Math::Category
1 like1,090 views
Math::Category is a Perl module that defines categories and mathematical structures like functors, natural transformations, and monads. It implements categories as objects with morphisms that have source and target objects. Functors map objects and morphisms between categories while natural transformations transform functors. The module also defines monads as endofunctors with additional structure and provides examples like the list and state monads. Kleisli categories are constructed from monads to model computations.
![(2): SubroutineMorphism(1)
✤ Perl
✤ Perl
✤
print
✤ sub_morph { $_[0] * 2 }
undef
✤ source target sub_morph { @_ } ( )
✤ ( )](https://image.slidesharecdn.com/slide-091120085005-phpapp01/85/Math-Category-7-320.jpg)
![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](https://image.slidesharecdn.com/slide-091120085005-phpapp01/85/Math-Category-24-320.jpg)
Math::Category
- 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. ✤ ✤ ✤ ✤ ( ) ✤