Mercurial > hg > Members > kono > Proof > category
annotate SetsCompleteness.agda @ 547:c0078b03201c
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Tue, 04 Apr 2017 13:51:48 +0900 |
| parents | 73a5606fa362 |
| children | 03b851adc4fb 97b98b81e990 |
| rev | line source |
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1 |
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2 open import Category -- https://github.com/konn/category-agda |
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3 open import Level |
| 535 | 4 open import Category.Sets renaming ( _o_ to _*_ ) |
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6 module SetsCompleteness where |
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8 |
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9 open import cat-utility |
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10 open import Relation.Binary.Core |
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11 open import Function |
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12 import Relation.Binary.PropositionalEquality |
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13 -- Extensionality a b = {A : Set a} {B : A → Set b} {f g : (x : A) → B x} → (∀ x → f x ≡ g x) → f ≡ g → ( λ x → f x ≡ λ x → g x ) |
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14 postulate extensionality : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → Relation.Binary.PropositionalEquality.Extensionality c₂ c₂ |
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| 520 | 16 ≡cong = Relation.Binary.PropositionalEquality.cong |
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17 |
| 524 | 18 lemma1 : { c₂ : Level } {a b : Obj (Sets { c₂})} {f g : Hom Sets a b} → |
| 19 Sets [ f ≈ g ] → (x : a ) → f x ≡ g x | |
| 20 lemma1 refl x = refl | |
| 503 | 21 |
| 504 | 22 record Σ {a} (A : Set a) (B : Set a) : Set a where |
| 503 | 23 constructor _,_ |
| 24 field | |
| 25 proj₁ : A | |
| 504 | 26 proj₂ : B |
| 503 | 27 |
| 28 open Σ public | |
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29 |
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30 |
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31 SetsProduct : { c₂ : Level} → CreateProduct ( Sets { c₂} ) |
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32 SetsProduct { c₂ } = record { |
| 504 | 33 product = λ a b → Σ a b |
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34 ; π1 = λ a b → λ ab → (proj₁ ab) |
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35 ; π2 = λ a b → λ ab → (proj₂ ab) |
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36 ; isProduct = λ a b → record { |
| 503 | 37 _×_ = λ f g x → record { proj₁ = f x ; proj₂ = g x } -- ( f x , g x ) |
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38 ; π1fxg=f = refl |
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39 ; π2fxg=g = refl |
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40 ; uniqueness = refl |
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41 ; ×-cong = λ {c} {f} {f'} {g} {g'} f=f g=g → prod-cong a b f=f g=g |
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42 } |
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43 } where |
