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annotate cat-utility.agda @ 790:1e7319868d77

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Sets is CCC
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Fri, 19 Apr 2019 23:42:19 +0900
parents 117e5b392673
children
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
56
83ff8d48fdca add unitility
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
1 module cat-utility where
83ff8d48fdca add unitility
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
2
83ff8d48fdca add unitility
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
3 -- Shinji KONO <kono@ie.u-ryukyu.ac.jp>
83ff8d48fdca add unitility
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
4
87
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
5 open import Category -- https://github.com/konn/category-agda
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
6 open import Level
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
7 --open import Category.HomReasoning
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
8 open import HomReasoning
56
83ff8d48fdca add unitility
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
9
87
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
10 open Functor
56
83ff8d48fdca add unitility
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
11
87
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
12 id1 : ∀{c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (a : Obj A ) → Hom A a a
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
13 id1 A a = (Id {_} {_} {_} {A} a)
253
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 202
diff changeset
14 -- We cannot make A implicit
56
83ff8d48fdca add unitility
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
15
731
117e5b392673 Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 695
diff changeset
16 record Iso {c₁ c₂ ℓ : Level} (C : Category c₁ c₂ ℓ)
117e5b392673 Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 695
diff changeset
17 (x y : Obj C )
117e5b392673 Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 695
diff changeset
18 : Set ( suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁)) where
117e5b392673 Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 695
diff changeset
19 field
117e5b392673 Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 695
diff changeset
20 ≅→ : Hom C x y
117e5b392673 Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 695
diff changeset
21 ≅← : Hom C y x
117e5b392673 Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 695
diff changeset
22 iso→ : C [ C [ ≅← o ≅→ ] ≈ id1 C x ]
117e5b392673 Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 695
diff changeset
23 iso← : C [ C [ ≅→ o ≅← ] ≈ id1 C y ]
117e5b392673 Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 695
diff changeset
24
117e5b392673 Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 695
diff changeset
25
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
26 record IsUniversalMapping {c₁ c₂ ℓ c₁' c₂' ℓ' : Level} (A : Category c₁ c₂ ℓ) (B : Category c₁' c₂' ℓ')
87
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
27 ( U : Functor B A )
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
28 ( F : Obj A → Obj B )
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
29 ( η : (a : Obj A) → Hom A a ( FObj U (F a) ) )
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
30 ( _* : { a : Obj A}{ b : Obj B} → ( Hom A a (FObj U b) ) → Hom B (F a ) b )
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
31 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁' ⊔ c₂' ⊔ ℓ' )) where
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
32 field
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
33 universalMapping : {a : Obj A} { b : Obj B } → { f : Hom A a (FObj U b) } →
101
0f7086b6a1a6 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
34 A [ A [ FMap U ( f * ) o η a ] ≈ f ]
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
35 uniquness : {a : Obj A} { b : Obj B } → { f : Hom A a (FObj U b) } → { g : Hom B (F a) b } →
101
0f7086b6a1a6 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
36 A [ A [ FMap U g o η a ] ≈ f ] → B [ f * ≈ g ]
56
83ff8d48fdca add unitility
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
37
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
38 record UniversalMapping {c₁ c₂ ℓ c₁' c₂' ℓ' : Level} (A : Category c₁ c₂ ℓ) (B : Category c₁' c₂' ℓ')
689
fb9fc9652c04 fix again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 688
diff changeset
39 ( U : Functor B A )
87
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
40 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁' ⊔ c₂' ⊔ ℓ' )) where
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
41 infixr 11 _*
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
42 field
690
3d41a8edbf63 fix universal mapping done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 689
diff changeset
43 F : Obj A → Obj B
3d41a8edbf63 fix universal mapping done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 689
diff changeset
44 η : (a : Obj A) → Hom A a ( FObj U (F a) )
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
45 _* : { a : Obj A}{ b : Obj B} → ( Hom A a (FObj U b) ) → Hom B (F a ) b
87
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
46 isUniversalMapping : IsUniversalMapping A B U F η _*
56
83ff8d48fdca add unitility
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
47
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
48 record coIsUniversalMapping {c₁ c₂ ℓ c₁' c₂' ℓ' : Level} (A : Category c₁ c₂ ℓ) (B : Category c₁' c₂' ℓ')
268
2ff44ee3cb32 co universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
49 ( F : Functor A B )
2ff44ee3cb32 co universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
50 ( U : Obj B → Obj A )
2ff44ee3cb32 co universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
