nLab
Spanier-Whitehead duality

Contents

Context

Stable Homotopy theory

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Duality

Contents

Idea

The stable homotopy category Ho(Spec)Ho(Spec) is a symmetric monoidal category via the symmetric smash product of spectra. Monoidal duality in Ho(Spec)Ho(Spec) is called Spanier-Whitehead duality or S-duality .

Examples

The explicit interpretation in terms of monoidal duality is (DoldPuppe, theorem 3.1).

Using this one shows that the trace on the identity on Σ + X\Sigma^\infty_+ X – its categorical dimension – is the Euler characteristic of XX.

References

The original references are

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.

Lecture notes include

Further discussion of Atiyah duals is in

For equivariant stable homotopy theory Spanier-Whitehead duality is discussed on pages 23 onwards of

The Spanier-Whitehead duality has been studied in the new category of cospectra with ideas close to the later ones of shape theory in

  • Elon L. Lima, The Spanier-Whitehead duality in new homotopy categories, Summa Brasil. Math. 4 1959 91–148 (1959) MR0116332

See also

  • 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)

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