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CATEGORY THEORY

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Curry's Paradox (CP) C. If this sentence is true then everything in ... Curry's (& Feys's) conclusions. We cannot 'explain' a paradox by running away from it ... – PowerPoint PPT presentation

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Title: CATEGORY THEORY


1
CATEGORY THEORY
  • Lecture 2, 5 May 2003
  • RSISE, ANU
  • NICTA, UNSW

2
A mathematical CRISIS!
  • Foundations of maths in 19th century
  • 1. The problem of infinite collections
  • 2. Georg Cantor invents set theory
  • 3. Naïve comprehension
  • To every property P there corresponds
  • x P x,
  • the set of items that have P

3
Russells Paradox (RP)
  • 1. Define R x not ? x x
  • Query ? R R?
  • 2. ? R R implies not ? R R (by 1, def R)
  • 3. So not ? R R (2, reductio ad absurdum)
  • 4. But then ? R R (3, 1, def R)
  • A CONTRADICTION!!!

4
Currys Paradox (CP)
  • C. If this sentence is true then everything in
    this lecture is true. In symbols,
  • 0. C If C is true then E is true
  • 1. C is true (hyp for conditional proof)
  • 2. C (1, Tarski convention T)
  • 3. If C is true then E is true (2, 0)
  • 4. E is true (3, 1, modus ponens)

5
CP (continued)
  • 5. If C is true then E is true (1-4, cond. pr.)
  • 6. C (5, definition 0 of C)
  • 7. C is true (6, Tarski convention T again)
  • 8. E is true (5, 7, modus ponens)
  • Caution Use of this argument, though clearly
    valid, is only moral when (e. g., in your thesis)
    everything you say is true!

6
REMEDIES FOR RP
  • 1. Theories of types (Whitehead-Russell)
  • 2. Limitation of size (Zermelo)
  • 3. Many-valued logic (Lukasiewicz)
  • 4. Inconsistent set theory (Brady)
  • 5. Lambda Calculus (LC) (Church)
  • 6. Combinatory Logic (CL) (Curry)
  • 7. Category theory? (You!!!)

7
Currys analysis of RP
  • Let F be this property of properties
  • 1. F(f) f(f)
  • Then, substituting F for f in 1, we have
  • 2. F(F) F(F)
  • From 4 motivating demands by Curry,
  • 3. The F defined by 1 is significant, and
  • 4. 2 is intuitively true

8
Currys ( Feyss) conclusions
  • We cannot explain a paradox by running away
    from it
  • We must stand and look it in the eye
  • We can subject paradoxes to analysis
  • F(F) does not belong to the propositions
  • H. B. Curry R. Feys, Combinatory logic vol. I
    (1958), N. Holland, pp. 4-5.

9
The fixed point finder Y
10
Sweating the small stuff
  • Limitation of size set theory
  • Let C be a category
  • The arrows (and/or objects) of C may be too many
    to form a set
  • But there are at least as many arrows as obs
    (each A comes with an identity idA).
  • If the arrows are a set, the cat C is small.

11
More small stuff
  • Notation C for a category (also D, etc.)
  • A, B for objects of C
  • Ob(C) for the collection of objects of C
  • Ar(C) for the collection of all arrows of C
  • HomC(A,B) for the arrows from A to B
  • C is locally small iff HomC(A,B) is a set, for
    all A, B in Ob(C). (usually assumed!)

12
Subcategories
  • D is a subcategory of C iff
  • 1. Ob(D) is a subset of Ob(C)
  • 2. Ar(D) is a subset of Ar(C)
  • 3. f in Ar(D) implies dom f, cod f in Ob(D)
  • 4. A in Ob(D) implies idA in Ar(D)
  • 5. f, g composable in Ar(D) implies that
  • g o f is in Ar(D)

13
Full subcategories
  • Let D be a subcategory of C
  • D is a full subcategory of C iff,
  • 1. for all A, B in Ob(C), if f is in HomC(A,B)
    then f is in Ar(D).
  • In a nutshell, full subcategories D have
    morphisms f between objects A, B of C whenever
    these objects A and B are themselves in D.

14
Products of categories
  • Let C, D be categories.
  • The product C?D is formed thus
  • Ob(C?D) consists of pairs (A, B), with A in Ob(C)
    and B in Ob(D).
  • Members of Ar(C?D) are the
  • (f,g) (A,B) ? (A,B)
  • with f A ? A and B B ? B.

15
The diagram commutes
  • A diagram commutes if any two paths between the
    same node compose to give the same morphism.

16
Slice categories
  • Let C be a cat, A an ob of C
  • Then the slice category C/A of objects of C over
    A is as follows
  • 1. Ob(C/A) consists of all arrows of C with
    target A.
  • 2. Ar(C/A) consists of all arrows h B ? B from
    f B ? A to g B ? A such that

17
this diagram commutes
h
B
B
B
g
f
A
18
How to think of C/A
  • Write h f ? g for h in Ar(C/A)
  • Think of Set/A as an A-indexed family of disjoint
    sets (the inverse images of the elements of A).
  • The commutativity of the diagram means that the
    function h is consistent with the decomposition
    of B and B into disjoint sets.

19
Element-free definitions
  • Cats make possible an element-free definition of
    mathematical properties via
  • (a) commutative diagrams
  • (b) limits
  • (c) adjoints
  • Well get to all those!

20
Isomorphism
  • Let f A ? B be in Ar(C)
  • f is an isomorphism iff there exists an inverse
    g B ? A of f, such that
  • 1. f o g idB
  • 2. g o f idA
  • I. e.,

21
another diagram commutes
f
A
B
g
id
id
A
B
f
22
Examples of isomorphisms
  • In Grp a group isomorphism
  • In Set a surjective injection
  • In Top a homeomorphism

23
Terminal objects
  • Let C be a category, T in Ob(C)
  • T is a terminal object in C iff
  • for every A in Ob(C),
  • there is exactly one f A ? T
  • Examples
  • One element sets in Set, Top, Grp

24
Duality
  • Let C be a category
  • Cop is the opposite category such that
  • 1. Ob(Cop) Ob(C)
  • 2. f B ? A in Ar(Cop) iff f A ? B in Ar(C)
  • In a nutshell, reverse all arrows

25
Initial objects
  • Let P be a property of objects or arrows
  • The dual of P is having P in Cop
  • Let C be a category, I in Ob(C)
  • I is an initial object in C iff
  • for every A in Ob(C),
  • there is exactly one f I ? A
  • Examples In Set and Top, the null set
  • In Grp, the one element group

26
Objects under A
  • This is the dual of the slice construction that
    produces C over A. That is,
  • 1. Ob(C under A) consists of all arrows of C with
    source A
  • 2. Ar(C under A) consists of all arrows
  • h B ? B from f A ? B to g A ? B such that
    h o f g
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