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44 prod-cong : ( a b : Obj (Sets {c₂}) ) {c : Obj (Sets {c₂}) } {f f' : Hom Sets c a } {g g' : Hom Sets c b } |
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45 → Sets [ f ≈ f' ] → Sets [ g ≈ g' ] |
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46 → Sets [ (λ x → f x , g x) ≈ (λ x → f' x , g' x) ] |
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47 prod-cong a b {c} {f} {.f} {g} {.g} refl refl = refl |
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48 |
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49 |
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50 record iproduct {a} (I : Set a) ( pi0 : I → Set a ) : Set a where |
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51 field |
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52 pi1 : ( i : I ) → pi0 i |
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53 |
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54 open iproduct |
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55 |
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56 SetsIProduct : { c₂ : Level} → (I : Obj Sets) (ai : I → Obj Sets ) |
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57 → IProduct ( Sets { c₂} ) I |
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58 SetsIProduct I fi = record { |
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59 ai = fi |
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60 ; iprod = iproduct I fi |
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61 ; pi = λ i prod → pi1 prod i |
| 509 | 62 ; isIProduct = record { |
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63 iproduct = iproduct1 |
| 509 | 64 ; pif=q = pif=q |
| 65 ; ip-uniqueness = ip-uniqueness | |
| 66 ; ip-cong = ip-cong | |
| 67 } | |
| 68 } where | |
| 69 iproduct1 : {q : Obj Sets} → ((i : I) → Hom Sets q (fi i)) → Hom Sets q (iproduct I fi) | |
| 70 iproduct1 {q} qi x = record { pi1 = λ i → (qi i) x } | |
| 71 pif=q : {q : Obj Sets} (qi : (i : I) → Hom Sets q (fi i)) {i : I} → Sets [ Sets [ (λ prod → pi1 prod i) o iproduct1 qi ] ≈ qi i ] | |
| 72 pif=q {q} qi {i} = refl | |
| 73 ip-uniqueness : {q : Obj Sets} {h : Hom Sets q (iproduct I fi)} → Sets [ iproduct1 (λ i → Sets [ (λ prod → pi1 prod i) o h ]) ≈ h ] | |
| 74 ip-uniqueness = refl | |
| 75 ipcx : {q : Obj Sets} {qi qi' : (i : I) → Hom Sets q (fi i)} → ((i : I) → Sets [ qi i ≈ qi' i ]) → (x : q) → iproduct1 qi x ≡ iproduct1 qi' x | |
| 76 ipcx {q} {qi} {qi'} qi=qi x = | |
| 77 begin | |
| 78 record { pi1 = λ i → (qi i) x } | |
| 520 | 79 ≡⟨ ≡cong ( λ QIX → record { pi1 = QIX } ) ( extensionality Sets (λ i → ≡cong ( λ f → f x ) (qi=qi i) )) ⟩ |
| 509 | 80 record { pi1 = λ i → (qi' i) x } |
| 81 ∎ where | |
| 82 open import Relation.Binary.PropositionalEquality | |
| 83 open ≡-Reasoning | |
| 84 ip-cong : {q : Obj Sets} {qi qi' : (i : I) → Hom Sets q (fi i)} → ((i : I) → Sets [ qi i ≈ qi' i ]) → Sets [ iproduct1 qi ≈ iproduct1 qi' ] | |
| 85 ip-cong {q} {qi} {qi'} qi=qi = extensionality Sets ( ipcx qi=qi ) | |
| 86 | |
| 87 | |
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88 -- |
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89 -- e f |
