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Monoidal Functor ( two functor remains )
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Wed, 22 Nov 2017 02:55:36 +0900
parents 7a729bb62ce3
children df3f1aae984f
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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696
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
1 open import Level
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
2 open import Level
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
3 open import Level
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
4 open import Category
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
5 module monoidal where
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
6
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
7 open import Data.Product renaming (_×_ to _*_)
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
8 open import Category.Constructions.Product
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
9 open import HomReasoning
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
10 open import cat-utility
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
11 open import Relation.Binary.Core
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
12 open import Relation.Binary
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
13
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
14 open Functor
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
15
698
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
16 record Iso {c₁ c₂ ℓ : Level} (C : Category c₁ c₂ ℓ)
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
17 (x y : Obj C )
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
18 : Set ( suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁)) where
696
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
19 field
698
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
20 ≅→ : Hom C x y
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
21 ≅← : Hom C y x
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
22 iso→ : C [ C [ ≅← o ≅→ ] ≈ id1 C x ]
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
23 iso← : C [ C [ ≅→ o ≅← ] ≈ id1 C y ]
696
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
24
698
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
25 record IsStrictMonoidal {c₁ c₂ ℓ : Level} (C : Category c₁ c₂ ℓ) (I : Obj C) ( BI : Functor ( C × C ) C )
696
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
26 : Set ( suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁)) where
700
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
27 infixr 9 _□_
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
28 _□_ : ( x y : Obj C ) → Obj C
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
29 _□_ x y = FObj BI ( x , y )
696
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
30 field
700
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
31 mα : {a b c : Obj C} → ( a □ b) □ c ≡ a □ ( b □ c )
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
32 mλ : (a : Obj C) → I □ a ≡ a
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
33 mρ : (a : Obj C) → a □ I ≡ a
698
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
34
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
35 record StrictMonoidal {c₁ c₂ ℓ : Level} (C : Category c₁ c₂ ℓ)
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
36 : Set ( suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁)) where
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
37 field
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
38 m-i : Obj C
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
39 m-bi : Functor ( C × C ) C
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
40 isMonoidal : IsStrictMonoidal C m-i m-bi
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
41
696
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
42
698
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
43 -- non strict version includes 5 naturalities
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
44 record IsMonoidal {c₁ c₂ ℓ : Level} (C : Category c₁ c₂ ℓ) (I : Obj C) ( BI : Functor ( C × C ) C )
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
45 : Set ( suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁)) where
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
46 open Iso
700
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
47 infixr 9 _□_ _■_
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
48 _□_ : ( x y : Obj C ) → Obj C
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
49 _□_ x y = FObj BI ( x , y )
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
50 _■_ : {a b c d : Obj C } ( f : Hom C a c ) ( g : Hom C b d ) → Hom C ( a □ b ) ( c □ d )
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
51 _■_ f g = FMap BI ( f , g )
698
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
52 field
700
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
53 mα-iso : {a b c : Obj C} → Iso C ( ( a □ b) □ c) ( a □ ( b □ c ) )
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
54 mλ-iso : {a : Obj C} → Iso C ( I □ a) a
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
