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 Specialist
Morgan State University
Planning & Information Technology
Baltimore, MD 21251
tel
:
443

885

4597
email
:
Sabah.Karam
@ morgan.edu
CDGO
2007
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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2007
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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2007
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
‘
¶)RUHDFKPRUSKLVPRQHREMHFW$LVWKH
domain
of f and another object, B, is the codomain
,
f: A
B.
a binary operation called
composition
. For each pair
of morphisms f: A
B and g: B
C, a composite
morphism, g
○
f: A
C is defined.
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2007
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 1
A
:
A
→
A
called the identity
morphism for A, such that for every morphism
f
:
A
→
B
, we have 1
A
○
f
=
f
=
f
○
1
A
.
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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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2007
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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2007
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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2007
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: X
Y a
morphism
F
(
f):
F
(X)
F
(Y)
such that the following two properties hold:
(i)
F
(
1
A
) =
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 = [s
1
, s
2
, s
3
, s
4
, …]
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(s
1
), f(s
2
), f(s
3
), f(s
4
), …]
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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2007
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_Order
CustomerID
customerName
dateShipped
dateReceived
datePayed
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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2007
η
B
F(f)
F(A)
F(B)
F(B)
η
A
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
) = 4
x
is one such
preserving map, since
f
(
x
+
y
) = 4(
x
+
y
) = 4
x
+ 4
y
=
f
(
x
) +
f
(
y
).
Consider f
(
a
+
b
) =
f
(
a
) *
f
(
b
), Then
f
(
x
) = e
x
satisfies this
condition
since 5 + 7 = 12 translates into e
5
* e
7
= e
12
.
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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2007
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)
succ
Int
: Int
Int
(successor)
toReal: Int
Real
(conversion)
Constants
: zero (Int) , true/false (Bool)
Composition
: false = not
○
true
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2007
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
X
T
A
Y
= min max
X
T
A
Y
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
删慮搠R湤 夠慲攠湯n

敭灴pⰠ捯湶數Ⱐ捯浰c捴c獥瑳t
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min
min
max
max
LOSS
GAIN
Optimal
solution
Original
problem
Primal/dual commutative diagram
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transpose
matrix A
new obj.
function
The original
objective function
Optimal
solution
max c
T
x
subject to Ax
≤ b
min b
T
y
subject to A
T
x
≥ 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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2007
succ
Real
toReal
Int
Int
Real
Real
succ
Int
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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2007
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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2007
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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2007
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/
÷
,
∫
/D
x
)
.
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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2007
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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2007
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” 2
nd
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” 3
rd
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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2007
End of Presentation
We would like to thank
Organizers:
Panos M. Pardalos & Altannar Chinchuluun
,
Univ. of Florida
Advisory Committee:
David Y. Gao & Hanif D. Sherali
,
Virginia Tech Univ.
And to everyone who attended this session
CDGO
2007
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