Sets is CCC
author Shinji KONO Fri, 19 Apr 2019 23:42:19 +0900 bded2347efa4
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open import Category -- https://github.com/konn/category-agda
open import Algebra
open import Level
open import Category.Sets
module monoid-monad {c : Level} where

open import Algebra.Structures
open import HomReasoning
open import cat-utility
open import Category.Cat
open import Data.Product
open import Relation.Binary.Core
open import Relation.Binary

-- open Monoid
open import Algebra.FunctionProperties using (Op₁; Op₂)

open import Relation.Binary.PropositionalEquality hiding ( [_] ; sym )

record ≡-Monoid c : Set (suc c) where
infixl 7 _*_
field
Carrier  : Set c
_*_      : Op₂ Carrier
ε        : Carrier   -- id in Monoid
isMonoid : IsMonoid _≡_ _*_ ε

postulate M : ≡-Monoid c
open ≡-Monoid

infixl 7 _∙_

_∙_   : ( m m' : Carrier M ) → Carrier M
_∙_ m m' = _*_ M m m'

A = Sets {c}

-- T : A → (M x A)

T : Functor A A
T = record {
FObj = λ a → (Carrier M) × a
; FMap = λ f p → (proj₁ p , f (proj₂ p ))
; isFunctor = record {
identity = IsEquivalence.refl (IsCategory.isEquivalence  ( Category.isCategory Sets ))
; distr = (IsEquivalence.refl (IsCategory.isEquivalence  ( Category.isCategory Sets )))
; ≈-cong = cong1
}
} where
cong1 : {ℓ′ : Level} → {a b : Set ℓ′} { f g : Hom (Sets {ℓ′}) a b} →
Sets [ f ≈ g ] → Sets [ map (λ (x : Carrier M) → x) f ≈ map (λ (x : Carrier M) → x) g ]
cong1 _≡_.refl = _≡_.refl

open Functor

Lemma-MM1 :  {a b : Obj A} {f : Hom A a b} →
A [ A [ FMap T f o (λ x → ε M , x) ] ≈
A [ (λ x → ε M , x) o f ] ]
Lemma-MM1 {a} {b} {f} = let open ≈-Reasoning A renaming ( _o_ to _*_ ) in
begin
FMap T f o (λ x → ε M , x)
≈⟨⟩
(λ x → ε M , x) o f
∎

-- η : a → (ε,a)
η : NTrans  A A identityFunctor T
η = record {
TMap = λ a → λ(x : a) → ( ε M , x ) ;
isNTrans = record {
commute = Lemma-MM1
}
}

-- μ : (m,(m',a)) → (m*m,a)

muMap : (a : Obj A  ) → FObj T ( FObj T a ) → Σ (Carrier M) (λ x → a )
muMap a ( m , ( m' , x ) ) = ( m ∙ m' ,  x )

Lemma-MM2 :  {a b : Obj A} {f : Hom A a b} →
A [ A [ FMap T f o (λ x → muMap a x) ] ≈
A [ (λ x → muMap b x) o FMap (T ○ T) f ] ]
Lemma-MM2 {a} {b} {f} =  let open ≈-Reasoning A renaming ( _o_ to _*_ ) in
begin
FMap T f o (λ x → muMap a x)
≈⟨⟩
(λ x → muMap b x) o FMap (T ○ T) f
∎

μ : NTrans  A A ( T ○ T ) T
μ = record {
TMap = λ a → λ x  → muMap a x ;
isNTrans = record {
commute = λ{a} {b} {f} → Lemma-MM2 {a} {b} {f}
}
}

open NTrans

Lemma-MM33 : {a : Level} {b : Level} {A : Set a} {B : A → Set b}  {x :  Σ A B } →  (((proj₁ x) , proj₂ x ) ≡ x )
Lemma-MM33 =  _≡_.refl

Lemma-MM34 : ∀( x : Carrier M  ) → ε M ∙ x ≡ x
Lemma-MM34  x = (( proj₁ ( IsMonoid.identity ( isMonoid M )) ) x )

Lemma-MM35 : ∀( x : Carrier M  ) → x ∙ ε M ≡ x
Lemma-MM35  x = ( proj₂  ( IsMonoid.identity ( isMonoid M )) ) x

Lemma-MM36 : ∀( x y z : Carrier M ) → (x ∙ y) ∙ z ≡ x ∙ (y ∙ z )
Lemma-MM36  x y z = ( IsMonoid.assoc ( isMonoid M ))  x y z

