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add universal mapping
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 22 Jul 2013 14:30:27 +0900
parents 98b8431a419b
children 83ff8d48fdca
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module nat  where 

-- Monad
-- Category A
-- A = Category
-- Functor T : A → A
--T(a) = t(a)
--T(f) = tf(f)

open import Category -- https://github.com/konn/category-agda
open import Level
open import Category.HomReasoning

--T(g f) = T(g) T(f)

open Functor
Lemma1 : {c₁ c₂ l : Level} {A : Category c₁ c₂ l} (T : Functor A A) →  {a b c : Obj A} {g : Hom A b c} { f : Hom A a b }
     → A [ ( FMap T (A [ g o f ] ))  ≈ (A [ FMap T g o FMap T f ]) ]
Lemma1 = \t → IsFunctor.distr ( isFunctor t )


open NTrans
Lemma2 : {c₁ c₂ l : Level} {A : Category c₁ c₂ l} {F G : Functor A A} 
      → (μ : NTrans A A F G) → {a b : Obj A} { f : Hom A a b }
      → A [ A [  FMap G f o TMap μ a ]  ≈ A [ TMap μ b o  FMap F f ] ]
Lemma2 = \n → IsNTrans.naturality ( isNTrans  n  )

open import Category.Cat

-- η :   1_A → T
-- μ :   TT → T
-- μ(a)η(T(a)) = a
-- μ(a)T(η(a)) = a
-- μ(a)(μ(T(a))) = μ(a)T(μ(a))

record IsMonad {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) 
                 ( T : Functor A A )
                 ( η : NTrans A A identityFunctor T )
                 ( μ : NTrans A A (T ○ T) T)
                 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ )) where
   field
      assoc  : {a : Obj A} → A [ A [ TMap μ a o TMap μ ( FObj T a ) ] ≈  A [  TMap μ a o FMap T (TMap μ a) ] ]
      unity1 : {a : Obj A} → A [ A [ TMap μ a o TMap η ( FObj T a ) ] ≈ Id {_} {_} {_} {A} (FObj T a) ]
      unity2 : {a : Obj A} → A [ A [ TMap μ a o (FMap T (TMap η a ))] ≈ Id {_} {_} {_} {A} (FObj T a) ]

record Monad {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (T : Functor A A) (η : NTrans A A identityFunctor T) (μ : NTrans A A (T ○ T) T)
       : Set (suc (c₁ ⊔ c₂ ⊔ ℓ )) where
  eta : NTrans A A identityFunctor T
  eta = η
  mu : NTrans A A (T ○ T) T
  mu = μ
  field
    isMonad : IsMonad A T η μ

open Monad
Lemma3 : {c₁ c₂ ℓ : Level} {A : Category c₁ c₂ ℓ}
                 { T : Functor A A }
                 { η : NTrans A A identityFunctor T }
                 { μ : NTrans A A (T ○ T) T }
                 { a : Obj A } →
                 ( M : Monad A T η μ )
    → A [ A [  TMap μ a o TMap μ ( FObj T a ) ] ≈  A [  TMap μ a o FMap T (TMap μ a) ] ]
Lemma3 = \m → IsMonad.assoc ( isMonad m )


Lemma4 : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) {a b : Obj A } { f : Hom A a b}
    → A [ A [ Id {_} {_} {_} {A} b o f ] ≈ f ]
Lemma4 = \a → IsCategory.identityL ( Category.isCategory a )

Lemma5 : {c₁ c₂ ℓ : Level} {A : Category c₁ c₂ ℓ}
                 { T : Functor A A }
                 { η : NTrans A A identityFunctor T }
                 { μ : NTrans A A (T ○ T) T }
                 { a : Obj A } →
                 ( M : Monad A T η μ )
    →  A [ A [ TMap μ a o TMap η ( FObj T a )  ] ≈ Id {_} {_} {_} {A} (FObj T a) ]
Lemma5 = \m → IsMonad.unity1 ( isMonad m )

Lemma6 : {c₁ c₂ ℓ : Level} {A : Category c₁ c₂ ℓ}
                 { T : Functor A A }
                 { η : NTrans A A identityFunctor T }
                 { μ : NTrans A A (T ○ T) T }
                 { a : Obj A } →
                 ( M : Monad A T η μ )
    →  A [ A [ TMap μ a o (FMap T (TMap η a )) ] ≈ Id {_} {_} {_} {A} (FObj T a) ]
Lemma6 = \m → IsMonad.unity2 ( isMonad m )

