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annotate limit-to.agda @ 472:f3d6d0275a0a

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discrete equality as a dom equality product from limit done
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Tue, 07 Mar 2017 03:21:46 +0900
parents 65ab0da524b8
children 2d32ded94aaf
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
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dd086f5fb29f same conflict again ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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1 open import Category -- https://github.com/konn/category-agda
350
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2 open import Level
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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3
403
375edfefbf6a maybe CAT
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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4 module limit-to where
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5
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6 open import cat-utility
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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7 open import HomReasoning
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8 open import Relation.Binary.Core
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9
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10 open import discrete
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531b739a1d7c add Nil
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11
365
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12
469
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
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13 --- Equalizer from Limit ( 2->A IdnexFunctor Γ and IndexNat : K ->　Γ)
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14 ---
fd79b6d9f350 fix comment
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15 ---
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16 --- f
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17 --- e -----→
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18 --- c -----→ a b A
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19 --- ^ / -----→
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20 --- |k h g
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dd086f5fb29f same conflict again ...
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21 --- | /
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1290d6876129 Functor MA → A
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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22 --- | / ^
1290d6876129 Functor MA → A
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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23 --- | / |
1290d6876129 Functor MA → A
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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24 --- |/ | Γ
1290d6876129 Functor MA → A
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 425
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25 --- d _ |
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26 --- |\ |
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27 --- \ K af
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28 --- \ -----→
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29 --- \ t0 t1 I
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30 --- -----→
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1290d6876129 Functor MA → A
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31 --- ag
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32 ---
c375d8f93a2c discrete category and product from a limit
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33 ---
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2f59c2db9d98 what is this ?
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34
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35 open Complete
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36 open Limit
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37 open NTrans
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38
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39 -- Functor Γ : TwoCat -> A
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40
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41 IndexFunctor : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) ( a b : Obj A) ( f g : Hom A a b ) → Functor (TwoCat {c₁} {c₂}) A
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parents: 443
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42 IndexFunctor {c₁} {c₂} {ℓ} A a b f g = record {
0ba86e29f492 limit-to and discrete clean up
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43 FObj = λ a → fobj a
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44 ; FMap = λ {a} {b} f → fmap {a} {b} f
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parents: 443
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45 ; isFunctor = record {
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46 identity = λ{x} → identity x
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47 ; distr = λ {a} {b} {c} {f} {g} → distr1 {a} {b} {c} {f} {g}
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parents: 443
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48 ; ≈-cong = λ {a} {b} {c} {f} → ≈-cong {a} {b} {c} {f}
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49 }
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50 } where
