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# Math::Category

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Perlを使って圏論概念をシミュレーション。

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### Math::Category

1. 1. Math::Category id:hiratara 2009-11-21
2. 2. Math::Category (1) ✤ Perl ✤ ✤ ✤
3. 3. Math::Category (2) ✤ ✤ dom, cod, comp ✤ ✤ Hom ✤
4. 4. (1) ✤ ✤ ✤ Morphism interface ✤ source: dom Morphism ✤ target: cod Morphism ✤ composition:
5. 5. (2) ✤ ✤ target source ✤ composition ✤ ✤ ✤
6. 6. (1): SimpleMorphism ✤ 2 1 ✤ simple_morph ‘1’ => ‘2’; ‘1’ ✤ source simple_morph ‘1’ => ‘1’ ‘2’ ✤ target simple_morph ‘2’ => ‘2’ ‘3’ ✤ (simple_morph ‘2’ => ‘3’) . (simple_morph ‘1’ => ‘2’) = (simple_morph ‘1’ => ‘3’)
7. 7. (2): SubroutineMorphism(1) ✤ Perl ✤ Perl ✤ print ✤ sub_morph { \$_[0] * 2 } undef ✤ source target sub_morph { @_ } ( ) ✤ ( )
8. 8. (2): SubroutineMorphism(2) ✤ \$sub2 . \$sub1 \$sub1 \$sub2 ✤ ✤ ✤
9. 9. (1) ✤ f.(g.h) (f . g) . h f g h ✤ ✤ id ( ) ✤ ✤ f×g: x → ( f(x), g(x) )
10. 10. (2) A×B πA πB C ( f, g ): f, g 2 f g ( f, g ) A πA A×B πB B (f,g) f g (f,g) (g,f)
11. 11. (3): ✤ : bi_morph \$morph1, \$morph2; (A1, A2) ✤ source target (f, g) (B1, B2) ✤ : op \$morph; B ✤ source targe f ✤ op op \$morph; \$morph A
12. 12. (1) ✤ Functor ✤ ✤ Morphism Morphism ✤ ( )
13. 13. (2) ✤ functor { ... }; ✤ (Morphism OK) ✤ ✤ ( \$functor2 . \$functor1 )
14. 14. (1): \$BI_FUNCTOR ✤ Hom(-, -) ✤ C^op × C Sets ✤ C^op × C ✤ Sets SubroutineMorphism ( Sets )
15. 15. (1) ✤ NaturalTransformation ✤ ✤ (Morphism)
16. 16. (2) ✤ nat { }; ✤ ✤ ✤ ( ) ✤
17. 17. (3) ✤ ✤ \$nat2 . \$nat1 ✤ \$funct . \$nat ✤ \$nat . \$funct
18. 18. (4): FunctorMorphism ✤ ✤ functor_morph nat { F my \$id = shift; ... ... return \$sum_morph τ }; ✤ source target source G target
19. 19. (2): \$YONEDA_EMBEDDING ✤ ✤ C^op Sets^C ✤ Hom(g, -)
20. 20. : CPS (1) ✤ uc CPS
21. 21. : CPS (2) Sets Hom( , -) Hom( , undef) ∈ print uc \$fun_morph \$cps_uc \$cps_uc->(print) ∈ Hom( , -) Hom( , undef)
22. 22. Monad (1) ✤ functor eta mu ✤ eta: I → T, mu: TT → T ✤ ✤ Haskell Monad ✤
23. 23. Monad (2) ✤ 1. \$monad->mu . (funct_nat \$monad->functor, \$monad->eta) \$monad->mu . (nat_funct \$monad->eta, \$monad->functor) T→T ✤ 2. \$monad->mu . (funct_nat \$monad->functor, \$monad->mu) \$monad->mu . (nat_funct \$monad->mu, \$monad->functor) ( TTT → T )
24. 24. Monad (1): \$LIST_MONAD ✤ ✤ [v1_1, v1_2, v1_3], [v2_1, v2_2], [v3_1, v3_2, v3_3, v3_4] ✤ map eta [] mu concat
25. 25. Monad (2): \$STATE_MONAD ✤ ✤ sub { my @states = @_; .. .. return ¥@values, ¥@new_states } 1 ✤ functor, eta, mu Haskell ( )
26. 26. Monad (3): Maybe ✤ Maybe nothing ✤ ( ) null ✤ List
27. 27. (5): KleisliMorphism ✤ m f: a -> m b b mma mmb mmc mg μc ma mb mc f g ηa ηb ηc a b c
28. 28. (5): KleisliMorphism ✤ m f: a -> m b b mma mmb mmc mg μc ma mb mc f g ηa ηb ηc a b c
29. 29. Kleisli : Maybe (1) ✤ HTML get_number: HTML (<span>3/10</span>) cut_tag: (3/10) parse_number: (3, 10) div: (0.3) ✤ NG
30. 30. Kleisli : Maybe (2) ✤
31. 31. Kleisli : Maybe (2) ✤
32. 32. Kleisli : Maybe (2) ✤ Maybe Kleisli
33. 33. Kleisli : Maybe (2) ✤ Maybe Kleisli
34. 34. ✤ ✤ ✤ ✤ ( ) ✤