# 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:

* Categorical duality/Dual category: reversing arrows
* Duality principle
* Double duality
* Triality
* Self-duality: concept whose dual is itself
* Harmonic duality
* topological duality: Conjugation
* Adjointness
* Hermitian duality
* Duality functors, (for example the duality functor $Hom_{k}(--,k)$ )
* Poincar\'e duality/Poincar\'e isomorphism
* Poincar\'e-Lefschetz duality, and Alexander-Lefschetz duality
* Alexander duality: J. W. Alexander's duality theory (cca. 1915)
* Serre duality :
example- in the proof of the Riemann-Roch theorem for curves.
* Logic duality: Dualities in logic, example: De Morgan dual, Boolean algebra
* Stone duality: Boolean algebras and Stone spaces
* Dual numbers- as in an associative algebra; (almost synonymous with double)
* 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.
* Hamilton--Lagrange duality in theoretical mechanics and optics
* Dual space
* Dual space example
* Dual homomorphisms
* Duality of Projective Geometry
* Analytic dualities
* Duals of an algebra/algebraic duality,
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
* cohomology theory duals: de Rham cohomology $\leftarrow \rightarrow$ Alexander-Spanier cohomology
* Hodge dual
* Duality of locally compact groups
* 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
groups form a braided monoidal category, whereas $\Pi (G)$ is a symmetric monoidal category.
* Tannaka duality: an extension of Tannakian duality by
Alexander Grothendieck to algebraic groups and Tannakian categories.
* Contravariant dualities
* Weak duality, example : weak duality theorem in linear programming;
dual problems in optimization theory
* Dual codes
* Duality in Electrical Engineering

#### Examples of duals:

* a category ${\mathcal {C}}$ and its dual ${\mathcal {C}}^{op}$
* the category of Hopf algebras over a field $k$ is (equivalent to) the opposite category of affine group schemes over the spectrum.
* Dual Abelian variety
* Example of a dual space theorem
* Example of Pontryagin duality
* initial and final object
* kernel and cokernel
* limit and colimit
* direct sum and product