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51 T = TwoCat {c₁} {c₂}
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52 fobj : Obj T → Obj A
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53 fobj t0 = a
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54 fobj t1 = b
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55 fmap : {x y : Obj T } → (Hom T x y ) → Hom A (fobj x) (fobj y)
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parents: 443
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56 fmap {t0} {t0} id-t0 = id1 A a
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57 fmap {t1} {t1} id-t1 = id1 A b
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parents: 443
diff changeset
58 fmap {t0} {t1} arrow-f = f
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parents: 443
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59 fmap {t0} {t1} arrow-g = g
0ba86e29f492 limit-to and discrete clean up
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parents: 443
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60 ≈-cong : {a : Obj T} {b : Obj T} {f g : Hom T a b} → T [ f ≈ g ] → A [ fmap f ≈ fmap g ]
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parents: 443
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61 ≈-cong {a} {b} {f} {.f} refl = let open ≈-Reasoning A in refl-hom
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62 identity : (x : Obj T ) -> A [ fmap (id1 T x) ≈ id1 A (fobj x) ]
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parents: 443
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63 identity t0 = let open ≈-Reasoning A in refl-hom
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parents: 443
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64 identity t1 = let open ≈-Reasoning A in refl-hom
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65 distr1 : {a : Obj T} {b : Obj T} {c : Obj T} {f : Hom T a b} {g : Hom T b c} →
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66 A [ fmap (T [ g o f ]) ≈ A [ fmap g o fmap f ] ]
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67 distr1 {t0} {t0} {t0} {id-t0 } { id-t0 } = let open ≈-Reasoning A in sym-hom idL
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parents: 443
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68 distr1 {t1} {t1} {t1} { id-t1 } { id-t1 } = let open ≈-Reasoning A in begin
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69 id b
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70 ≈↑⟨ idL ⟩
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71 id b o id b
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72 ∎
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73 distr1 {t0} {t0} {t1} { id-t0 } { arrow-f } = let open ≈-Reasoning A in begin
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parents: 461
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74 fmap (T [ arrow-f o id-t0 ] )
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75 ≈⟨⟩
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76 fmap arrow-f
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77 ≈↑⟨ idR ⟩
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78 fmap arrow-f o id a
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79 ∎
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80 distr1 {t0} {t0} {t1} { id-t0 } { arrow-g } = let open ≈-Reasoning A in begin
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81 fmap (T [ arrow-g o id-t0 ] )
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82 ≈⟨⟩
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83 fmap arrow-g
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84 ≈↑⟨ idR ⟩
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85 fmap arrow-g o id a
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86 ∎
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87 distr1 {t0} {t1} {t1} { arrow-f } { id-t1 } = let open ≈-Reasoning A in begin
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88 fmap (T [ id-t1 o arrow-f ] )
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89 ≈⟨⟩
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90 fmap arrow-f
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91 ≈↑⟨ idL ⟩
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92 id b o fmap arrow-f
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93 ∎
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94 distr1 {t0} {t1} {t1} { arrow-g } { id-t1 } = let open ≈-Reasoning A in begin
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95 fmap (T [ id-t1 o arrow-g ] )
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96 ≈⟨⟩
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97 fmap arrow-g
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98 ≈↑⟨ idL ⟩
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99 id b o fmap arrow-g
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100 ∎
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101
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102 --- Nat for Limit
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103 --
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104 -- Nat : K -> IndexFunctor
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105 --
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106
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107 open Functor
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108
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109 IndexNat : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ)
