CATEGORY THEORY

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CATEGORY THEORY

• Lecture 3, 12 May 2003
• RSISE, ANU
• NICTA, UNSW

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Only TRUTHS today

• If this sentence is true then
• EVERYTHING
• in this lecture is
• TRUE

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Currys analysis of RP

• Let F be this property of properties
• 1. F(x) x(x)
• Then, substituting F for x 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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Nothing special about

• Let now g be any function. Consider
• 1. G(x) g(x(x))
• Then, substituting G for x in 1, we have
• 2. G(G) g(G(G))
• I. e., the reasoning producing RP yields
• a fixed point
• for any unary operation g

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Hey, wait a minute!

• Some functions naturally have fixed points
• E.g., 0 (and 1) for ?z.z?z Z ? Z
• But others resist fixed points
• E.g., ?z.z1 Z ? Z
• And (we devoutly hope)
• ?z.z Proposition ? Proposition
• Yes, exactly! So functions are typed

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The fixed point finder Y

• Recapitulating, let us define
• Df1. G ?z.g(zz)
• Then let us consider
• Df2. Y ?g.GG
• Observe that all variables in Y are bound
• 3. Yf (?z.f(zz))(?z.f(zz))
• f((?z.f(zz))(?z.f(zz))) f(Yf)
• i.e., Y has found a fixed point for f

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Whats a function in Set?

• How parse f A?B in Set?
• The Cartesian product approach
• f is triple (A, G, B), where
• G ? A?B, and
• ?a(a ? A ??!b(b ? B (a,b) ? G) )
• That is, a mapping from A to B in Set is a graph
  which picks out a unique b in B for every a in A

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Whats a function in Set?

• The disjoint union approach
• Copy A and B in different colours to get AB (we
  paint a from A red, b from B green)
• f is triple (A, G, B), where
• G is quotient of AB on equivalence ?
• Idea a ? b iff f(a) b
• E.g, 2 ? -2 ? 4 if f is the square function
• ?b(b ? B ??!c(b ? c c is ? class ) )

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Say this in CAT

• Barr Wells, p. 8
• G is the graph of the function f
• G is the cograph of f
• G and G are (mutually) dual

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Functors

• F C ? D is a functor iff
• if f A?B in C then
• Ff FA ? FB in D
• F(idA) idFA
• if g B?C in C then F(g o f) Fg o Ff

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Varieties of functors

• F Cop ? D is contravariant from C to D
• (Then Fop C ? Dop )
• F C ? D is covariant from C to D
• F C ? D is faithful iff F is injective when
  restricted to each homset.
• F is full iff F is surjective on each homset
• I.e., ?A?B(A,B ? Ob(C) ? every arrow in
  Hom(FA,FB) is Ff for some f in Hom(A,B) )

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Preserving and Reflecting

• F preserves a property P iff Ff also has P
  whenever f has P
• F reflects P iff whenever Ff has P so also f has P

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Examples of functors

• Identity idC C ? C for every category C
• Forgetful functors U C ? Set
• Free functors F Set ? C
• Powerset functors P Set ? Set
• Contravariant Let f A ? B, B0 ? B
• Pf(B0) x ? A f(x) ? B0
• Covariant Exercise 6 (at least 2 ways)

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Hom functors (I)

• HomC Hom is a functor in each variable
• Hom(A,f) Hom(A,B) ? Hom(A,C) is defined, for
  fixed A and each f B?C, by
• Hom(A,f)(g) g o f, for g in Hom(A,B)
• Hom(h,B) is defined, for fixed B and each f B?C,
  by Hom(h,B)(f) f o h, for h in Hom(A,B)
• Hom(A,-) is the covariant hom functor

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Hom functors (II)

• Hom(A, -) C ? Set is a functor
• Hom (-, B) Cop ? Set is the contravariant hom
  functor
• Hom(-, -) Cop ? C ? Set

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Isomorphism Equivalence

• Cat is a category
• C and D are isomorphic iff there is a functor F
  C ? D which has an inverse G D ? C
• F C ? D is an equivalence iff
• F is full and faithful and
• For any B in Ob(D) there is an A in Ob(C) such
  that F(A) is isomorphic to B

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Comma categories

• Let F C?A, G C?B be functors
• The comma category (F,G) has
• Objects (C, f, D) with f FC ? GD an arrow of A,
  C in Ob(C), D in Ob(D)
• Arrows (h,k) (C,f,D) ? (C,f,D) such that h
  C?C, k D?D, (Gk)of fo(Fh)
• That is, the diagram commutes