-- T = M x A
-- nat of η
-- g ○ f = μ(c) T(g) f
-- η(b) ○ f = f
-- f ○ η(a) = f
-- h ○ (g ○ f) = (h ○ g) ○ f

record Kleisli  { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ )
                 ( T : Functor A A )
                 ( η : NTrans A A identityFunctor T )
                 ( μ : NTrans A A (T ○ T) T )
                 ( M : Monad A T η μ ) : Set (suc (c₁ ⊔ c₂ ⊔ ℓ )) where
     monad : Monad A T η μ 
     monad = M
     -- g ○ f = μ(c) T(g) f
     join : { a b : Obj A } → ( c : Obj A ) →
                      ( Hom A b ( FObj T c )) → (  Hom A a ( FObj T b)) → Hom A a ( FObj T c )
     join c g f = A [ TMap μ c  o A [ FMap T g  o f ] ]

lemma12 :  {c₁ c₂ ℓ : Level} (L : Category c₁ c₂ ℓ) { a b c : Obj L } → 
       ( x : Hom L c a ) → ( y : Hom L b c ) → L [ L [ x o y ] ≈ L [ x o y ] ]
lemma12 L x y =  
   let open ≈-Reasoning ( L )  in 
      begin  L [ x o y ]  ∎


open Kleisli
-- η(b) ○ f = f
Lemma7 : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) →
                 ( T : Functor A A )
                 ( η : NTrans A A identityFunctor T )
                 { μ : NTrans A A (T ○ T) T }
                 { a : Obj A } ( b : Obj A ) 
                 ( f : Hom A a ( FObj T b) )
                 ( m : Monad A T η μ )
                 ( k : Kleisli A T η μ m) 
                      → A  [ join k b (TMap η b) f  ≈ f ]
Lemma7 c T η b f m k = 
  let open ≈-Reasoning (c) 
      μ = mu ( monad k )
  in 
     begin  
         join k b (TMap η b) f 
     ≈⟨ refl-hom ⟩
         c [ TMap μ b  o c [ FMap T ((TMap η b)) o f ] ]  
     ≈⟨ IsCategory.associative (Category.isCategory c) ⟩
        c [ c [ TMap μ b  o  FMap T ((TMap η b)) ] o f ]
     ≈⟨ car ( IsMonad.unity2 ( isMonad ( monad k )) )  ⟩
        c [  id (FObj T b)   o f ]
     ≈⟨ IsCategory.identityL (Category.isCategory c) ⟩
        f
     ∎

-- f ○ η(a) = f
Lemma8 : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ)
                 ( T : Functor A A )
                 ( η : NTrans A A identityFunctor T )
                 { μ : NTrans A A (T ○ T) T }
                 ( a  : Obj A )  ( b : Obj A )
                 ( f : Hom A a ( FObj T b) )
                 ( m : Monad A T η μ )
                 ( k : Kleisli A T η μ m) 
                      → A  [ join k b f (TMap η a)  ≈ f ]
Lemma8 c T η a b f m k = 
  begin
     join k b f (TMap η a) 
  ≈⟨ refl-hom ⟩
     c [ TMap μ b  o c [  FMap T f o (TMap η a) ] ]  
  ≈⟨ cdr  ( IsNTrans.naturality ( isNTrans η  )) ⟩
     c [ TMap μ b  o c [ (TMap η ( FObj T b)) o f ] ] 
  ≈⟨ IsCategory.associative (Category.isCategory c) ⟩
     c [ c [ TMap μ b  o (TMap η ( FObj T b)) ] o f ] 
  ≈⟨ car ( IsMonad.unity1 ( isMonad ( monad k )) ) ⟩
     c [ id (FObj T b)  o f ]
  ≈⟨  IsCategory.identityL (Category.isCategory c)  ⟩
     f
  ∎  where 
      open ≈-Reasoning (c) 
      μ = mu ( monad k )