51 ( ε : (b : Obj B) → Hom B ( FObj F (U b) ) b )
2ff44ee3cb32 co universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
52 ( _*' : { b : Obj B}{ a : Obj A} → ( Hom B (FObj F a) b ) → Hom A a (U b ) )
2ff44ee3cb32 co universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
53 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁' ⊔ c₂' ⊔ ℓ' )) where
2ff44ee3cb32 co universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
54 field
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
55 couniversalMapping : {b : Obj B} { a : Obj A } → { f : Hom B (FObj F a) b } →
268
2ff44ee3cb32 co universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
56 B [ B [ ε b o FMap F ( f *' ) ] ≈ f ]
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
57 couniquness : {b : Obj B} { a : Obj A } → { f : Hom B (FObj F a) b } → { g : Hom A a (U b) } →
268
2ff44ee3cb32 co universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
58 B [ B [ ε b o FMap F g ] ≈ f ] → A [ f *' ≈ g ]
2ff44ee3cb32 co universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
59
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
60 record coUniversalMapping {c₁ c₂ ℓ c₁' c₂' ℓ' : Level} (A : Category c₁ c₂ ℓ) (B : Category c₁' c₂' ℓ')
689
fb9fc9652c04 fix again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 688
diff changeset
61 ( F : Functor A B )
268
2ff44ee3cb32 co universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
62 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁' ⊔ c₂' ⊔ ℓ' )) where
2ff44ee3cb32 co universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
63 infixr 11 _*'
2ff44ee3cb32 co universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
64 field
690
3d41a8edbf63 fix universal mapping done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 689
diff changeset
65 U : Obj B → Obj A
3d41a8edbf63 fix universal mapping done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 689
diff changeset
66 ε : (b : Obj B) → Hom B ( FObj F (U b) ) b
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
67 _*' : { b : Obj B}{ a : Obj A} → ( Hom B (FObj F a) b ) → Hom A a (U b )
268
2ff44ee3cb32 co universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
68 iscoUniversalMapping : coIsUniversalMapping A B F U ε _*'
2ff44ee3cb32 co universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
69
87
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
70 open NTrans
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
71 open import Category.Cat
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
72 record IsAdjunction {c₁ c₂ ℓ c₁' c₂' ℓ' : Level} (A : Category c₁ c₂ ℓ) (B : Category c₁' c₂' ℓ')
87
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
73 ( U : Functor B A )
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
74 ( F : Functor A B )
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
75 ( η : NTrans A A identityFunctor ( U ○ F ) )
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
76 ( ε : NTrans B B ( F ○ U ) identityFunctor )
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
77 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁' ⊔ c₂' ⊔ ℓ' )) where
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
78 field
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
79 adjoint1 : { b : Obj B } →
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
80 A [ A [ ( FMap U ( TMap ε b )) o ( TMap η ( FObj U b )) ] ≈ id1 A (FObj U b) ]
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
81 adjoint2 : {a : Obj A} →
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
82 B [ B [ ( TMap ε ( FObj F a )) o ( FMap F ( TMap η a )) ] ≈ id1 B (FObj F a) ]
56
83ff8d48fdca add unitility
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
83
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
84 record Adjunction {c₁ c₂ ℓ c₁' c₂' ℓ' : Level} (A : Category c₁ c₂ ℓ) (B : Category c₁' c₂' ℓ')
87
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
85 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁' ⊔ c₂' ⊔ ℓ' )) where
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
86 field
688
a5f2ca67e7c5 fix monad/adjunction definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 681
diff changeset
87 U : Functor B A
a5f2ca67e7c5 fix monad/adjunction definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 681
diff changeset
88 F : Functor A B
a5f2ca67e7c5 fix monad/adjunction definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 681
diff changeset
89 η : NTrans A A identityFunctor ( U ○ F )
a5f2ca67e7c5 fix monad/adjunction definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 681
diff changeset
90 ε : NTrans B B ( F ○ U ) identityFunctor
87
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
91 isAdjunction : IsAdjunction A B U F η ε
202
58ee98bbb333 remove an extensionality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 176
diff changeset
92 U-functor = U
58ee98bbb333 remove an extensionality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 176
diff changeset
93 F-functor = F
58ee98bbb333 remove an extensionality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 176
diff changeset
94 Eta = η
58ee98bbb333 remove an extensionality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 176
diff changeset
95 Epsiron = ε
56
83ff8d48fdca add unitility
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
96
83ff8d48fdca add unitility
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
97
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
98 record IsMonad {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ)
87
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
99 ( T : Functor A A )
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
100 ( η : NTrans A A identityFunctor T )
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
101 ( μ : NTrans A A (T ○ T) T)