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110 → {a b : Obj A} (f g : Hom A a b )
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parents: 467
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111 (d : Obj A) → (h : Hom A d a ) → A [ A [ f o h ] ≈ A [ g o h ] ] →
461
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parents: 460
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112 NTrans TwoCat A (K A TwoCat d) (IndexFunctor {c₁} {c₂} {ℓ} A a b f g)
460
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parents: 458
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113 IndexNat {c₁} {c₂} {ℓ} A {a} {b} f g d h fh=gh = record {
457
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parents: 443
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114 TMap = λ x → nmap x d h ;
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115 isNTrans = record {
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116 commute = λ {x} {y} {f'} → commute1 {x} {y} {f'} d h fh=gh
457
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117 }
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118 } where
461
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parents: 460
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119 I = TwoCat {c₁} {c₂}
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02eefa110eae nat commute in limit-to
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parents: 428
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120 Γ : Functor I A
457
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121 Γ = IndexFunctor {c₁} {c₂} {ℓ} A a b f g
431
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parents: 430
diff changeset
122 nmap : (x : Obj I ) ( d : Obj (A) ) (h : Hom A d a ) → Hom A (FObj (K A I d) x) (FObj Γ x)
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
123 nmap t0 d h = h
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
124 nmap t1 d h = A [ f o h ]
431
a15f52456c78 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 430
diff changeset
125 commute1 : {x y : Obj I} {f' : Hom I x y} (d : Obj A) (h : Hom A d a ) → A [ A [ f o h ] ≈ A [ g o h ] ]
427
531b739a1d7c add Nil
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 426
diff changeset
126 → A [ A [ FMap Γ f' o nmap x d h ] ≈ A [ nmap y d h o FMap (K A I d) f' ] ]
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
127 commute1 {t0} {t1} {arrow-f} d h fh=gh = let open ≈-Reasoning A in begin
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
128 f o h
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
129 ≈↑⟨ idR ⟩
467
ba042c2d3ff5 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 466
diff changeset
130 (f o h ) o id d
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
131 ∎
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
132 commute1 {t0} {t1} {arrow-g} d h fh=gh = let open ≈-Reasoning A in begin
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
133 g o h
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
134 ≈↑⟨ fh=gh ⟩
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
135 f o h
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
136 ≈↑⟨ idR ⟩
467
ba042c2d3ff5 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 466
diff changeset
137 (f o h ) o id d
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
138 ∎
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
139 commute1 {t0} {t0} {id-t0} d h fh=gh = let open ≈-Reasoning A in begin
467
ba042c2d3ff5 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 466
diff changeset
140 id a o h
429
02eefa110eae nat commute in limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 428
diff changeset
141 ≈⟨ idL ⟩
02eefa110eae nat commute in limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 428
diff changeset
142 h
02eefa110eae nat commute in limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 428
diff changeset
143 ≈↑⟨ idR ⟩
467
ba042c2d3ff5 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 466
diff changeset
144 h o id d
429
02eefa110eae nat commute in limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 428
diff changeset
145 ∎
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
146 commute1 {t1} {t1} {id-t1} d h fh=gh = let open ≈-Reasoning A in begin
467
ba042c2d3ff5 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 466
diff changeset
147 id b o ( f o h )
429
02eefa110eae nat commute in limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 428
diff changeset
148 ≈⟨ idL ⟩
428
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 427
diff changeset
149 f o h
429
02eefa110eae nat commute in limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 428
diff changeset
150 ≈↑⟨ idR ⟩
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
151 ( f o h ) o id d
428
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 427
diff changeset
152 ∎
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
153
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
154
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
155 equlimit : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) {a b : Obj A} -> (f g : Hom A a b) (comp : Complete A TwoCat ) ->
461
8436a018f88a clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
156 Hom A ( limit-c comp (IndexFunctor A a b f g)) a
8436a018f88a clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
157 equlimit A {a} {b} f g comp = TMap (limit-u comp (IndexFunctor A a b f g)) t0
460
961c236807f1 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 458
diff changeset