-- Functional Extensionality Axiom (We cannot prove this in Agda / Coq, just assumming )
import Relation.Binary.PropositionalEquality
-- postulate extensionality : { a b : Obj A } {f g : Hom A a b } →  Relation.Binary.PropositionalEquality.Extensionality c c
postulate extensionality : Relation.Binary.PropositionalEquality.Extensionality c c

-- Multi Arguments Functional Extensionality
extensionality30 : {f g : Carrier M → Carrier M → Carrier M → Carrier M } →
(∀ x y z  → f x y z ≡ g x y z )  → ( f ≡ g )
extensionality30 eq = extensionality ( λ x → extensionality ( λ y → extensionality (eq x y) ) )

Lemma-MM9  :  (λ(x : Carrier M) → ( ε M ∙ x ))  ≡ ( λ(x : Carrier M) → x  )
Lemma-MM9  = extensionality Lemma-MM34

Lemma-MM10 : ( λ x →   (x ∙ ε M))  ≡ ( λ x → x )
Lemma-MM10  = extensionality Lemma-MM35

Lemma-MM11 : (λ x y z → ((x ∙ y ) ∙ z))  ≡ (λ x y z → ( x ∙ (y ∙ z ) ))
Lemma-MM11 = extensionality30 Lemma-MM36

T = T
; η = η
; μ = μ
unity1 = Lemma-MM3 ;
unity2 = Lemma-MM4 ;
assoc  = Lemma-MM5
}
} where
Lemma-MM3 : {a : Obj A} → A [ A [ TMap μ a o TMap η ( FObj T a ) ] ≈ Id {_} {_} {_} {A} (FObj T a) ]
Lemma-MM3 {a} = let open ≈-Reasoning (A) renaming ( _o_ to _*_ ) in
begin
TMap μ a o TMap η ( FObj T a )
≈⟨⟩
( λ x →   ε M ∙ (proj₁ x) , proj₂ x )
≈⟨  cong ( λ f → ( λ x →  ( ( f (proj₁ x) ) , proj₂ x ))) ( Lemma-MM9 )  ⟩
( λ (x : FObj T a) → (proj₁ x) , proj₂ x )
≈⟨⟩
( λ x → x )
≈⟨⟩
Id {_} {_} {_} {A} (FObj T a)
∎
Lemma-MM4 : {a : Obj A} → A [ A [ TMap μ a o (FMap T (TMap η a ))] ≈ Id {_} {_} {_} {A} (FObj T a) ]
Lemma-MM4 {a} = let open ≈-Reasoning (A) renaming ( _o_ to _*_ ) in
begin
TMap μ a o (FMap T (TMap η a ))
≈⟨⟩
( λ x → ( proj₁ x ∙ (ε M) , proj₂ x ))
≈⟨  cong ( λ f → ( λ x →  ( f (proj₁ x) ) , proj₂ x )) ( Lemma-MM10 )  ⟩
( λ x → (proj₁ x) , proj₂ x )
≈⟨⟩
( λ x → x )
≈⟨⟩
Id {_} {_} {_} {A} (FObj T a)
∎
Lemma-MM5 : {a : Obj A} → A [ A [ TMap μ a o TMap μ ( FObj T a ) ] ≈  A [  TMap μ a o FMap T (TMap μ a) ] ]
Lemma-MM5 {a} = let open ≈-Reasoning (A) renaming ( _o_ to _*_ ) in
begin
TMap μ a o TMap μ ( FObj T a )
≈⟨⟩
( λ x → (proj₁ x) ∙ (proj₁ (proj₂ x)) ∙ (proj₁ (proj₂ (proj₂ x))) , proj₂ (proj₂ (proj₂ x)))
≈⟨ cong ( λ f → ( λ x →
(( f ( proj₁ x ) ((proj₁ (proj₂ x))) ((proj₁ (proj₂ (proj₂ x)))  )) ,  proj₂ (proj₂ (proj₂ x)) )
)) Lemma-MM11  ⟩
( λ x → ( proj₁ x) ∙(( proj₁ (proj₂ x)) ∙ (proj₁ (proj₂ (proj₂ x)))) , proj₂ (proj₂ (proj₂ x)))
≈⟨⟩
TMap μ a o FMap T (TMap μ a)
∎

F  : (m : Carrier M) → { a b : Obj A } → ( f : a → b ) → Hom A a ( FObj T b )
F m {a} {b} f =  λ (x : a ) → ( m , ( f x) )

postulate m m' : Carrier M
postulate a b c' : Obj A
postulate f :  b → c'
postulate g :  a → b