CDGO 2007: 2nd International Conference on Complementarity,

Duality and Global Optimization in Science and Engineering

February 28-March 2, 2007

Industrial and Systems Engineering Department

A Category-Theoretic Approach to Duality

Sabah E. Karam, Information SpecialistMorgan State UniversityPlanning & Information TechnologyBaltimore, MD 21251

tel: 443-885-4597 email: Sabah.Karam@ morgan.edu

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Historical notes

Categories were first introduced by S.Eilenberg and S. MacLane during the years 1942-1945, in connection with algebraic topology, a branch of mathematics in which tools from abstract algebra are used to study topological spaces.

Category theory has come to occupy a central position in pure mathematics and theoretical computer science.

Categories are algebraic structures with many complementary natures, e.g., geometric, logical, computational, combinatorial.

Category Theory is an alternative to classical set theory as a foundation for mathematics. The primitive, set-theoretic concept of "element" or "membership" is replaced by that of "function."

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Applications of CT

Mathematics and Computer Science Quantum Physics and Tensor CT Genomes and Computational Biology Information Systems (databases, OOT) Unified Modeling Language and Software Engineering Compiler Optimization Logic and Philosophy Natural Transformation Models in Molecular Biology Neural Network Analysis and Design

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Reasons to use CT

it is a unifying language for discussing different mathematical models and other logic-based structures,

it reveals common structures in seemingly unrelated systems and a framework for comparing them,

it reveals invertible structures, i.e. for every categorical construct there is a dual formed by reversing all the transformations,

it consolidates the description of similar operations such as 'products' found in set theory, group theory, linear algebra, and topology, and

it produces graphical models which are intuitive, formal, declarative, and subject to further analysis.

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A category consists of 3 entities: objects, morphisms, and compositions

a class of objects (A, B, C, …)

a class of morphisms between objects symbolized by ‘’ For each morphism one object, A, is the domain of f and another object, B, is the codomain, f: AB.

a binary operation called composition. For each pair of morphisms f: AB and g: BC, a composite morphism, g ○ f: AC is defined.

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Morphisms have two properties

Associativity: If f : A → B, g : B → C and h : C → D then h ○ (g ○ f ) = (h ○ g) ○ f , and

Identity : For every object A, there exists a morphism 1A : A → A called the identity morphism for A, such that for every morphism f : A → B, we have 1A ○ f = f = f ○ 1A.

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SOME MORE TERMINOLOGY

Every morphism has a source object, called the domain, and a target object, called the codomain. If f is a morphism with X as its source and Y as its target, we write f: X → Y.

We write Hom(X,Y) for the set of morphisms from X to

Y. In traditional set theory morphisms are nothing more than the set of functions from X to Y.

Hom( ) is short for Homology.

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What is a homology?

A correspondence or structural parallel.

In biology, two or more structures are said to be homologous if they are alike because of shared ancestry. This could be evolutionary ancestry, e.g. the wings of bats and the arms of humans, or developmental ancestry, e.g. the ovaries of female humans and the testicles of males.

Scientists use physical structures to reconstruct evolutionary history.

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What is a homology (cond’t)?

In mathematics, especially algebraic topology and abstract algebra, homology is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group).

In anthropology and archaeology, homology refers to a type of analogy whereby two human beliefs, practices or arte-facts are separated by time but share similarities due to genetic or historical connections.

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Elementary Example

Any partial ordering, sequencing, or arrangement of the elements of a set (a) Objects are the elements of the partial order; numbers, sets, points in a plane, integers, people in a genealogy relationship,

(b) Morphisms: , , divisibility relationship.(c) Composition works because of transitivity.

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Mathematical Categories

Set = sets with linear transfomations Vect = vector spaces with linear transfomations Poset = partially ordered sets with monotone functions Grp = groups with group homomorphisms Top = topological spaces with continuous functions Diff = smooth manifolds with smooth maps Ring = rings with ring homomorphisms Met = metric spaces with contraction maps

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Functors, Natural Transformations, and Adjoints

Saunders MacLane, one of the founders of category theory, remarked, "I didn't invent categories to study functors; I invented them to study natural transformations." Also called natural equivalence or isomorphism of functors.

The context of Mac Lane's remark was the axiomatic theory of homology. With the language of natural transformations he could easily express: (i) how homology groups are compatible with morphisms between objects and (ii) how two equivalent homology theories not only have the same homology groups but also the same morphisms between those groups.