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
102 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ )) where
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
103 field
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
104 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) ] ]
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
105 unity1 : {a : Obj A} → A [ A [ TMap μ a o TMap η ( FObj T a ) ] ≈ Id {_} {_} {_} {A} (FObj T a) ]
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
106 unity2 : {a : Obj A} → A [ A [ TMap μ a o (FMap T (TMap η a ))] ≈ Id {_} {_} {_} {A} (FObj T a) ]
56
83ff8d48fdca add unitility
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
107
688
a5f2ca67e7c5 fix monad/adjunction definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 681
diff changeset
108 record Monad {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ)
87
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
109 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ )) where
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
110 field
688
a5f2ca67e7c5 fix monad/adjunction definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 681
diff changeset
111 T : Functor A A
a5f2ca67e7c5 fix monad/adjunction definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 681
diff changeset
112 η : NTrans A A identityFunctor T
a5f2ca67e7c5 fix monad/adjunction definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 681
diff changeset
113 μ : NTrans A A (T ○ T) T
87
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
114 isMonad : IsMonad A T η μ
130
5f331dfc000b remove Kleisli record
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 101
diff changeset
115 -- g ○ f = μ(c) T(g) f
5f331dfc000b remove Kleisli record
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 101
diff changeset
116 join : { a b : Obj A } → { c : Obj A } →
5f331dfc000b remove Kleisli record
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 101
diff changeset
117 ( Hom A b ( FObj T c )) → ( Hom A a ( FObj T b)) → Hom A a ( FObj T c )
5f331dfc000b remove Kleisli record
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 101
diff changeset
118 join {_} {_} {c} g f = A [ TMap μ c o A [ FMap T g o f ] ]
5f331dfc000b remove Kleisli record
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 101
diff changeset
119
56
83ff8d48fdca add unitility
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
120
87
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
121 Functor*Nat : {c₁ c₂ ℓ c₁' c₂' ℓ' c₁'' c₂'' ℓ'' : Level} (A : Category c₁ c₂ ℓ) {B : Category c₁' c₂' ℓ'} (C : Category c₁'' c₂'' ℓ'')
300
d6a6dd305da2 arrow and lambda fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
diff changeset
122 (F : Functor B C) → { G H : Functor A B } → ( n : NTrans A B G H ) → NTrans A C (F ○ G) (F ○ H)
87
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
123 Functor*Nat A {B} C F {G} {H} n = record {
300
d6a6dd305da2 arrow and lambda fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
diff changeset
124 TMap = λ a → FMap F (TMap n a)
87
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
125 ; isNTrans = record {
130
5f331dfc000b remove Kleisli record
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 101
diff changeset
126 commute = commute
87
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
127 }
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
128 } where
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
129 commute : {a b : Obj A} {f : Hom A a b}
87
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
130 → C [ C [ (FMap F ( FMap H f )) o ( FMap F (TMap n a)) ] ≈ C [ (FMap F (TMap n b )) o (FMap F (FMap G f)) ] ]
130
5f331dfc000b remove Kleisli record
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 101
diff changeset
131 commute {a} {b} {f} = let open ≈-Reasoning (C) in
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
132 begin
87
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
133 (FMap F ( FMap H f )) o ( FMap F (TMap n a))
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
134 ≈⟨ sym (distr F) ⟩
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
135 FMap F ( B [ (FMap H f) o TMap n a ])
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
136 ≈⟨ IsFunctor.≈-cong (isFunctor F) ( nat n ) ⟩
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
137 FMap F ( B [ (TMap n b ) o FMap G f ] )
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
138 ≈⟨ distr F ⟩
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
139 (FMap F (TMap n b )) o (FMap F (FMap G f))
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
140 ∎
56
83ff8d48fdca add unitility
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
141
87
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
142 Nat*Functor : {c₁ c₂ ℓ c₁' c₂' ℓ' c₁'' c₂'' ℓ'' : Level} (A : Category c₁ c₂ ℓ) {B : Category c₁' c₂' ℓ'} (C : Category c₁'' c₂'' ℓ'')
300
d6a6dd305da2 arrow and lambda fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
diff changeset
143 { G H : Functor B C } → ( n : NTrans B C G H ) → (F : Functor A B) → NTrans A C (G ○ F) (H ○ F)
87
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
144 Nat*Functor A {B} C {G} {H} n F = record {
300
d6a6dd305da2 arrow and lambda fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
diff changeset
145 TMap = λ a → TMap n (FObj F a)
87
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
146 ; isNTrans = record {
130
5f331dfc000b remove Kleisli record
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 101
diff changeset
147 commute = commute
87
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
148 }
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
149 } where
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
150 commute : {a b : Obj A} {f : Hom A a b}
87
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