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90 -- c -------→ a ---------→ b f ( f' |
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91 -- ^ . ---------→ |
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92 -- | . g |
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93 -- |k . |
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94 -- | . h |
| 514 | 95 --y : d |
| 509 | 96 |
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97 -- cf. https://github.com/danr/Agda-projects/blob/master/Category-Theory/Equalizer.agda |
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98 |
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99 data sequ {c : Level} (A B : Set c) ( f g : A → B ) : Set c where |
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100 elem : (x : A ) → (eq : f x ≡ g x) → sequ A B f g |
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101 |
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102 equ : { c₂ : Level} {a b : Obj (Sets {c₂}) } { f g : Hom (Sets {c₂}) a b } → ( sequ a b f g ) → a |
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103 equ (elem x eq) = x |
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104 |
| 533 | 105 fe=ge0 : { c₂ : Level} {a b : Obj (Sets {c₂}) } { f g : Hom (Sets {c₂}) a b } → |
| 106 (x : sequ a b f g) → (Sets [ f o (λ e → equ e) ]) x ≡ (Sets [ g o (λ e → equ e) ]) x | |
| 107 fe=ge0 (elem x eq ) = eq | |
| 108 | |
| 541 | 109 irr : { c₂ : Level} {d : Set c₂ } { x y : d } ( eq eq' : x ≡ y ) → eq ≡ eq' |
| 110 irr refl refl = refl | |
| 111 | |
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112 open sequ |
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113 |
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114 -- equalizer-c = sequ a b f g |
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115 -- ; equalizer = λ e → equ e |
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116 |
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117 SetsIsEqualizer : { c₂ : Level} → (a b : Obj (Sets {c₂}) ) (f g : Hom (Sets {c₂}) a b) → IsEqualizer Sets (λ e → equ e )f g |
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118 SetsIsEqualizer {c₂} a b f g = record { |
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119 fe=ge = fe=ge |
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120 ; k = k |
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121 ; ek=h = λ {d} {h} {eq} → ek=h {d} {h} {eq} |
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122 ; uniqueness = uniqueness |
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123 } where |
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124 fe=ge : Sets [ Sets [ f o (λ e → equ e ) ] ≈ Sets [ g o (λ e → equ e ) ] ] |
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125 fe=ge = extensionality Sets (fe=ge0 ) |
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126 k : {d : Obj Sets} (h : Hom Sets d a) → Sets [ Sets [ f o h ] ≈ Sets [ g o h ] ] → Hom Sets d (sequ a b f g) |
| 520 | 127 k {d} h eq = λ x → elem (h x) ( ≡cong ( λ y → y x ) eq ) |
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128 ek=h : {d : Obj Sets} {h : Hom Sets d a} {eq : Sets [ Sets [ f o h ] ≈ Sets [ g o h ] ]} → Sets [ Sets [ (λ e → equ e ) o k h eq ] ≈ h ] |
