Stabilizing tensor products

Author:
Harold M. Hastings

Journal:
Proc. Amer. Math. Soc. **49** (1975), 1-7

MSC:
Primary 55E10; Secondary 18D10

DOI:
https://doi.org/10.1090/S0002-9939-1975-0362298-3

MathSciNet review:
0362298

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Abstract: Let be a symmetric monoidal category with a suspension, and let *SC* be the resulting stable category. We shall give necessary and sufficient conditions for extending the symmetric monoidal structure to a monoidal structure on *SC*. These imply that the usual smash product on finite pointed CW complexes cannot be extended to a smash product (with as unit) on finite spectra, hence not on Boardman spectra. This confirms a conjecture of Alex Heller.

**[1]**J. F. Adams,*Stable homotopy and generalized homology*, notes by R. Ming, Univ. of Chicago Math. Lecture Notes, 1971 (mimeographed).**[2]**J. M. Boardman,*Stable homotopy theory*, University of Warwick, 1965 onwards (mimeographed).**[3]**J. M. Boardman and R. M. Vogt,*Homotopy-everything -spaces*, Bull. Amer. Math. Soc.**74**(1968), 1117-1122. MR**38**#5215. MR**0236922 (38:5215)****[4]**S. Eilenberg and G. M. Kelly,*Closed categories*, Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), Springer, New York, 1966, pp. 421-562. MR**37**#1432. MR**0225841 (37:1432)****[5]**H. M. Hastings,*A smash product for spectra*, Bull. Amer. Math. Soc.**79**(1973), 946-952. MR**0343257 (49:8001)****[6]**A. Heller,*Stable homotopy categories*, Bull. Amer. Math. Soc.**74**(1968), 28-63. MR**36**#7137. MR**0224090 (36:7137)****[7]**J. P. May,*The stable homotopy category*(to appear).**[8]**D. Puppe,*On the stable homotopy category*, Math. Institut, Heidelberg, 1972 (mimeographed). MR**0383401 (52:4282)****[9]**E. H. Spanier and J. H. C. Whitehead,*Duality in homotopy theory*, Mathematika**2**(1955), 56-80. MR**17**, 653. MR**0074823 (17:653b)****[10]**R. Vogt,*Boardman's stable homotopy category*, Aarhus Universitet Lecture Notes, no. 21, 1971 (mimeographed).

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1975-0362298-3

Keywords:
Smash product,
stable smash product,
Spanier-Whitehead category,
stable category,
Boardman spectra,
monoidal category,
symmetric monoidal category,
category with suspension

Article copyright:
© Copyright 1975
American Mathematical Society