Perlを使って圏論概念をシミュレーション。

1. Math::Category
id:hiratara
2009-11-21

2. Math::Category (1)
✤ Perl
✤
✤
✤

3. Math::Category (2)
✤
✤ dom, cod, comp
✤
✤ Hom
✤

4. (1)
✤
✤
✤ Morphism interface
✤ source: dom Morphism
✤ target: cod Morphism
✤ composition:

5. (2)
✤
✤ target source
✤ composition
✤
✤
✤

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. (2): SubroutineMorphism(1)
✤ Perl
✤ Perl
✤
print
✤ sub_morph { $_[0] * 2 }
undef
✤ source target sub_morph { @_ } ( )
✤ ( )

8. (2): SubroutineMorphism(2)
✤ $sub2 . $sub1 $sub1 $sub2
✤
✤
✤

9. (1)
✤ f.(g.h) (f . g) . h f g h
✤
✤ id ( )
✤
✤ f×g: x → ( f(x), g(x) )

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. (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. (1)
✤ Functor
✤
✤ Morphism Morphism
✤
( )

13. (2)
✤ functor { ... };
✤
(Morphism OK)
✤
✤ ( $functor2 . $functor1 )

14. (1): $BI_FUNCTOR
✤ Hom(-, -)
✤ C^op × C Sets
✤ C^op × C
✤ Sets SubroutineMorphism
( Sets )

15. (1)
✤ NaturalTransformation
✤
✤ (Morphism)

16. (2)
✤ nat { };
✤
✤
✤ ( )
✤

17. (3)
✤
✤ $nat2 . $nat1
✤ $funct . $nat
✤ $nat . $funct

18. (4): FunctorMorphism
✤
✤ functor_morph nat { F
my $id = shift;
... ...
return $sum_morph
τ
};
✤ source target source
G
target

19. (2): $YONEDA_EMBEDDING
✤
✤ C^op Sets^C
✤ Hom(g, -)

20. : CPS (1)
✤ uc CPS

21. : CPS (2)
Sets
Hom( , -) Hom( , undef)
∈
print
uc $fun_morph $cps_uc
$cps_uc->(print)
∈
Hom( , -) Hom( , undef)

22. Monad (1)
✤ functor eta mu
✤ eta: I → T, mu: TT → T
✤
✤ Haskell Monad
✤

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. 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. Monad (2): $STATE_MONAD
✤
✤
sub { my @states = @_; .. .. return ¥@values, ¥@new_states }
1
✤ functor, eta, mu Haskell ( )

26. Monad (3): Maybe
✤ Maybe nothing
✤ ( ) null
✤ List

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. (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. Kleisli : Maybe (1)
✤ HTML
get_number: HTML (<span>3/10</span>)
cut_tag: (3/10)
parse_number: (3, 10)
div: (0.3)
✤ NG

30. Kleisli : Maybe (2)
✤

31. Kleisli
: Maybe (2)
✤

32. Kleisli : Maybe (2)
✤ Maybe Kleisli

33. Kleisli : Maybe (2)
✤ Maybe Kleisli

34. ✤
✤
✤
✤ ( )
✤