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129 ek=h {d} {h} {eq} = refl |
| 523 | 130 injection : { c₂ : Level } {a b : Obj (Sets { c₂})} (f : Hom Sets a b) → Set c₂ |
| 131 injection f = ∀ x y → f x ≡ f y → x ≡ y | |
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132 elm-cong : (x y : sequ a b f g) → equ x ≡ equ y → x ≡ y |
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133 elm-cong ( elem x eq ) (elem .x eq' ) refl = ≡cong ( λ ee → elem x ee ) ( irr eq eq' ) |
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134 lemma5 : {d : Obj Sets} {h : Hom Sets d a} {fh=gh : Sets [ Sets [ f o h ] ≈ Sets [ g o h ] ]} {k' : Hom Sets d (sequ a b f g)} → |
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135 Sets [ Sets [ (λ e → equ e) o k' ] ≈ h ] → (x : d ) → equ (k h fh=gh x) ≡ equ (k' x) |
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136 lemma5 refl x = refl -- somehow this is not equal to lemma1 |
| 512 | 137 uniqueness : {d : Obj Sets} {h : Hom Sets d a} {fh=gh : Sets [ Sets [ f o h ] ≈ Sets [ g o h ] ]} {k' : Hom Sets d (sequ a b f g)} → |
| 138 Sets [ Sets [ (λ e → equ e) o k' ] ≈ h ] → Sets [ k h fh=gh ≈ k' ] | |
| 525 | 139 uniqueness {d} {h} {fh=gh} {k'} ek'=h = extensionality Sets ( λ ( x : d ) → begin |
| 140 k h fh=gh x | |
| 141 ≡⟨ elm-cong ( k h fh=gh x) ( k' x ) (lemma5 {d} {h} {fh=gh} {k'} ek'=h x ) ⟩ | |
| 142 k' x | |
| 143 ∎ ) where | |
| 144 open import Relation.Binary.PropositionalEquality | |
| 145 open ≡-Reasoning | |
| 146 | |
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148 open Functor |
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149 |
| 538 | 150 ---- |
| 151 -- C is locally small i.e. Hom C i j is a set c₁ | |
| 152 -- | |
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153 record Small { c₁ c₂ ℓ : Level} ( C : Category c₁ c₂ ℓ ) ( I : Set c₁ ) |
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154 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ )) where |
| 507 | 155 field |
| 540 | 156 hom→ : {i j : Obj C } → Hom C i j → I |
| 157 hom← : {i j : Obj C } → ( f : I ) → Hom C i j | |
| 158 hom-iso : {i j : Obj C } → { f : Hom C i j } → hom← ( hom→ f ) ≡ f | |
| 536 | 159 -- ≈-≡ : {a b : Obj C } { x y : Hom C a b } → (x≈y : C [ x ≈ y ] ) → x ≡ y |
| 507 | 160 |
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161 open Small |
| 507 | 162 |
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163 ΓObj : { c₁ c₂ ℓ : Level} { C : Category c₁ c₂ ℓ } { I : Set c₁ } ( s : Small C I ) ( Γ : Functor C ( Sets { c₁} )) |
| 538 | 164 (i : Obj C ) → Set c₁ |
| 165 ΓObj s Γ i = FObj Γ i | |
| 507 | 166 |
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167 ΓMap : { c₁ c₂ ℓ : Level} { C : Category c₁ c₂ ℓ } { I : Set c₁ } ( s : Small C I ) ( Γ : Functor C ( Sets { c₁} )) |
| 539 | 168 {i j : Obj C } → ( f : I ) → ΓObj s Γ i → ΓObj s Γ j |
| 540 | 169 ΓMap s Γ {i} {j} f = FMap Γ ( hom← s f ) |
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170 |
| 538 | 171 record snat { c₂ } { I OC : Set c₂ } ( sobj : OC → Set c₂ ) |
| 539 | 172 ( smap : { i j : OC } → (f : I )→ sobj i → sobj j ) : Set c₂ where |
| 527 | 173 field |
| 538 | 174 snmap : ( i : OC ) → sobj i |
