A category (范畴) \( \mathcal{C} = (\text{Ob}\mathcal{C}, \text{Mor}\mathcal{C}, \circ) \) consists of a class of objects (物件), a class of morphisms (态射) from one object to another, and a composition (复合) operator on compatible pairs of morphisms, which satisfies:

  1. Identity morphisms: \( \forall A \in \text{Ob}\mathcal{C}, \exists 1_A \in \text{Mor}\mathcal{C}, 1_A: A \to A \), so that \( \forall \alpha: X \to A, \beta: A \to Y\), \(1_A \circ \alpha = \alpha, \beta \circ 1_b = \beta \);
  2. Closed under composition: \( \forall \alpha, \beta \in \text{Mor}\mathcal{C}, \alpha: A \to B, \beta: B \to C \), \( \exists \beta \circ \alpha \in \text{Mor}\mathcal{C}, \beta \circ \alpha: A \to C \);
  3. Associative composition: Given \( \alpha: A \to B, \beta: B \to C, \gamma: C \to D \), \( (\gamma \circ \beta) \circ \alpha = \gamma \circ (\beta \circ \alpha) \);

A category \( \mathcal{C} \) is said to be small if \( \text{Ob}\mathcal{C} \) and \( \text{Mor}\mathcal{C} \) are sets. Given two objects \( A,B \in \text{Ob}\mathcal{C} \), their set of morphisms \( H(A,B) \) in category \( \mathcal{C} \) is defined such that \( \alpha \in H(A,B) \iff \alpha: A \to B, \alpha \in \text{Mor}\mathcal{C} \). A morphism \( f: A \to B \) is an isomorphism (同构) if it has an inverse: \( \exists f^{-1}: B \to A \), \( f \circ f^{-1} = 1_B, f^{-1} \circ f = 1_A \). Two objects are isomorphic if there is an isomorphism between them.

In most concrete categories over sets, an object is some mathematical structure; a morphism is a map between two objects; and the composition is just function composition.

Duality is a very pervasive and important concept in (modern) mathematics. See Encyclopedia of Math article for a list

Given two (interchangeable) sets of mathematical objects (concepts, operations, propositions, etc.), duality means they are isomorphic(?). the mathematical structure of the objects are preserved all propositions that hold in one formulation also hold in the dual formulation

A dual operation \( * \) is often (but not always) an involution (对合): \( x^{ ** } = x \). An element is self-dual to a dual operation if it is a fixed point of the dual operation: \( x^* = x \).

Dual object itself does not carry any extra value than its primal object, but it may be much easier to understand, or make certain otherwise unthinkable calculations possible, which is the major motivation of studying dual objects.

Mathematical Logic

Logic:

  • operation: negation
  • pairs: <proposition, negation>; <conjunction, disjunction>; <material implication, converse implication>

Bi-implication is self-dual.

Set theory:

  • operation: complement
  • pairs: <set, complement set>; <union, intersection>; <contained in, contains>

Geometry

In projective geometry, line and point are dual concepts. In three-dimensional projective geometry, point and plane are dual, while line is self-dual. In projective geometry, once you have proved a theorem you get a dual theorem for free, unless the theorem is self-dual or the dual is trivial. A pair of dual theorems is "Two points determine a line" and "Two lines determine a point".

Differential geometry: k-form and k-dimensional surface.

Algebra

Linear algebra, only considering finite dimensional vector spaces:

  • Operation: bilinear map from pairs of vector spaces to scalars
    • inner product: \( \langle , \rangle: V \times V \to \mathbb{R} \)
  • Pairs: <vector, linear functional>, <vector space, dual vector space>, <linear map, adjoint linear map>, <surjection, injection>, <closed convex set, polar>

A function from object A to object B very often gives rise to a function from the dual of B to the dual of A. One kind of problem (existence; surjection) is converted into a different kind (uniqueness; injection) in the dual formulation.

Abstract Algebra: Pontryagin duality, Abelian groups;

Topology

In topology, closed and open are dual concepts.

Banach space:

  • operation: continuous bilinear map <,): : X×Y→R
  • pairs: <element, continuous linear functional>, <Banach space, dual Banach space>

Whenever we have two mathematical objects A and B, a set S of "scalars" of some kind, and a function β: A×B → S that is a structure-preserving map in each variable separately, we can think of the elements of A as elements of the dual of B, and vice versa. Functions like β are called pairings. {III.19 Gowers}

Structure: linearity, topology, etc.

Manifold: Poincare duality

  • operation: intersection number \(H_i(X) \times H_i(X) \rightarrow Z\)
  • pairs: <homology, cohomology>

Analysis

The importance of the duality concept in functional analysis relies chiefly in the possibility of relating properties of representations in one space to those in its dual, a space that can be shown to possess certain analytical regularity properties, such as closure, even if its domain space does not. {Red-horse, R.G., 2009}

Functional analysis duality pairing: \( \langle , \rangle \). Some properties of a function are naturally expressed in the dual formulation.

  • Algebraic dual of linear vector space: the set of all linear functionals defined on the space
  • Topological dual of linear topological vector space: the set of all continuous linear functionals defined on this space.
    • Also called phase space.
    • a topology is naturally induced in the dual space
    • weak topology in the original space: topology induced by continuous linear functionals
    • the dual of a normed space is a Banach space
    • the dual of a Hilbert space is itself a Hilbert space
    • weak integral (Pettis integral)
    • strong integral
  • Distribution and test function
  • Dual spaces in Fourier transform: time domain and frequency domain

It is not always easy to translate a statement about a function into an equivalent statement about its Fourier transform.

Optimization

In mathematical optimization theory, duality means that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. The solution to the dual problem provides a lower bound to the solution of the primal (minimization) problem. The difference is called the duality gap.

von Neumann conjectured the duality theorem for linear optimization, realizing that two person zero sum matrix game (where minimax = maximin) was equivalent to linear programming.

Graph theory

edge and vertex (undirected graph and its line graph)

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