CATEGORY THEORY

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CATEGORY THEORY

• Lecture 4, 19 May 2003
• RSISE, ANU
• NICTA, UNSW

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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

• A generalization of the slice A/A.
• 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

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More on Comma Cats

• Identify A in Ob(A) with functor A 1 ? A
• Then slice A/A is the comma cat (IdA, A)
• Every comma category has 2 projections
• p1 (F, G) ? C projects to 1st coordinates
• p2 (F, G) ? D projects objects onto 3rd
  coordinates and arrows onto 2nd coordinates

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What is a Natural transformation?

• Let FC ? D and GC ? D be functors
• Think of each of F and G as projecting a picture
  of the cat C inside the cat D
• Natural transformations arise when we imagine
  sliding the picture defined by F onto the picture
  defined by G (Pierce,p.41)

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Natural transformations as maps

• Let µ be a map from Ob(C) to hom(D) such that,
  for each C, C and gC ? C,
• Gg o µC µC o Fg
• I. e., the diagram (on p. 16) commutes
• The arrows µC are the components of µ
• µ is a natural equivalence if each component of µ
  is an isomorphism in D

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Functor categories

• Let C and D be categories, C small
• Func(C, D) is then a category
• (also written DC or even C ? D ),
• whose objects are functors from C to D
• whose arrows are natural transformations
• Note natural transformations compose by setting
  (µo?)C µC o ?C
• Notation Nat(F,G) for hom functor in DC

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More NATURAL(?!) notation

• gt in diagrams for NatTrans (µ F gt G)
• Let µ F ? G be a natural transformation
• Two natural transformations induced
• Kµ KF ? KG
• The component KµC for C in Ob(C) is
• KµC KFC ? KGC
• Kµ is a natural transformation because the
  functor K takes CommDis to CommDis

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Natural notation continued

• µH FH ? GH
• The component µH for B in Ob(B) is the component
  of µ at HB. µH is a natural transformation since
  H defined on arrows
• and 2) define different notions
• Yet their formal properties are the same
• Thus advances mathematics