| 539 | 175 sncommute : { i j : OC } → ( f : I ) → smap f ( snmap i ) ≡ snmap j |
| 507 | 176 |
| 534 | 177 open snat |
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178 |
| 546 | 179 snat1 : { c₂ : Level } { I OC : Set c₂ } ( sobj : OC → Set c₂ ) ( smap : { i j : OC } → (f : I )→ sobj i → sobj j ) |
| 180 ( snm : ( i : OC ) → sobj i ) → (( snm1 : ( i : OC ) → sobj i ) ( i j : OC ) → (f : I ) → smap f ( snm1 i ) ≡ snm1 j ) → snat sobj smap | |
| 181 snat1 SO SM snm eq = record { snmap = λ i → snm i ; sncommute = λ {i} {j} f → eq snm i j f } | |
| 182 | |
| 541 | 183 ≡cong2 : { c : Level } { A B C : Set c } { a a' : A } { b b' : B } ( f : A → B → C ) |
| 184 → a ≡ a' | |
| 185 → b ≡ b' | |
| 186 → f a b ≡ f a' b' | |
| 187 ≡cong2 _ refl refl = refl | |
| 188 | |
| 543 | 189 snat-cong : { c : Level } { I OC : Set c } ( SObj : OC → Set c ) ( SMap : { i j : OC } → (f : I )→ SObj i → SObj j) |
| 190 { s t : snat SObj SMap } | |
| 191 → (( i : OC ) → snmap s i ≡ snmap t i ) | |
| 192 → ( ( i j : OC ) ( f : I ) → SMap {i} {j} f ( snmap s i ) ≡ snmap s j ) | |
| 193 → ( ( i j : OC ) ( f : I ) → SMap {i} {j} f ( snmap t i ) ≡ snmap t j ) | |
| 194 → s ≡ t | |
| 546 | 195 snat-cong {_} {I} {OC} SO SM {s} {t} eq1 eq2 eq3 = ≡cong2 ( λ x y → snat1 SO SM x y ) |
| 543 | 196 ( extensionality Sets ( λ i → eq1 i ) ) |
| 546 | 197 ( extensionality Sets ( λ snm → |
| 198 ( extensionality Sets ( λ i → | |
| 199 ( extensionality Sets ( λ j → | |
| 547 | 200 ( extensionality Sets ( λ f → irr2 snm i j f ( eq2s i j f (eq1 i) snm (eq2 i j f ) ) {!!} )))))))) |
| 543 | 201 where |
| 547 | 202 eq2s1 : { i j : OC } { f : I } → {si : SO i} { sj : SO j } |
| 203 ( snm : ( i : OC ) → SO i ) → ( si ≡ snm i ) → ( sj ≡ snm j ) → SM {i} {j} f si ≡ sj → SM f ( snm i) ≡ snm j | |
| 204 eq2s1 {i} {j} {f} {si} {sj} snm eq1 eq2 eq3 = begin | |
| 205 SM f ( snm i) | |
| 206 ≡⟨ ≡cong (λ x → SM f x ) (sym eq1) ⟩ | |
| 207 SM f si | |
| 208 ≡⟨ eq3 ⟩ | |
| 209 sj | |
| 210 ≡⟨ eq2 ⟩ | |
| 211 snm j | |
| 212 ∎ where | |
| 213 open import Relation.Binary.PropositionalEquality | |
| 214 open ≡-Reasoning | |
| 215 eq2s : ( i j : OC ) ( f : I ) → ( snmap s i ≡ snmap t i ) → | |
| 216 ( snm : ( i : OC ) → SO i ) → SM {i} {j} f ( snmap s i ) ≡ snmap s j → SM f ( snm i) ≡ snm j | |
| 217 eq2s i j f eq1 snm eq2 = eq2s1 snm {!!} {!!} eq2 | |
| 218 irr3 : { i j : OC } { f : I } → {snmapsi : SO i } → {snmapsj : SO j } → | |
| 219 ( es et : SM f ( snmapsi ) ≡ snmapsj ) → es ≡ et | |
| 220 irr3 refl refl = refl | |
| 546 | 221 irr2 : ( snm : ( i : OC ) → SO i ) ( i j : OC ) (f : I ) → |
| 222 ( es et : SM f (snm i ) ≡ snm j ) → es ≡ et | |
| 547 | 223 irr2 snm i j f es et = irr3 es et |
| 541 | 224 |
| 543 | 225 |
| 530 | 226 open import HomReasoning |
| 227 open NTrans | |
| 533 | 228 |
| 535 | 229 Cone : { c₁ c₂ ℓ : Level} ( C : Category c₁ c₂ ℓ ) ( I : Set c₁ ) ( s : Small C I ) ( Γ : Functor C (Sets {c₁} ) ) |
| 534 | 230 → NTrans C Sets (K Sets C (snat (ΓObj s Γ) (ΓMap s Γ) ) ) Γ |
| 535 | 231 Cone C I s Γ = record { |
| 538 | 232 TMap = λ i → λ sn → snmap sn i |
| 531 | 233 ; isNTrans = record { commute = comm1 } |
| 530 | 234 } where |
| 534 | 235 comm1 : {a b : Obj C} {f : Hom C a b} → |
