CATEGORY THEORY

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CATEGORY THEORY

• Lecture 2, 5 May 2003
• RSISE, ANU
• NICTA, UNSW

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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

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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!!!

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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)

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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!

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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!!!)

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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

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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.

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The fixed point finder Y
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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.

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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!)

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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)

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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.

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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.

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The diagram commutes

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

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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

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this diagram commutes
h
B
B
B
g
f
A
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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.

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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!

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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.,

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another diagram commutes
f
A
B
g
id
id
A
B
f
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Examples of isomorphisms

• In Grp a group isomorphism
• In Set a surjective injection
• In Top a homeomorphism

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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

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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

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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

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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