PlanetPhysics/Duality and Triality

Duality in Mathematics and Categorical Physics
The following is a contributed mathematical, physical mathematics and engineering topic entry (rather than a philosophical one), concerning different types of duality encountered in different areas of mathematics and categorical/algebraic theoretical physics; accordingly there is a string of distinct definitions associated with this topic rather than a single, general definition, although some of the linked definitions, that is, categorical duality, are more general than others.

Duality definitions in mathematics:
example- in the proof of the Riemann-Roch theorem for curves. the Legendre transformation -an application of the duality between points and lines; generalized Legendre, that is, the Legendre-Fenchel transformation. for example, dual pairs of Hopf *-algebras and duality of cross products of C*-algebras #Tangled, or Mirror, duality: interchanging morphisms and objects #Duality as a homological mirror symmetry groups form a braided monoidal category, whereas $$\Pi(G)$$ is a symmetric monoidal category. Alexander Grothendieck to algebraic groups and Tannakian categories. dual problems in optimization theory
 * 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 $$Hom_k(--,k)$$ )
 * 11) Poincar\'e duality/Poincar\'e isomorphism
 * 12) Poincar\'e-Lefschetz duality, and Alexander-Lefschetz duality
 * 13) Alexander duality: J. W. Alexander's duality theory (cca. 1915)
 * 14) Serre duality :
 * 1) Logic duality: Dualities in logic, example: De Morgan dual, Boolean algebra
 * 2) Stone duality: Boolean algebras and Stone spaces
 * 3) Dual numbers- as in an associative algebra; (almost synonymous with double)
 * 4) Geometric dualities: dual polyhedron, dual of a planar graph, duality in order theory,
 * 1) Hamilton--Lagrange duality in theoretical mechanics and optics
 * 2) Dual space
 * 3) Dual space example
 * 4) Dual homomorphisms
 * 5) Duality of Projective Geometry
 * 6) Analytic dualities
 * 7) Duals of an algebra/algebraic duality,
 * 1) cohomology theory duals: de Rham cohomology $$\leftarrow \rightarrow$$ Alexander-Spanier cohomology
 * 2) Hodge dual
 * 3) Duality of locally compact groups
 * 4) Pontryagin duality, for locally compact commutative topological groups and their linear representations #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 $$G$$ from its category of representations $$\Pi(G)$$; Krein's theorem shows which categories arise as a dual object to a compact group; the finite-dimensional representations of Drinfel'd 's quantum
 * 1) Tannaka duality: an extension of Tannakian duality by
 * 1) Contravariant dualities
 * 2) Weak duality, example : weak duality theorem in linear programming;
 * 1) Dual codes
 * 2) Duality in Electrical Engineering

Examples of duals:

 * 1) a category $$\mathcal{C}$$ and its dual $$\mathcal{C}^{op}$$
 * 2) the category of Hopf algebras over a field $$k$$ 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