| 538 | 236 Sets [ Sets [ FMap Γ f o (λ sn → snmap sn a) ] ≈ |
| 237 Sets [ (λ sn → (snmap sn b)) o FMap (K Sets C (snat (ΓObj s Γ) (ΓMap s Γ))) f ] ] | |
| 534 | 238 comm1 {a} {b} {f} = extensionality Sets ( λ sn → begin |
| 538 | 239 FMap Γ f (snmap sn a ) |
| 540 | 240 ≡⟨ ≡cong ( λ f → ( FMap Γ f (snmap sn a ))) (sym ( hom-iso s )) ⟩ |
| 241 FMap Γ ( hom← s ( hom→ s f)) (snmap sn a ) | |
| 535 | 242 ≡⟨⟩ |
| 540 | 243 ΓMap s Γ (hom→ s f) (snmap sn a ) |
| 244 ≡⟨ sncommute sn (hom→ s f) ⟩ | |
| 538 | 245 snmap sn b |
| 534 | 246 ∎ ) where |
| 247 open import Relation.Binary.PropositionalEquality | |
| 248 open ≡-Reasoning | |
| 249 | |
| 530 | 250 |
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251 SetsLimit : { c₁ c₂ ℓ : Level} ( C : Category c₁ c₂ ℓ ) ( I : Set c₁ ) ( small : Small C I ) ( Γ : Functor C (Sets {c₁} ) ) |
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252 → Limit Sets C Γ |
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253 SetsLimit { c₂} C I s Γ = record { |
| 534 | 254 a0 = snat (ΓObj s Γ) (ΓMap s Γ) |
| 535 | 255 ; t0 = Cone C I s Γ |
| 523 | 256 ; isLimit = record { |
| 530 | 257 limit = limit1 |
| 539 | 258 ; t0f=t = λ {a t i } → t0f=t {a} {t} {i} |
| 259 ; limit-uniqueness = λ {a t i } → limit-uniqueness {a} {t} {i} | |
| 260 } | |
| 523 | 261 } where |
| 527 | 262 a0 : Obj Sets |
| 534 | 263 a0 = snat (ΓObj s Γ) (ΓMap s Γ) |
| 539 | 264 comm2 : { a : Obj Sets } {x : a } {i j : Obj C} (t : NTrans C Sets (K Sets C a) Γ) (f : I) |
| 538 | 265 → ΓMap s Γ f (TMap t i x) ≡ TMap t j x |
| 266 comm2 {a} {x} t f = ≡cong ( λ f → f x ) ( IsNTrans.commute ( isNTrans t ) ) | |
| 534 | 267 limit1 : (a : Obj Sets) → NTrans C Sets (K Sets C a) Γ → Hom Sets a (snat (ΓObj s Γ) (ΓMap s Γ)) |
| 538 | 268 limit1 a t = λ x → record { snmap = λ i → ( TMap t i ) x ; |
| 537 | 269 sncommute = λ f → comm2 t f } |
| 539 | 270 t0f=t : {a : Obj Sets} {t : NTrans C Sets (K Sets C a) Γ} {i : Obj C} → Sets [ Sets [ TMap (Cone C I s Γ) i o limit1 a t ] ≈ TMap t i ] |
| 540 | 271 t0f=t {a} {t} {i} = extensionality Sets ( λ x → begin |
| 272 ( Sets [ TMap (Cone C I s Γ) i o limit1 a t ]) x | |
| 273 -- ≡⟨⟩ | |
| 274 -- snmap ( record { snmap = λ i → ( TMap t i ) x ; sncommute = λ {i j} f → comm2 {a} {x} {i} {j} t f } ) i | |
| 275 ≡⟨⟩ | |
| 276 TMap t i x | |
| 277 ∎ ) where | |
| 278 open import Relation.Binary.PropositionalEquality | |
| 279 open ≡-Reasoning | |
| 539 | 280 limit-uniqueness : {a : Obj Sets} {t : NTrans C Sets (K Sets C a) Γ} {f : Hom Sets a (snat (ΓObj s Γ) (ΓMap s Γ))} → |
| 281 ({i : Obj C} → Sets [ Sets [ TMap (Cone C I s Γ) i o f ] ≈ TMap t i ]) → Sets [ limit1 a t ≈ f ] | |
| 540 | 282 limit-uniqueness {a} {t} {f} cif=t = extensionality Sets ( λ x → begin |
| 283 limit1 a t x | |
| 284 ≡⟨⟩ | |
| 285 record { snmap = λ i → ( TMap t i ) x ; sncommute = λ f → comm2 t f } | |
| 544 | 286 ≡⟨ snat-cong (ΓObj s Γ) (ΓMap s Γ) {!!} {!!} {!!} ⟩ |
| 541 | 287 record { snmap = λ i → snmap (f x ) i ; sncommute = sncommute (f x ) } |
| 540 | 288 ≡⟨⟩ |
| 289 f x | |
| 290 ∎ ) where | |
| 291 open import Relation.Binary.PropositionalEquality | |
| 292 open ≡-Reasoning | |
| 537 | 293 |
| 294 | |
| 295 | |
| 296 | |
| 297 | |
| 298 | |
| 299 | |
| 539 | 300 |
| 301 |