151 → C [ C [ ( FMap H (FMap F f )) o ( TMap n (FObj F a)) ] ≈ C [ (TMap n (FObj F b )) o (FMap G (FMap F f)) ] ]
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
152 commute {a} {b} {f} = IsNTrans.commute ( isNTrans n)
56
83ff8d48fdca add unitility
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
153
87
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
154 -- T ≃ (U_R ○ F_R)
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
155 -- μ = U_R ε F_R
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
156 -- nat-ε
4690953794c4 MEsolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 86
diff changeset
157 -- nat-η -- same as η but has different types
84
ee25f96ee8cc record Resolution
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 80
diff changeset
158
688
a5f2ca67e7c5 fix monad/adjunction definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 681
diff changeset
159 record MResolution {c₁ c₂ ℓ c₁' c₂' ℓ' : Level} (A : Category c₁ c₂ ℓ) ( B : Category c₁' c₂' ℓ' ) ( M : Monad A )
a5f2ca67e7c5 fix monad/adjunction definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 681
diff changeset
160 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁' ⊔ c₂' ⊔ ℓ' )) where
a5f2ca67e7c5 fix monad/adjunction definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 681
diff changeset
161 field
a5f2ca67e7c5 fix monad/adjunction definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 681
diff changeset
162 UR : Functor B A
a5f2ca67e7c5 fix monad/adjunction definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 681
diff changeset
163 FR : Functor A B
a5f2ca67e7c5 fix monad/adjunction definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 681
diff changeset
164 ηR : NTrans A A identityFunctor ( UR ○ FR )
a5f2ca67e7c5 fix monad/adjunction definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 681
diff changeset
165 εR : NTrans B B ( FR ○ UR ) identityFunctor
a5f2ca67e7c5 fix monad/adjunction definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 681
diff changeset
166 μR : NTrans A A ( (UR ○ FR) ○ ( UR ○ FR )) ( UR ○ FR )
a5f2ca67e7c5 fix monad/adjunction definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 681
diff changeset
167 Adj : IsAdjunction A B UR FR ηR εR
a5f2ca67e7c5 fix monad/adjunction definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 681
diff changeset
168 T=UF : Monad.T M ≃ (UR ○ FR)
a5f2ca67e7c5 fix monad/adjunction definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 681
diff changeset
169 μ=UεF : {x : Obj A } → A [ TMap μR x ≈ FMap UR ( TMap εR ( FObj FR x ) ) ]
300
d6a6dd305da2 arrow and lambda fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
diff changeset
170 -- ηR=η : {x : Obj A } → A [ TMap ηR x ≈ TMap η x ] -- We need T → UR FR conversion
d6a6dd305da2 arrow and lambda fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
diff changeset
171 -- μR=μ : {x : Obj A } → A [ TMap μR x ≈ TMap μ x ]
86
be4e3b073e0d resosultion
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 84
diff changeset
172
88
419923b149ca on going
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
173
350
c483374f542b try equalizer from limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 312
diff changeset
174 --
c483374f542b try equalizer from limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 312
diff changeset
175 -- e f
c483374f542b try equalizer from limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 312
diff changeset
176 -- c -------→ a ---------→ b
c483374f542b try equalizer from limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 312
diff changeset
177 -- ^ . ---------→
c483374f542b try equalizer from limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 312
diff changeset
178 -- | . g
c483374f542b try equalizer from limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 312
diff changeset
179 -- |k .
c483374f542b try equalizer from limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 312
diff changeset
180 -- | . h
c483374f542b try equalizer from limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 312
diff changeset
181 -- d
c483374f542b try equalizer from limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 312
diff changeset
182
443
f526f4b68565 fix IsEqualizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
183 record IsEqualizer { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {c a b : Obj A} (e : Hom A c a) (f g : Hom A a b) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where
260
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
184 field
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
185 fe=ge : A [ A [ f o e ] ≈ A [ g o e ] ]
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
186 k : {d : Obj A} (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d c
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
187 ek=h : {d : Obj A} → ∀ {h : Hom A d a} → {eq : A [ A [ f o h ] ≈ A [ g o h ] ] } → A [ A [ e o k {d} h eq ] ≈ h ]
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
188 uniqueness : {d : Obj A} → ∀ {h : Hom A d a} → {eq : A [ A [ f o h ] ≈ A [ g o h ] ] } → {k' : Hom A d c } →
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
189 A [ A [ e o k' ] ≈ h ] → A [ k {d} h eq ≈ k' ]
443
f526f4b68565 fix IsEqualizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
190 equalizer1 : Hom A c a
f526f4b68565 fix IsEqualizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
191 equalizer1 = e
260
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
192
443
f526f4b68565 fix IsEqualizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
193 record Equalizer { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a b : Obj A} (f g : Hom A a b) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where
440
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
194 field
443
