With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
Examples
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
In QFT and String theory
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 $X$.
The original references are
Edwin Spanier, ; Henry Whitehead, (1953), A first approximation to homotopy theory, Proc. Nat. Acad. Sci. U.S.A. 39: 655–660, MR0056290
Edwin Spanier, ; Henry Whitehead, (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
An English version of that paper is:
Atiyah duality is due to
Lecture notes include
Frank Adams, part III, section 5 of Stable homotopy and generalised homology, 1974
Michael Hopkins (notes by Akhil Mathew), Lecture 22 in: Spectra and stable homotopy theory, 2012 (pdf, pdf)
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
See also
Last revised on April 4, 2021 at 09:40:45. See the history of this page for a list of all contributions to it.