158
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
159 lim-to-equ : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ)
461
8436a018f88a clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
160 (comp : Complete A TwoCat )
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
161 → {a b : Obj A} (f g : Hom A a b )
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
162 → (fe=ge : A [ A [ f o (equlimit A f g comp) ] ≈ A [ g o (equlimit A f g comp) ] ] )
460
961c236807f1 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 458
diff changeset
163 → IsEqualizer A (equlimit A f g comp) f g
961c236807f1 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 458
diff changeset
164 lim-to-equ {c₁} {c₂} {ℓ} A comp {a} {b} f g fe=ge = record {
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
165 fe=ge = fe=ge
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
166 ; k = λ {d} h fh=gh → k {d} h fh=gh
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
167 ; ek=h = λ {d} {h} {fh=gh} → ek=h d h fh=gh
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
168 ; uniqueness = λ {d} {h} {fh=gh} {k'} → uniquness d h fh=gh k'
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
169 } where
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
170 I : Category c₁ c₂ c₂
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
171 I = TwoCat
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
172 Γ : Functor I A
461
8436a018f88a clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
173 Γ = IndexFunctor A a b f g
8436a018f88a clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
174 e : Hom A (limit-c comp (IndexFunctor A a b f g)) a
460
961c236807f1 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 458
diff changeset
175 e = equlimit A f g comp
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
176 c : Obj A
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
177 c = limit-c comp Γ
466
44bd77c80555 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 462
diff changeset
178 lim : Limit A I Γ c ( limit-u comp Γ )
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
179 lim = isLimit comp Γ
466
44bd77c80555 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 462
diff changeset
180 inat : (d : Obj A) (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → NTrans I A (K A I d) Γ
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
181 inat = IndexNat A f g
350
c483374f542b try equalizer from limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
182 k : {d : Obj A} (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d c
466
44bd77c80555 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 462
diff changeset
183 k {d} h fh=gh = limit lim d (inat d h fh=gh )
431
a15f52456c78 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 430
diff changeset
184 ek=h : (d : Obj A ) (h : Hom A d a ) → ( fh=gh : A [ A [ f o h ] ≈ A [ g o h ] ] ) → A [ A [ e o k h fh=gh ] ≈ h ]
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
185 ek=h d h fh=gh = let open ≈-Reasoning A in begin
430
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
186 e o k h fh=gh
460
961c236807f1 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 458
diff changeset
187 ≈⟨⟩
961c236807f1 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 458
diff changeset
188 TMap (limit-u comp Γ) t0 o k h fh=gh
466
44bd77c80555 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 462
diff changeset
189 ≈⟨ t0f=t lim {d} {inat d h fh=gh } {t0} ⟩
44bd77c80555 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 462
diff changeset
190 TMap (inat d h fh=gh) t0
460
961c236807f1 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 458
diff changeset
191 ≈⟨⟩
430
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
192 h
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
193 ∎
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
194 uniq-nat : {i : Obj I} → (d : Obj A ) (h : Hom A d a ) ( k' : Hom A d c )
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
195 ( fh=gh : A [ A [ f o h ] ≈ A [ g o h ] ]) → A [ A [ e o k' ] ≈ h ] →
466
44bd77c80555 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 462
diff changeset
196 A [ A [ TMap (limit-u comp Γ) i o k' ] ≈ TMap (inat d h fh=gh) i ]
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
197 uniq-nat {t0} d h k' fh=gh ek'=h = let open ≈-Reasoning A in begin
460
961c236807f1 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 458
diff changeset
198 TMap (limit-u comp Γ) t0 o k'
430
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
199 ≈⟨⟩
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
200 e o k'
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
201 ≈⟨ ek'=h ⟩
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
202 h
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
203 ≈⟨⟩
466
44bd77c80555 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 462
diff changeset
204 TMap (inat d h fh=gh) t0
430
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
205 ∎
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
206 uniq-nat {t1} d h k' fh=gh ek'=h = let open ≈-Reasoning A in begin
460
961c236807f1 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 458
diff changeset
207 TMap (limit-u comp Γ) t1 o k'
961c236807f1 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 458
diff changeset
208 ≈↑⟨ car (idR) ⟩
467
ba042c2d3ff5 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 466
diff changeset
209 ( TMap (limit-u comp Γ) t1 o id c ) o k'
460
961c236807f1 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 458