f526f4b68565 fix IsEqualizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
195 equalizer-c : Obj A
f526f4b68565 fix IsEqualizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
196 equalizer : Hom A equalizer-c a
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
197 isEqualizer : IsEqualizer A equalizer f g
440
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
198
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
199 --
260
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
200 -- Product
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
201 --
264
78ce12f8e6b6 pullback done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 260
diff changeset
202 -- c
78ce12f8e6b6 pullback done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 260
diff changeset
203 -- f | g
78ce12f8e6b6 pullback done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 260
diff changeset
204 -- |f×g
78ce12f8e6b6 pullback done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 260
diff changeset
205 -- v
300
d6a6dd305da2 arrow and lambda fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
diff changeset
206 -- a <-------- ab ---------→ b
264
78ce12f8e6b6 pullback done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 260
diff changeset
207 -- π1 π2
260
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
208
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
209
672
749df4959d19 fix completeness
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 669
diff changeset
210 record IsProduct { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) (a b ab : Obj A)
260
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
211 ( π1 : Hom A ab a )
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
212 ( π2 : Hom A ab b )
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
213 : Set (ℓ ⊔ (c₁ ⊔ c₂)) where
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
214 field
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
215 _×_ : {c : Obj A} ( f : Hom A c a ) → ( g : Hom A c b ) → Hom A c ab
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
216 π1fxg=f : {c : Obj A} { f : Hom A c a } → { g : Hom A c b } → A [ A [ π1 o ( f × g ) ] ≈ f ]
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
217 π2fxg=g : {c : Obj A} { f : Hom A c a } → { g : Hom A c b } → A [ A [ π2 o ( f × g ) ] ≈ g ]
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
218 uniqueness : {c : Obj A} { h : Hom A c ab } → A [ ( A [ π1 o h ] ) × ( A [ π2 o h ] ) ≈ h ]
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
219 ×-cong : {c : Obj A} { f f' : Hom A c a } → { g g' : Hom A c b } → A [ f ≈ f' ] → A [ g ≈ g' ] → A [ f × g ≈ f' × g' ]
440
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
220
681
bd8f7346f252 fix Product and pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 676
diff changeset
221 record Product { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) ( a b : Obj A )
440
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
222 : Set (ℓ ⊔ (c₁ ⊔ c₂)) where
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
223 field
681
bd8f7346f252 fix Product and pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 676
diff changeset
224 product : Obj A
bd8f7346f252 fix Product and pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 676
diff changeset
225 π1 : Hom A product a
bd8f7346f252 fix Product and pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 676
diff changeset
226 π2 : Hom A product b
bd8f7346f252 fix Product and pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 676
diff changeset
227 isProduct : IsProduct A a b product π1 π2
260
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
228
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
229 -----
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
230 --
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
231 -- product on arbitrary index
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
232 --
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
233
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 690
diff changeset
234 record IsIProduct { c c₁ c₂ ℓ : Level} ( I : Set c) ( A : Category c₁ c₂ ℓ )
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
235 ( p : Obj A ) -- product
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
236 ( ai : I → Obj A ) -- families
508
3ce21b2a671a IProduct is written in Sets
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 495
diff changeset
237 ( pi : (i : I ) → Hom A p ( ai i ) ) -- projections
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
238 : Set (c ⊔ ℓ ⊔ (c₁ ⊔ c₂)) where
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
239 field
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
240 iproduct : {q : Obj A} → ( qi : (i : I) → Hom A q (ai i) ) → Hom A q p
676
faf48511f69d two product as in CWM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 673
diff changeset
241 pif=q : {q : Obj A} → { qi : (i : I) → Hom A q (ai i) }
faf48511f69d two product as in CWM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 673
diff changeset
242 → ∀ { i : I } → A [ A [ ( pi i ) o ( iproduct qi ) ] ≈ (qi i) ]
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
243 ip-uniqueness : {q : Obj A} { h : Hom A q p } → A [ iproduct ( λ (i : I) → A [ (pi i) o h ] ) ≈ h ]
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
244 ip-cong : {q : Obj A} → { qi : (i : I) → Hom A q (ai i) } → { qi' : (i : I) → Hom A q (ai i) }
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
245 → ( ∀ (i : I ) → A [ qi i ≈ qi' i ] ) → A [ iproduct qi ≈ iproduct qi' ]
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
246 -- another form of uniquness
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
247 ip-uniqueness1 : {q : Obj A} → ( qi : (i : I) → Hom A q (ai i) ) → ( product' : Hom A q p )
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