55 mρ-iso : {a : Obj C} → Iso C ( a □ I) a
698
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
56 mα→nat1 : {a a' b c : Obj C} → ( f : Hom C a a' ) →
700
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
57 C [ C [ ( f ■ id1 C ( b □ c )) o ≅→ (mα-iso {a} {b} {c}) ] ≈
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
58 C [ ≅→ (mα-iso ) o ( (f ■ id1 C b ) ■ id1 C c ) ] ]
698
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
59 mα→nat2 : {a b b' c : Obj C} → ( f : Hom C b b' ) →
700
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
60 C [ C [ ( id1 C a ■ ( f ■ id1 C c ) ) o ≅→ (mα-iso {a} {b} {c} ) ] ≈
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
61 C [ ≅→ (mα-iso ) o ( (id1 C a ■ f ) ■ id1 C c ) ] ]
698
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
62 mα→nat3 : {a b c c' : Obj C} → ( f : Hom C c c' ) →
700
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
63 C [ C [ ( id1 C a ■ ( id1 C b ■ f ) ) o ≅→ (mα-iso {a} {b} {c} ) ] ≈
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
64 C [ ≅→ (mα-iso ) o ( id1 C ( a □ b ) ■ f ) ] ]
698
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
65 mλ→nat : {a a' : Obj C} → ( f : Hom C a a' ) →
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
66 C [ C [ f o ≅→ (mλ-iso {a} ) ] ≈
700
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
67 C [ ≅→ (mλ-iso ) o ( id1 C I ■ f ) ] ]
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
68 mρ→nat : {a a' : Obj C} → ( f : Hom C a a' ) →
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
69 C [ C [ f o ≅→ (mρ-iso {a} ) ] ≈
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
70 C [ ≅→ (mρ-iso ) o ( f ■ id1 C I ) ] ]
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
71 αABC□1D : {a b c d e : Obj C } → Hom C (((a □ b) □ c ) □ d) ((a □ (b □ c)) □ d)
701
7a729bb62ce3 Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 700
diff changeset
72 αABC□1D {a} {b} {c} {d} {e} = ( ≅→ mα-iso ■ id1 C d )
700
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
73 αAB□CD : {a b c d e : Obj C } → Hom C ((a □ (b □ c)) □ d) (a □ ((b □ c ) □ d))
701
7a729bb62ce3 Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 700
diff changeset
74 αAB□CD {a} {b} {c} {d} {e} = ≅→ mα-iso
700
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
75 1A□BCD : {a b c d e : Obj C } → Hom C (a □ ((b □ c ) □ d)) (a □ (b □ ( c □ d) ))
701
7a729bb62ce3 Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 700
diff changeset
76 1A□BCD {a} {b} {c} {d} {e} = ( id1 C a ■ ≅→ mα-iso )
700
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
77 αABC□D : {a b c d e : Obj C } → Hom C (a □ (b □ ( c □ d) )) ((a □ b ) □ (c □ d))
701
7a729bb62ce3 Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 700
diff changeset
78 αABC□D {a} {b} {c} {d} {e} = ≅← mα-iso
700
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
79 αA□BCD : {a b c d e : Obj C } → Hom C (((a □ b) □ c ) □ d) ((a □ b ) □ (c □ d))
701
7a729bb62ce3 Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 700
diff changeset
80 αA□BCD {a} {b} {c} {d} {e} = ≅→ mα-iso
700
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
81 αAIB : {a b : Obj C } → Hom C (( a □ I ) □ b ) (a □ ( I □ b ))
701
7a729bb62ce3 Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 700
diff changeset
82 αAIB {a} {b} = ≅→ mα-iso
700
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
83 1A□λB : {a b : Obj C } → Hom C (a □ ( I □ b )) ( a □ b )
701
7a729bb62ce3 Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 700
diff changeset
84 1A□λB {a} {b} = id1 C a ■ ≅→ mλ-iso
700
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
85 ρA□IB : {a b : Obj C } → Hom C (( a □ I ) □ b ) ( a □ b )
701
7a729bb62ce3 Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 700
diff changeset
86 ρA□IB {a} {b} = ≅→ mρ-iso ■ id1 C b
700
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
87
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
88 field
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
89 comm-penta : {a b c d e : Obj C}
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
90 → C [ C [ αABC□D {a} {b} {c} {d} {e} o C [ 1A□BCD {a} {b} {c} {d} {e} o C [ αAB□CD {a} {b} {c} {d} {e} o αABC□1D {a} {b} {c} {d} {e} ] ] ]
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
91 ≈ αA□BCD {a} {b} {c} {d} {e} ]
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
92 comm-unit : {a b : Obj C}
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
93 → C [ C [ 1A□λB {a} {b} o αAIB ] ≈ ρA□IB {a} {b} ]
696
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
94
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
95 record Monoidal {c₁ c₂ ℓ : Level} (C : Category c₁ c₂ ℓ)
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
96 : Set ( suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁)) where
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
97 field
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
98 m-i : Obj C
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
99 m-bi : Functor ( C × C ) C
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
100 isMonoidal : IsMonoidal C m-i m-bi
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