diff changeset
210 ≈⟨⟩
961c236807f1 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 458
diff changeset
211 ( TMap (limit-u comp Γ) t1 o FMap (K A I c) arrow-f ) o k'
961c236807f1 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 458
diff changeset
212 ≈↑⟨ car ( nat1 (limit-u comp Γ) arrow-f ) ⟩
961c236807f1 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 458
diff changeset
213 ( FMap Γ arrow-f o TMap (limit-u comp Γ) t0 ) o k'
430
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
214 ≈⟨⟩
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
215 (f o e ) o k'
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
216 ≈↑⟨ assoc ⟩
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
217 f o ( e o k' )
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
218 ≈⟨ cdr ek'=h ⟩
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
219 f o h
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
220 ≈⟨⟩
466
44bd77c80555 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 462
diff changeset
221 TMap (inat d h fh=gh) t1
430
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
222 ∎
431
a15f52456c78 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 430
diff changeset
223 uniquness : (d : Obj A ) (h : Hom A d a ) → ( fh=gh : A [ A [ f o h ] ≈ A [ g o h ] ] ) →
372
b4855a3ebd34 add more lemma in limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 371
diff changeset
224 ( k' : Hom A d c )
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
225 → A [ A [ e o k' ] ≈ h ] → A [ k h fh=gh ≈ k' ]
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
226 uniquness d h fh=gh k' ek'=h = let open ≈-Reasoning A in begin
430
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
227 k h fh=gh
461
8436a018f88a clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
228 ≈⟨ limit-uniqueness lim k' ( λ{i} → uniq-nat {i} d h k' fh=gh ek'=h ) ⟩
430
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
229 k'
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
230 ∎
368
b18585089d2e add more parameter to nat in lim-to-equ
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 367
diff changeset
231
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
232
469
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
233 --- Product from Limit ( given Discrete->A functor Γ and pnat : K ->　Γ)
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
234
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
235 open DiscreteHom
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
236
469
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
237 plimit : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (S : Set c₁) (comp : Complete A ( DiscreteCat S ) )
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
238 → ( Γ : Functor (DiscreteCat S ) A ) → Obj A
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
239 plimit A S comp Γ = limit-c comp Γ
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
240
472
f3d6d0275a0a discrete equality as a dom equality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 469
diff changeset
241 discrete-identity : { c₁ : Level} { S : Set c₁} { a : DiscreteObj {c₁} S } → (f : DiscreteHom a a ) → (DiscreteCat S) [ f ≈ id1 (DiscreteCat S) a ]
f3d6d0275a0a discrete equality as a dom equality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 469
diff changeset
242 discrete-identity f = refl
f3d6d0275a0a discrete equality as a dom equality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 469
diff changeset
243
469
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
244 pnat : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (S : Set c₁) (comp : Complete A ( DiscreteCat S ) )
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
245 → (Γ : Functor (DiscreteCat S ) A )
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
246 → (q : Obj A ) ( qi : (i : Obj ( DiscreteCat S)) → Hom A q (FObj Γ i) )
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
247 → NTrans (DiscreteCat S )A (K A (DiscreteCat S) q) Γ
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
248 pnat A S comp Γ q qi = record {
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
249 TMap = qi ;
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
250 isNTrans = record { commute = λ {a} {b} {f} → commute {a} {b} {f} }
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
251 } where
469
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
252 commute : {a b : Obj (DiscreteCat S) } {f : Hom ( DiscreteCat S) a b} →
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
253 A [ A [ FMap Γ f o qi a ] ≈
469
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
254 A [ qi b o FMap (K A (DiscreteCat S )q) f ] ]
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
255 commute {a} {b} {f} with discrete f
472
f3d6d0275a0a discrete equality as a dom equality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 469
diff changeset
256 commute {a} {.a} {f} | refl = let open ≈-Reasoning A in begin
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
257 FMap Γ f o qi a
472
f3d6d0275a0a discrete equality as a dom equality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 469
diff changeset
258 ≈⟨ car ( fcong Γ (discrete-identity f )) ⟩
f3d6d0275a0a discrete equality as a dom equality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 469
diff changeset
259 FMap Γ (id1 (DiscreteCat S) a ) o qi a
f3d6d0275a0a discrete equality as a dom equality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 469
diff changeset