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Definition of Functor

Let C and D be categories. A functor F from C to D is a mapping that

(a) associates with each object X ε C an object F(X) ε D, and(b) associates with each morphism f: XY a morphism F(f): F(X) F(Y) such that the following two properties hold:

(i) F(1A) = 1 F(A) for every object, and

(ii) F(g ○ f ) = F( g) ○ F (f )

That is to say, functors preserve identity morphisms and composition of morphisms.

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Example of a functor

Given a set S = {a, b, c, 1, 2, 3, @, # ,$)

Objects: List(S) = {a, b2, c$, 3#a3, ...}, L = [s1, s2, s3, s4, …]

Morphisms: f: S S’ (e.g. a sort routine)

Identity: we also need to define an associative binary concatenation operator,call it *, and an identity operator, call it [ ], such that [ ] * L = L = l * [ ].

Functor: F(f): List(S) List(S’)List(f)( L) = [ f(s1), f(s2), f(s3), f(s4), …]

Equivalent to the java class mapList. It can be used to create a dictionary by reading a collection of words and definitions.

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Object-Oriented (OO) Technology

Objects are the principle building blocks of object-oriented programs. Each object is a programming unit consisting of data (instance variables) and functionality (instance methods).

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Customer_OrderCustomerIDcustomerNamedateShippeddateReceiveddatePayed

checkInventory( )contactCustomer( )Ship( )refund( )calculateSale( )

Definition of Natural Transformation

Let X and Z be two categories and let F and G be two functors F: X Z, and G: X Z. Let f: A B

η is a NT from F to G, written η:F G, if the diagram commutes.

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ηB

F(f)

F(A)

F(B) F(B)

ηA

G(f)

F(A)

Examples of Natural Transformations (NT)

NT’s are structure preserving mappings from one functor to another functor.

Two decks of playing cards, all analog wrist watches, and all tie shoes with the same number of holes are isomorphic.

Consider f(x + y) = f(x) + f(y). Then f(x) = 4x is one such preserving map, since f(x + y) = 4(x + y) = 4x + 4y = f(x) + f(y).

Consider f(a + b) = f(a) * f(b), Then f(x) = ex satisfies this condition since 5 + 7 = 12 translates into e5 * e7 = e12.

In group theory, every group is naturally isomorphic to its opposite group in which the preserving map, F(a*b) = b*a, inverts the binary operation.

The dual of dual is a “natural transformation” in category theory

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Analogous electric and mechanical systems

Electrical and mechanical system have differential equations of the same form and can be considered isomorphic.

Electrical: e = iR e = voltage, i = current, R = resistance

L = inductance, C = capacitance, Q = charge LQ'' + RQ' + Q/C = 0 Mechanical: f = vB v = velocity, f = force, B = friction, M = mass

spring-mass differential eq. mx'' + bx' + kx = 0

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Functional programming language

Objects: Int, Real, Bool, Char, Ref Morphisms: isZero: Int Bool (test for

zero)not: Bool Bool (negation)succInt: Int Int (successor)toReal: Int Real (conversion)

Constants: zero (Int) , true/false (Bool) Composition: false = not ○ true

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Contravariant functors & Dual Spaces

There are many constructions in mathematics which would be functors but for the fact that they "turn morphisms around" and reverse the direction of

composition F(g ○ f ) = F( f ) ○ F (g ).

Dual vector spaces, maps which assign to every vector space its dual space reflect, in an abstract way, the relationship between row vectors and column vectors. The dual or transpose is a contravariant functor from the category of all vector spaces over a fixed field to itself.

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Primal/Dual & Minimax Theorem

A dual mathematical program has the property that its objective is always a bound on the original mathematical program, called the primal

Minimax theorem proven by von Neumann in 1928, it is a cornerstone of duality and of game theory

Let X and Y be mixed strategies for players A and B. Let A be the payoff matrix. Then

max min XTAY = min max XTAY X Y Y X

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Commutative diagrams and LP

min{max F(x,y): y in Y}: x in X}

= max{min F(x,y): x in X}: y in Y} F: X*Y R and X and Y are non-empty, convex, compact sets.

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min

min

maxmax

LOSS

GAINOptimal solution

Original problem

Primal/dual commutative diagram

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transposematrix A

new obj. function

The originalobjective function

Optimal solution

max cTxsubject to Ax ≤ b

min bTysubject to ATx ≥ b

Commutative diagrams as Proofs

Commutative diagrams play the role in category theory that equations play in algebra. Commutative diagrams can be used to assert the validity of program transformations.