248 → ( ∀ { i : I } → A [ A [ ( pi i ) o product' ] ≈ (qi i) ] )
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
249 → A [ product' ≈ iproduct qi ]
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
250 ip-uniqueness1 {a} qi product' eq = let open ≈-Reasoning (A) in begin
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
251 product'
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
252 ≈↑⟨ ip-uniqueness ⟩
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
253 iproduct (λ i₁ → A [ pi i₁ o product' ])
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
254 ≈⟨ ip-cong ( λ i → begin
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
255 pi i o product'
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
256 ≈⟨ eq {i} ⟩
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
257 qi i
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
258 ∎ ) ⟩
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
259 iproduct qi
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
260 ∎
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
261
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 690
diff changeset
262 record IProduct { c c₁ c₂ ℓ : Level} ( I : Set c) ( A : Category c₁ c₂ ℓ ) (ai : I → Obj A) : Set (c ⊔ ℓ ⊔ (c₁ ⊔ c₂)) where
508
3ce21b2a671a IProduct is written in Sets
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 495
diff changeset
263 field
3ce21b2a671a IProduct is written in Sets
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 495
diff changeset
264 iprod : Obj A
3ce21b2a671a IProduct is written in Sets
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 495
diff changeset
265 pi : (i : I ) → Hom A iprod ( ai i )
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 690
diff changeset
266 isIProduct : IsIProduct I A iprod ai pi
508
3ce21b2a671a IProduct is written in Sets
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 495
diff changeset
267
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
268
260
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
269 -- Pullback
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
270 -- f
300
d6a6dd305da2 arrow and lambda fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
diff changeset
271 -- a ------→ c
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
272 -- ^ ^
260
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
273 -- π1 | |g
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
274 -- | |
300
d6a6dd305da2 arrow and lambda fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
diff changeset
275 -- ab ------→ b
260
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
276 -- ^ π2
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
277 -- |
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
278 -- d
260
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
279 --
681
bd8f7346f252 fix Product and pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 676
diff changeset
280 record IsPullback { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a b c ab : Obj A}
260
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
281 ( f : Hom A a c ) ( g : Hom A b c )
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
282 ( π1 : Hom A ab a ) ( π2 : Hom A ab b )
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
283 : Set (ℓ ⊔ (c₁ ⊔ c₂)) where
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
284 field
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
285 commute : A [ A [ f o π1 ] ≈ A [ g o π2 ] ]
681
bd8f7346f252 fix Product and pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 676
diff changeset
286 pullback : { d : Obj A } → { π1' : Hom A d a } { π2' : Hom A d b } → A [ A [ f o π1' ] ≈ A [ g o π2' ] ] → Hom A d ab
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
287 π1p=π1 : { d : Obj A } → { π1' : Hom A d a } { π2' : Hom A d b } → { eq : A [ A [ f o π1' ] ≈ A [ g o π2' ] ] }
681
bd8f7346f252 fix Product and pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 676
diff changeset
288 → A [ A [ π1 o pullback eq ] ≈ π1' ]
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
289 π2p=π2 : { d : Obj A } → { π1' : Hom A d a } { π2' : Hom A d b } → { eq : A [ A [ f o π1' ] ≈ A [ g o π2' ] ] }
681
bd8f7346f252 fix Product and pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 676
diff changeset
290 → A [ A [ π2 o pullback eq ] ≈ π2' ]
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
291 uniqueness : { d : Obj A } → ( p' : Hom A d ab ) → { π1' : Hom A d a } { π2' : Hom A d b } → { eq : A [ A [ f o π1' ] ≈ A [ g o π2' ] ] }
260
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
292 → { π1p=π1' : A [ A [ π1 o p' ] ≈ π1' ] }
a87d3ea9efe4 pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
293 → { π2p=π2' : A [ A [ π2 o p' ] ≈ π2' ] }
681
bd8f7346f252 fix Product and pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 676
diff changeset
294 → A [ pullback eq ≈ p' ]
bd8f7346f252 fix Product and pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 676
diff changeset
295
bd8f7346f252 fix Product and pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 676
diff changeset
296 record Pullback { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a b c : Obj A}
bd8f7346f252 fix Product and pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 676
diff changeset
297 ( f : Hom A a c ) ( g : Hom A b c )
bd8f7346f252 fix Product and pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 676
diff changeset
298 : Set (ℓ ⊔ (c₁ ⊔ c₂)) where
bd8f7346f252 fix Product and pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 676
diff changeset
299 field
bd8f7346f252 fix Product and pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 676
diff changeset
300 ab : Obj A
bd8f7346f252 fix Product and pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 676
diff changeset
301 π1 : Hom A ab a
bd8f7346f252 fix Product and pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 676
