category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
The stable homotopy category $Ho(Spec)$ is a symmetric monoidal category via the symmetric smash product of spectra. Monoidal duality in $Ho(Spec)$ 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 $\Sigma^\infty_+ X$ – its categorical dimension – is the Euler characteristic of $X$.
Brown-Comentz duality?
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
Atiyah duality is due to
Further discussion of Atiyah duals is in
For equivariant stable homotopy theory Spanier-Whitehead duality is discussed on pages 23 onwards of
See also