Diagram chasing is a method of mathematical proof especially in homological algebra and computer science.

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succReal

toReal

Int Int

Real Real

succInt

toReal

Relationship between M-theory and Type IIA supergravity/string theory

In the strong coupling limit type IIA string theory approaches an 11 dimensional Lorentz invariant theory.

Commutative diagrams are used to assert the validity of relationships between theories.

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S1 compactification

low energy limit

M theory Type IIA string theory

D=11 supergravity

Type IIA supergravity

S1 compactification

low energy limit

Classification of Duals

Dorn's dual primal/dual are convex quadratic programs Fenchel's Conjugate Dual Generalized penalty-function/surrogate Dual Geometric dual Inference dual Lagrangian Dual LP Dual. This is the cornerstone of duality. In canonical

form: Primal: Min{cx: x >= 0, Ax >= b}. Dual: Max{yb: y >= 0, yA <= c}.

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Duals (cond’t)

Self Dual when a dual is equivalent to its primal - LP problems Semi-infinite Dual Superadditive Dual Surrogate Dual Symmetric Dual Wolfe's Dual Hooker: A relaxation dual in which there is a finite algorithm for

solving the relaxation is an inference dual. An inference dual in which the proofs are parameterized is a relaxation dual.

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Natural Transformation Models in Molecular Biology

Molecular models in terms of categories, functors and natural transformations are introduced for: (a)unimolecular chemical transformations, (b) multi-molecular chemical, and (c) biochemical transformations

Several applications of such natural transformations are then presented to analyze (a) protein biosynthesis, (b) embryogenesis and (c) nuclear transplant experiments

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Complex graph matching problems as combinatorial optimization

Phletora of confusing terms: computational complexity; neural networks; linear programming; weighted graph matching; quadratic optimization; simplex-based algorithm; Hungarian method; eigendecomposition; pattern recognition; symmetric polynomial transform; genetic algorithms; probabilistic relaxation; clustering techniques.

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Duality, polarity, complementarity

Electronics: two devices or two circuits having mathematical descriptions that are identical except that voltages in one formula correspond to currents in the other formula.

Chemistry: Conjugate Acid/Base pairs Electromagnetic theory: electric fields are dual to magnetic fields. Meterology: precipitation/evaporation Mathematics: projective geometry, category theory, Morgans laws

(logic), set theory, operations (+/-, x/÷, ∫/Dx). Biology: dualism is the theory that blood cells have two origins,

from the lymphatic system and from the bone marrow. Physics: particle/wave nature of light, electrical-mechanical

duality of the differential equations.

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Duality, polarity, complementarity

Genetics: DNA base-pairing (A-T, C-G) Philosophy: yin-yang basis of Chinese medicine Endocrinology: metabolic processes that assemble/ disassemble

(anabolic/catabolic) molecules in the body = a hormonal process Theology: Koranic verses describe created pairs Molecular biology: code-duality (analog/digital) Language: structure from which meaning is derived Psychology: referent and probe in judgment General theory of relativity: 4 elementary forces Quantum field theory: fermion-boson duality

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Establishing a curriculum based on a Duality Principle

Dr. Glenda Prime, Coordinator, Doctoral Programs in Mathematics and Science Education

Glenda.Prime@morgan.edu

The School of Education and Urban Studies, through the Doctoral Programs in Mathematics Education and Science Education seeks to enhance the quality of science and mathematics education by preparing a cadre of highly qualified mathematics and science educators, supervisors and curriculum specialists. 

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References

S. Eilenberg and S. MacLane, "Natural Isomorphisms in Group Theory," Proceedings of the National Academy of Sciences, 28, (1942), pp. 537-543

S. Mac Lane, “Categories for the Working Mathematician” 2nd edition, Springer (2000)

J.N. Hooker, Duality in Optimization and Constraint Satisfaction, Carnegie Mellon Univ., Pittsburgh, PA (2006) http://wpweb2.tepper.cmu.edu/jnh/duals.pdf

M. Barr and C. Wells, “Category Theory for Computing Science” 3rd edition, CRM (1999)

Proceedings SIAM & Society for Mathematical Biology Meeting N/A(3), pages pp. 230-232, Colorado, 1983.

http://glossary.computing.society.informs.org/second.php?page=duals.html

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End of Presentation

We would like to thankOrganizers: Panos M. Pardalos & Altannar Chinchuluun, Univ. of FloridaAdvisory Committee: David Y. Gao & Hanif D. Sherali, Virginia Tech Univ.

And to everyone who attended this session

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