diff changeset
302 π2 : Hom A ab b
bd8f7346f252 fix Product and pullback
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 676
diff changeset
303 isPullback : IsPullback A f g π1 π2
312
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 300
diff changeset
304
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 300
diff changeset
305 --
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 300
diff changeset
306 -- Limit
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 300
diff changeset
307 --
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 300
diff changeset
308 -----
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 300
diff changeset
309
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 300
diff changeset
310 -- Constancy Functor
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 300
diff changeset
311
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 690
diff changeset
312 K : { c₁' c₂' ℓ' : Level} (I : Category c₁' c₂' ℓ') { c₁'' c₂'' ℓ'' : Level} ( A : Category c₁'' c₂'' ℓ'' )
312
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 300
diff changeset
313 → ( a : Obj A ) → Functor I A
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 690
diff changeset
314 K I A a = record {
312
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 300
diff changeset
315 FObj = λ i → a ;
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 300
diff changeset
316 FMap = λ f → id1 A a ;
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 300
diff changeset
317 isFunctor = let open ≈-Reasoning (A) in record {
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 300
diff changeset
318 ≈-cong = λ f=g → refl-hom
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 300
diff changeset
319 ; identity = refl-hom
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 300
diff changeset
320 ; distr = sym idL
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 300
diff changeset
321 }
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 300
diff changeset
322 }
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 300
diff changeset
323
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 300
diff changeset
324
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 690
diff changeset
325 record IsLimit { c₁ c₂ ℓ : Level} { c₁' c₂' ℓ' : Level} ( I : Category c₁ c₂ ℓ ) ( A : Category c₁' c₂' ℓ' ) ( Γ : Functor I A )
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 690
diff changeset
326 (a0 : Obj A ) (t0 : NTrans I A ( K I A a0 ) Γ ) : Set (suc (c₁' ⊔ c₂' ⊔ ℓ' ⊔ c₁ ⊔ c₂ ⊔ ℓ )) where
312
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 300
diff changeset
327 field
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 690
diff changeset
328 limit : ( a : Obj A ) → ( t : NTrans I A ( K I A a ) Γ ) → Hom A a a0
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 690
diff changeset
329 t0f=t : { a : Obj A } → { t : NTrans I A ( K I A a ) Γ } → ∀ { i : Obj I } →
312
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 300
diff changeset
330 A [ A [ TMap t0 i o limit a t ] ≈ TMap t i ]
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 690
diff changeset
331 limit-uniqueness : { a : Obj A } → { t : NTrans I A ( K I A a ) Γ } → { f : Hom A a a0 } → ( ∀ { i : Obj I } →
312
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 300
diff changeset
332 A [ A [ TMap t0 i o f ] ≈ TMap t i ] ) → A [ limit a t ≈ f ]
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 300
diff changeset
333
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 690
diff changeset
334 record Limit { c₁ c₂ ℓ : Level} { c₁' c₂' ℓ' : Level} ( I : Category c₁ c₂ ℓ ) ( A : Category c₁' c₂' ℓ' ) ( Γ : Functor I A )
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
335 : Set (suc (c₁' ⊔ c₂' ⊔ ℓ' ⊔ c₁ ⊔ c₂ ⊔ ℓ )) where
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
336 field
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
337 a0 : Obj A
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 690
diff changeset
338 t0 : NTrans I A ( K I A a0 ) Γ
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 690
diff changeset
339 isLimit : IsLimit I A Γ a0 t0
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
340
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 690
diff changeset
341 LimitNat : { c₁' c₂' ℓ' : Level} (I : Category c₁' c₂' ℓ') { c₁ c₂ ℓ : Level} ( B : Category c₁ c₂ ℓ ) { c₁'' c₂'' ℓ'' : Level}
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
342 ( C : Category c₁'' c₂'' ℓ'' )
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
343 ( Γ : Functor I B )
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 690
diff changeset
344 ( lim : Obj B ) ( tb : NTrans I B ( K I B lim ) Γ ) →
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 690
diff changeset
345 ( U : Functor B C) → NTrans I C ( K I C (FObj U lim) ) (U ○ Γ)
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 690
diff changeset
346 LimitNat I B C Γ lim tb U = record {
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
347 TMap = TMap (Functor*Nat I C U tb) ;
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
348 isNTrans = record { commute = λ {a} {b} {f} → let open ≈-Reasoning (C) in begin
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
349 FMap (U ○ Γ) f o TMap (Functor*Nat I C U tb) a
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
350 ≈⟨ nat ( Functor*Nat I C U tb ) ⟩
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 690
diff changeset
351 TMap (Functor*Nat I C U tb) b o FMap (U ○ K I B lim) f
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
352 ≈⟨ cdr (IsFunctor.identity (isFunctor U) ) ⟩
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 690
diff changeset
353 TMap (Functor*Nat I C U tb) b o FMap (K I C (FObj U lim)) f
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
354 ∎