101
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
102
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
103 open import Category.Cat
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
104
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
105 record IsMonoidalFunctor {c₁ c₂ ℓ : Level} {C D : Category c₁ c₂ ℓ} ( M : Monoidal C ) ( N : Monoidal D )
698
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
106 ( MF : Functor C D )
702
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
107 ( ψ : Hom D (Monoidal.m-i N) (FObj MF (Monoidal.m-i M) ) )
696
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
108 : Set ( suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁)) where
698
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
109 _⊗_ : (x y : Obj C ) → Obj C
700
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
110 _⊗_ x y = (IsMonoidal._□_ (Monoidal.isMonoidal M) ) x y
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
111 _□_ : {a b c d : Obj C } ( f : Hom C a c ) ( g : Hom C b d ) → Hom C ( a ⊗ b ) ( c ⊗ d )
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
112 _□_ f g = FMap (Monoidal.m-bi M) ( f , g )
698
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
113 _●_ : (x y : Obj D ) → Obj D
700
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
114 _●_ x y = (IsMonoidal._□_ (Monoidal.isMonoidal N) ) x y
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
115 _■_ : {a b c d : Obj D } ( f : Hom D a c ) ( g : Hom D b d ) → Hom D ( a ● b ) ( c ● d )
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
116 _■_ f g = FMap (Monoidal.m-bi N) ( f , g )
702
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
117 F● : Functor ( C × C ) D
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
118 F● = record {
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
119 FObj = λ x → (FObj MF (proj₁ x) ) ● (FObj MF (proj₂ x) )
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
120 ; FMap = λ {x : Obj ( C × C ) } {y} f → FMap (Monoidal.m-bi N) ( FMap MF (proj₁ f ) , FMap MF (proj₂ f) )
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
121 ; isFunctor = record {
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
122 ≈-cong = {!!}
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
123 ; identity = {!!}
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
124 ; distr = {!!}
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
125 }
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
126 }
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
127 F⊗ : Functor ( C × C ) D
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
128 F⊗ = record {
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
129 FObj = λ x → FObj MF ( proj₁ x ⊗ proj₂ x )
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
130 ; FMap = λ {a} {b} f → FMap MF ( FMap (Monoidal.m-bi M) ( proj₁ f , proj₂ f) )
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
131 ; isFunctor = record {
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
132 ≈-cong = {!!}
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
133 ; identity = {!!}
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
134 ; distr = {!!}
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
135 }
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
136 }
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
137 field
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
138 φab : NTrans ( C × C ) D F● F⊗
698
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
139 open Iso
d648ebb8ac29 Monoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 697
diff changeset
140 open Monoidal
701
7a729bb62ce3 Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 700
diff changeset
141 open IsMonoidal hiding ( _■_ ; _□_ )
699
10ab28030c20 add definitions
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 698
diff changeset
142 αC : {a b c : Obj C} → Hom C (( a ⊗ b ) ⊗ c ) ( a ⊗ ( b ⊗ c ) )
10ab28030c20 add definitions
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 698
diff changeset
143 αC {a} {b} {c} = ≅→ (mα-iso (isMonoidal M) {a} {b} {c})
10ab28030c20 add definitions
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 698
diff changeset
144 αD : {a b c : Obj D} → Hom D (( a ● b ) ● c ) ( a ● ( b ● c ) )
10ab28030c20 add definitions
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 698
diff changeset
145 αD {a} {b} {c} = ≅→ (mα-iso (isMonoidal N) {a} {b} {c})
701
7a729bb62ce3 Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 700
diff changeset
146 F : Obj C → Obj D
7a729bb62ce3 Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 700
diff changeset
147 F x = FObj MF x
702
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
148 φ : ( x y : Obj C ) → Hom D ( FObj F● (x , y) ) ( FObj F⊗ ( x , y ))
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
149 φ x y = NTrans.TMap φab ( x , y )
701
7a729bb62ce3 Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 700
diff changeset
150 1●φBC : {a b c : Obj C} → Hom D ( F a ● ( F b ● F c ) ) ( F a ● ( F ( b ⊗ c ) ))
702
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
151 1●φBC {a} {b} {c} = id1 D (F a) ■ φ b c
701