260 ≈⟨ car ( IsFunctor.identity (isFunctor Γ) ) ⟩
f3d6d0275a0a discrete equality as a dom equality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 469
diff changeset
261 id1 A (FObj Γ a) o qi a
f3d6d0275a0a discrete equality as a dom equality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 469
diff changeset
262 ≈⟨ idL ⟩
f3d6d0275a0a discrete equality as a dom equality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 469
diff changeset
263 qi a
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
264 ≈↑⟨ idR ⟩
472
f3d6d0275a0a discrete equality as a dom equality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 469
diff changeset
265 qi a o id q
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
266 ≈⟨⟩
472
f3d6d0275a0a discrete equality as a dom equality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 469
diff changeset
267 qi a o FMap (K A (DiscreteCat S) q) f
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
268 ∎
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
269
469
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
270 lim-to-product : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) ( S : Set c₁ )
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
271 (comp : Complete A ( DiscreteCat S ) )
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
272 → ( Γ : Functor (DiscreteCat S ) A )
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
273 → IProduct A (Obj ( DiscreteCat S ) )(plimit A S comp Γ) (λ i → FObj Γ i ) ( λ i → TMap (limit-u comp Γ) i )
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
274 lim-to-product A S comp Γ = record {
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
275 iproduct = λ {q} qi → iproduct {q} qi ;
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
276 pif=q = λ {q } qi { i } → pif=q {q} qi {i} ;
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
277 ip-uniqueness = λ {q } { h } → ip-uniqueness {q} {h} ;
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
278 ip-cong = λ {q } { qi } { qi' } qi=qi' → ip-cong {q} {qi} {qi'} qi=qi'
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
279 } where
469
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
280 D = DiscreteCat S
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
281 I = Obj ( DiscreteCat S )
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
282 lim = isLimit comp Γ
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
283 ai = λ i → FObj Γ i
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
284 p = limit-c comp Γ
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
285 pi = λ i → TMap (limit-u comp Γ) i
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
286 iproduct : {q : Obj A} → ( qi : (i : I) → Hom A q (ai i) ) → Hom A q p
469
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
287 iproduct {q} qi = limit lim q (pnat A S comp Γ q qi )
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
288 pif=q : {q : Obj A} → ( qi : (i : I) → Hom A q (ai i) ) → ∀ { i : I } → A [ A [ ( pi i ) o ( iproduct qi ) ] ≈ (qi i) ]
469
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
289 pif=q {q} qi {i} = t0f=t lim {q} {pnat A S comp Γ q qi } {i}
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
290 ipu : {i : Obj D} → (q : Obj A) (h : Hom A q p ) → A [ A [ TMap (limit-u comp Γ) i o h ] ≈ A [ pi i o h ] ]
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
291 ipu {i} q h = let open ≈-Reasoning A in refl-hom
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
292 ip-uniqueness : {q : Obj A} { h : Hom A q p } → A [ iproduct ( λ (i : I) → A [ (pi i) o h ] ) ≈ h ]
469
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
293 ip-uniqueness {q} {h} = limit-uniqueness lim {q} {pnat A S comp Γ q (λ i → A [ pi i o h ] )} h (ipu q h)
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
294 ipc : {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: 467
diff changeset
295 → (i : I ) → A [ qi i ≈ qi' i ] →
469
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
296 A [ A [ TMap (limit-u comp Γ) i o iproduct qi' ] ≈ TMap (pnat A S comp Γ q qi) i ]
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
297 ipc {q} {qi} {qi'} i qi=qi' = let open ≈-Reasoning A in begin
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
298 TMap (limit-u comp Γ) i o iproduct qi'
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
299 ≈⟨⟩
469
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
300 TMap (limit-u comp Γ) i o limit lim q (pnat A S comp Γ q qi' )
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
301 ≈⟨ t0f=t lim {q} {pnat A S comp Γ q qi'} {i} ⟩
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
302 TMap (pnat A S comp Γ q qi') i
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
303 ≈⟨⟩
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
304 qi' i
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
305 ≈↑⟨ qi=qi' ⟩
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
306 qi i
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
307 ≈⟨⟩
469
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
308 TMap (pnat A S comp Γ q qi) i
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
309 ∎
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
310 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: 467
diff changeset
311 → ( ∀ (i : I ) → A [ qi i ≈ qi' i ] ) → A [ iproduct qi ≈ iproduct qi' ]
469
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
312 ip-cong {q} {qi} {qi'} qi=qi' = limit-uniqueness lim {q} {pnat A S comp Γ q qi} (iproduct qi' ) (λ {i} → ipc {q} {qi} {qi'} i (qi=qi' i))