Stable Homotopy theory
With duals for objects
With duals for morphisms
Special sorts of products
In higher category theory
The stable homotopy category is a symmetric monoidal category via the symmetric smash product of spectra. Monoidal duality in is called Spanier-Whitehead duality or S-duality .
The explicit interpretation in terms of monoidal duality is (DoldPuppe, theorem 3.1).
Using this one shows that the trace on the identity on – its categorical dimension – is the Euler characteristic of .
The original references are
Spanier, E. H.; Whitehead, J. H. C. (1953), A first approximation to homotopy theory , Proc. Nat. Acad. Sci. U.S.A. 39: 655–660, MR0056290
Spanier, E. H.; Whitehead, J. H. C. (1955), Duality in homotopy theory , Mathematika 2: 56–80, MR0074823
The interpretation of the duality as ordinary monoidal duality in the stable homotopy category is apparently due to
- Albrecht Dold, Dieter Puppe, Duality, trace and transfer , Proceedings of the Steklov Institute of Mathematics, (1984), issue 4
Atiyah duality is due to
- Michael Atiyah, Thom complexes , Proc. London Math. Soc. (3) , no. 11 (1961), 291–310.
Further discussion of Atiyah duals is in
For equivariant stable homotopy theory Spanier-Whitehead duality is discussed on pages 23 onwards of
- Roy Joshua, Spanier-Whitehead duality in etale homotopy Transactions of the American Mathematical Society Vol. 296, No. 1 (Jul., 1986), pp. 151-166 (article consists of 16 pages) (jstor)
Revised on April 3, 2013 21:37:08
by Urs Schreiber