Duality in Mathematics and Categorical Physics

Duality definitions in mathematics:

1. Categorical duality/Dual category: reversing arrows
2. Duality principle
3. Double duality
4. Triality
5. Self-duality: concept whose dual is itself
6. Harmonic duality
7. topological duality: Conjugation
8. Adjointness
9. Hermitian duality
10. Duality functors, (for example the duality functor )
11. Poincaré duality/Poincaré isomorphism
12. Poincaré-Lefschetz duality, and Alexander-Lefschetz duality
13. Alexander duality: J. W. Alexander's duality theory (cca. 1915)
14. Serre duality : example- in the proof of the Riemann-Roch theorem for curves.
15. Logic duality: Dualities in logic, example: De Morgan dual, Boolean algebra
16. Stone duality: Boolean algebras and Stone spaces
17. Dual numbers- as in an associative algebra; (almost synonymous with double)
18. Geometric dualities: dual polyhedron, dual of a planar graph, duality in order theory, the Legendre transformation -an application of the duality between points and lines; generalized Legendre, that is, the Legendre-Fenchel transformation.
19. Hamilton–Lagrange duality in theoretical mechanics and optics
20. Dual space
21. Dual space example
22. Dual homomorphisms
23. Duality of Projective Geometry
24. Analytic dualities
25. Duals of an algebra/algebraic duality, for example, dual pairs of Hopf *-algebras and duality of cross products of C*-algebras
26. Tangled, or Mirror, duality: interchanging morphisms and objects
27. Duality as a homological mirror symmetry
28. cohomology theory duals: de Rham cohomology Alexander-Spanier cohomology
29. Hodge dual
30. Duality of locally compact groups
31. Pontryagin duality, for locally compact commutative topological groups and their linear representations
32. Tannaka-Krein duality: for compact matrix pseudogroups and non-commutative topological groups; its generalization leads to quantum groups in quantum theories; Tannaka's theorem provides the means to reconstruct a compact group from its category of representations ; Krein's theorem shows which categories arise as a dual object to a compact group; the finite-dimensional representations of Drinfel'd 's quantum groups form a braided monoidal category, whereas is a symmetric monoidal category.
33. Tannaka duality: an extension of Tannakian duality by Alexander Grothendieck to algebraic groups and Tannakian categories.
34. Contravariant dualities
35. Weak duality, example : weak duality theorem in linear programming; dual problems in optimization theory
36. Dual codes
37. Duality in Electrical Engineering

Examples of duals:

1. a category and its dual
2. the category of Hopf algebras over a field is (equivalent to) the opposite category of affine group schemes over the spectrum.
3. Dual Abelian variety
4. Example of a dual space theorem
5. Example of Pontryagin duality
6. initial and final object
7. kernel and cokernel
8. limit and colimit
9. direct sum and product

Bibliography

1

S. Doplicher and J. Roberts. A new duality theory for compact groups. Inventiones Mathematicae, 98:157–218, 1989.

2

André Joyal and Ross Street, An introduction to Tannaka duality and quantum groups, in Part II of Category Theory, Proceedings, Como 1990, eds. A. Carboni, M. C. Pedicchio and G. Rosolini, Lectures Notes in Mathematics No.1488, Springer, Berlin, 1991, 411-492.