Schwede and Brooke Shipley, Algebras and modules in monoidal model categories, to appear in Proc. London Math. Soc.
 [Shi]
 B. Shipley, Symmetric spectra and topological Hochschild homology, to appear in Ktheory.
 [Voe97]
 V. Voevodsky, The Milnor conjecture, preprint, 1997.
 [Vog70]
 R. Vogt, Boardman's stable homotopy category, Aarhus Univ. Lect. Notes, vol. 21, Aarhus University, 1970. MR 43:1187
 [Wal85]
 F. Waldhausen, Algebraic Ktheory of spaces, Lecture Notes in Mathematics, vol. 1126, SpringerVerlag, 1985, pp. 318419. MR 86m:18011
 [Whi62]
 G. W. Whitehead, Generalized homology theories, Trans. Amer. Math. Soc. 102 (1962), 227283. MR 25:573
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Additional Information
Mark Hovey
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email:
hovey@member.ams.org
Brooke Shipley
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email:
bshipley@math.purdue.edu
Jeff Smith
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email:
jhs@math.purdue.edu
DOI:
http://dx.doi.org/10.1090/S0894034799003203
PII:
S 08940347(99)003203
Received by editor(s):
March 31, 1998
Received by editor(s) in revised form:
July 7, 1999
Published electronically:
September 22, 1999
Additional Notes:
The first two authors were partially supported by NSF Postdoctoral Fellowships
The third author was partially supported by an NSF Grant.
Article copyright:
© Copyright 1999 American Mathematical Society