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
355 } }
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
356
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
357 open Limit
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 690
diff changeset
358 record LimitPreserve { c₁ c₂ ℓ : Level} { c₁' c₂' ℓ' : Level} ( I : Category c₁ c₂ ℓ ) ( A : Category c₁' c₂' ℓ' )
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
359 { c₁'' c₂'' ℓ'' : Level} ( C : Category c₁'' c₂'' ℓ'' )
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
360 (G : Functor A C) : Set (suc (c₁' ⊔ c₂' ⊔ ℓ' ⊔ c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁'' ⊔ c₂'' ⊔ ℓ'' )) where
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
361 field
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 690
diff changeset
362 preserve : ( Γ : Functor I A ) → ( limita : Limit I A Γ ) →
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 690
diff changeset
363 IsLimit I C (G ○ Γ) (FObj G (a0 limita ) ) (LimitNat I A C Γ (a0 limita ) (t0 limita) G )
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 690
diff changeset
364 plimit : { Γ : Functor I A } → ( limita : Limit I A Γ ) → Limit I C (G ○ Γ )
492
c7b8017bcd4d on going..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
365 plimit {Γ} limita = record { a0 = (FObj G (a0 limita ))
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 690
diff changeset
366 ; t0 = LimitNat I A C Γ (a0 limita ) (t0 limita) G
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
367 ; isLimit = preserve Γ limita }
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
368
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
369 record Complete { c₁' c₂' ℓ' : Level} { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) ( I : Category c₁' c₂' ℓ' )
440
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
370 : Set (suc (c₁' ⊔ c₂' ⊔ ℓ' ⊔ c₁ ⊔ c₂ ⊔ ℓ )) where
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
371 field
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 690
diff changeset
372 climit : ( Γ : Functor I A ) → Limit I A Γ
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 690
diff changeset
373 cproduct : ( I : Set c₁' ) (fi : I → Obj A ) → IProduct I A fi -- c₁ should be a free level
672
749df4959d19 fix completeness
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 669
diff changeset
374 cequalizer : {a b : Obj A} (f g : Hom A a b) → Equalizer A f g
484
fcae3025d900 fix Limit pu a0 and t0 in record definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
375 open Limit
672
749df4959d19 fix completeness
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 669
diff changeset
376 limit-c : ( Γ : Functor I A ) → Obj A
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
377 limit-c Γ = a0 ( climit Γ)
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 690
diff changeset
378 limit-u : ( Γ : Functor I A ) → NTrans I A ( K I A (limit-c Γ )) Γ
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
379 limit-u Γ = t0 ( climit Γ)
672
749df4959d19 fix completeness
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 669
diff changeset
380 open Equalizer
749df4959d19 fix completeness
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 669
diff changeset
381 equalizer-p : {a b : Obj A} (f g : Hom A a b) → Obj A
749df4959d19 fix completeness
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 669
diff changeset
382 equalizer-p f g = equalizer-c (cequalizer f g )
749df4959d19 fix completeness
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 669
diff changeset
383 equalizer-e : {a b : Obj A} (f g : Hom A a b) → Hom A (equalizer-p f g) a
749df4959d19 fix completeness
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 669
diff changeset
384 equalizer-e f g = equalizer (cequalizer f g )
526
b035cd3be525 Small Category for Sets Limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 508
diff changeset
385
662
e1d54c0f73a7 move InitialObject to cat-utility
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 526
diff changeset
386
e1d54c0f73a7 move InitialObject to cat-utility
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 526
diff changeset
387 -- initial object
e1d54c0f73a7 move InitialObject to cat-utility
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 526
diff changeset
388
695
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 691
diff changeset
389 record HasInitialObject {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (i : Obj A) : Set (suc ℓ ⊔ (suc c₁ ⊔ suc c₂)) where
662
e1d54c0f73a7 move InitialObject to cat-utility
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 526
diff changeset
390 field
e1d54c0f73a7 move InitialObject to cat-utility
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 526
diff changeset
391 initial : ∀( a : Obj A ) → Hom A i a
e1d54c0f73a7 move InitialObject to cat-utility
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 526
diff changeset
392 uniqueness : { a : Obj A } → ( f : Hom A i a ) → A [ f ≈ initial a ]
e1d54c0f73a7 move InitialObject to cat-utility
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 526
diff changeset
393
688
a5f2ca67e7c5 fix monad/adjunction definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 681
diff changeset
394 record InitialObject {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) : Set (suc ℓ ⊔ (suc c₁ ⊔ suc c₂)) where
a5f2ca67e7c5 fix monad/adjunction definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 681
diff changeset
395 field
a5f2ca67e7c5 fix monad/adjunction definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 681
diff changeset
396 initialObject : Obj A
695
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 691
diff changeset
397 hasInitialObject : HasInitialObject A initialObject
688
a5f2ca67e7c5 fix monad/adjunction definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 681
diff changeset
398