7a729bb62ce3 Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 700
diff changeset
152 φAB⊗C : {a b c : Obj C} → Hom D ( F a ● ( F ( b ⊗ c ) )) (F ( a ⊗ ( b ⊗ c )))
702
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
153 φAB⊗C {a} {b} {c} = φ a (b ⊗ c )
701
7a729bb62ce3 Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 700
diff changeset
154 φAB●1 : {a b c : Obj C} → Hom D ( ( F a ● F b ) ● F c ) ( F ( a ⊗ b ) ● F c )
702
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
155 φAB●1 {a} {b} {c} = φ a b ■ id1 D (F c)
701
7a729bb62ce3 Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 700
diff changeset
156 φA⊗BC : {a b c : Obj C} → Hom D ( F ( a ⊗ b ) ● F c ) (F ( (a ⊗ b ) ⊗ c ))
702
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
157 φA⊗BC {a} {b} {c} = φ ( a ⊗ b ) c
701
7a729bb62ce3 Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 700
diff changeset
158 FαC : {a b c : Obj C} → Hom D (F ( (a ⊗ b ) ⊗ c )) (F ( a ⊗ ( b ⊗ c )))
7a729bb62ce3 Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 700
diff changeset
159 FαC {a} {b} {c} = FMap MF ( ≅→ (mα-iso (isMonoidal M) {a} {b} {c}) )
702
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
160 1●ψ : { a b : Obj C } → Hom D (F a ● Monoidal.m-i N ) ( F a ● F ( Monoidal.m-i M ) )
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
161 1●ψ{a} {b} = id1 D (F a) ■ ψ
701
7a729bb62ce3 Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 700
diff changeset
162 φAIC : { a b : Obj C } → Hom D ( F a ● F ( Monoidal.m-i M ) ) (F ( a ⊗ Monoidal.m-i M ))
702
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
163 φAIC {a} {b} = φ a ( Monoidal.m-i M )
701
7a729bb62ce3 Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 700
diff changeset
164 FρC : { a b : Obj C } → Hom D (F ( a ⊗ Monoidal.m-i M ))( F a )
7a729bb62ce3 Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 700
diff changeset
165 FρC {a} {b} = FMap MF ( ≅→ (mρ-iso (isMonoidal M) {a} ) )
7a729bb62ce3 Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 700
diff changeset
166 ρD : { a b : Obj C } → Hom D (F a ● Monoidal.m-i N ) ( F a )
7a729bb62ce3 Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 700
diff changeset
167 ρD {a} {b} = ≅→ (mρ-iso (isMonoidal N) {F a} )
702
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
168 ψ●1 : { a b : Obj C } → Hom D (Monoidal.m-i N ● F b ) ( F ( Monoidal.m-i M ) ● F b )
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
169 ψ●1 {a} {b} = ψ ■ id1 D (F b)
701
7a729bb62ce3 Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 700
diff changeset
170 φICB : { a b : Obj C } → Hom D ( F ( Monoidal.m-i M ) ● F b ) ( F ( ( Monoidal.m-i M ) ⊗ b ) )
702
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
171 φICB {a} {b} = φ ( Monoidal.m-i M ) b
701
7a729bb62ce3 Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 700
diff changeset
172 FλD : { a b : Obj C } → Hom D ( F ( ( Monoidal.m-i M ) ⊗ b ) ) (F b )
7a729bb62ce3 Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 700
diff changeset
173 FλD {a} {b} = FMap MF ( ≅→ (mλ-iso (isMonoidal M) {b} ) )
7a729bb62ce3 Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 700
diff changeset
174 λD : { a b : Obj C } → Hom D (Monoidal.m-i N ● F b ) (F b )
7a729bb62ce3 Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 700
diff changeset
175 λD {a} {b} = ≅→ (mλ-iso (isMonoidal N) {F b} )
696
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
176 field
700
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
177 comm1 : {a b c : Obj C } → D [ D [ φAB⊗C {a} {b} {c} o D [ 1●φBC o αD ] ] ≈ D [ FαC o D [ φA⊗BC o φAB●1 ] ] ]
702
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
178 comm-idr : {a b : Obj C } → D [ D [ FρC {a} {b} o D [ φAIC {a} {b} o 1●ψ{a} {b} ] ] ≈ ρD {a} {b} ]
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
179 comm-idl : {a b : Obj C } → D [ D [ FλD {a} {b} o D [ φICB {a} {b} o ψ●1 {a} {b} ] ] ≈ λD {a} {b} ]
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
180 where
696
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
181
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
182 record MonoidalFunctor {c₁ c₂ ℓ : Level} {C D : Category c₁ c₂ ℓ} ( M : Monoidal C ) ( N : Monoidal D )
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
183 : Set ( suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁)) where
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
184 _⊗_ : (x y : Obj C ) → Obj C
700
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
185 _⊗_ x y = (IsMonoidal._□_ (Monoidal.isMonoidal M) ) x y
696
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
186 _●_ : (x y : Obj D ) → Obj D
700
cfd2d402c486 monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 699
diff changeset
187 _●_ x y = (IsMonoidal._□_ (Monoidal.isMonoidal N) ) x y
696
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
188 field
10ccac3bc285 Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
189 MF : Functor C D
702
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
190 ψ : Hom D (Monoidal.m-i N) (FObj MF (Monoidal.m-i M) )
d16327b48b83 Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 701
diff changeset
191 isMonodailFunctor : IsMonoidalFunctor M N MF ψ