-- h ○ (g ○ f) = (h ○ g) ○ f
Lemma9 : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ)
                 ( T : Functor A A )
                 ( η : NTrans A A identityFunctor T )
                 ( μ : NTrans A A (T ○ T) T )
                 ( a b c d : Obj A )
                 ( f : Hom A a ( FObj T b) )
                 ( g : Hom A b ( FObj T c) ) 
                 ( h : Hom A c ( FObj T d) )
                 ( m : Monad A T η μ )
                 ( k : Kleisli A T η μ m)
                      → A  [ join k d h (join k c g f)  ≈ join k d ( join k d h g) f ]
Lemma9 A T η μ a b c d f g h m k = 
  begin 
     join k d h (join k c g f)  
   ≈⟨⟩
     join k d h (                  ( TMap μ c o ( FMap T g o f ) ) )
   ≈⟨ refl-hom ⟩
     ( TMap μ d  o ( FMap T h  o  ( TMap μ c o ( FMap T g  o f ) ) ) )
   ≈⟨ cdr ( cdr ( assoc )) ⟩
     ( TMap μ d  o ( FMap T h o ( ( TMap μ c o  FMap T g )  o f ) ) )
   ≈⟨ assoc  ⟩    ---   ( f  o  ( g  o  h ) ) = ( ( f  o  g )  o  h )
     (     ( TMap μ d o  FMap T h ) o  ( (TMap μ c   o  FMap T g )    o f ) )
   ≈⟨ assoc  ⟩
     ( ( ( TMap μ d o      FMap T h ) o  (TMap μ c   o  FMap T g ) )  o f )
   ≈⟨ car (sym assoc) ⟩
     ( ( TMap μ d o  ( FMap T h     o   ( TMap μ c   o  FMap T g ) ) ) o f )
   ≈⟨ car ( cdr (assoc) ) ⟩
     ( ( TMap μ d o  ( ( FMap T h o       TMap μ c ) o  FMap T g )   ) o f )
   ≈⟨ car assoc ⟩
     ( ( ( TMap μ d o  ( FMap T h   o   TMap μ c ) ) o  FMap T g ) o f )
   ≈⟨ car (car ( cdr ( begin 
           ( FMap T h o TMap μ c )
       ≈⟨ nat A μ ⟩
           ( TMap μ (FObj T d) o FMap T (FMap T h) )
        ∎ 
   )))  ⟩
      ( ( ( TMap μ d  o  ( TMap μ ( FObj T d)  o FMap T (  FMap T h ) ) )  o FMap T g )  o f )
   ≈⟨ car (sym assoc) ⟩
     ( ( TMap μ d  o  ( ( TMap μ ( FObj T d)   o FMap T (  FMap T h ) )   o FMap T g ) )   o f )
   ≈⟨ car ( cdr (sym assoc) ) ⟩
     ( ( TMap μ d  o  ( TMap μ ( FObj T d)   o ( FMap T (  FMap T h ) o FMap T g ) ) )   o f )
   ≈⟨ car ( cdr (cdr (sym (distr T )))) ⟩
     ( ( TMap μ d  o  ( TMap μ ( FObj T d)     o FMap T ( ( FMap T h  o g ) ) ) )   o f )
   ≈⟨ car assoc ⟩
     ( ( ( TMap μ d  o  TMap μ ( FObj T d)  )  o FMap T ( ( FMap T h  o g ) ) )    o f )
   ≈⟨ car ( car (
      begin
         ( TMap μ d o TMap μ (FObj T d) )
      ≈⟨ IsMonad.assoc ( isMonad m) ⟩
         ( TMap μ d o FMap T (TMap μ d) )
      ∎
   )) ⟩
     ( ( ( TMap μ d  o    FMap T ( TMap μ d) ) o FMap T ( ( FMap T h  o g ) ) )    o f )
   ≈⟨ car (sym assoc) ⟩
     ( ( TMap μ d  o  ( FMap T ( TMap μ d )    o FMap T ( ( FMap T h  o g ) ) ) )  o f )
   ≈⟨ sym assoc ⟩
     ( TMap μ d  o  ( ( FMap T ( TMap μ d )    o FMap T ( ( FMap T h  o g ) ) )  o f ) )
   ≈⟨ cdr ( car ( sym ( distr T )))   ⟩
     ( TMap μ d  o ( FMap T ( ( ( TMap μ d )   o ( FMap T h  o g ) ) ) o f ) )
   ≈⟨ refl-hom ⟩
     join k d ( ( TMap μ d  o ( FMap T h  o g ) ) ) f
   ≈⟨ refl-hom ⟩
     join k d ( join k d h g) f 
  ∎ where open ≈-Reasoning (A) 



-- Kleisli :
-- Kleisli = record { Hom = 
--                 ; Hom = _⟶_
--                  ; Id = IdProd
--                  ; _o_ = _∘_
--                  ; _≈_ = _≈_
--                  ; isCategory = record { isEquivalence = record { refl  = λ {φ} → ≈-refl {φ = φ}
--                                                                 ; sym   = λ {φ ψ} → ≈-symm {φ = φ} {ψ}
--                                                                 ; trans = λ {φ ψ σ} → ≈-trans {φ = φ} {ψ} {σ}
--                                                                 }
--                                        ; identityL = identityL
--                                        ; identityR = identityR
--                                        ; o-resp-≈ = o-resp-≈
--                                        ; associative = associative
--                                        }
--                  }