Mercurial > hg > Members > kono > Proof > category
annotate SetsCompleteness.agda @ 664:9e8276b89cd0
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Sun, 22 Oct 2017 23:36:30 +0900 |
| parents | 855e497a9c8f |
| children | 9904edf40547 |
| rev | line source |
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| 606 | 1 open import Category -- https://github.com/konn/category-agda |
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2 open import Level |
| 535 | 3 open import Category.Sets renaming ( _o_ to _*_ ) |
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4 |
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5 module SetsCompleteness where |
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6 |
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7 open import cat-utility |
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8 open import Relation.Binary.Core |
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9 open import Function |
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10 import Relation.Binary.PropositionalEquality |
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11 -- 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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12 postulate extensionality : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → Relation.Binary.PropositionalEquality.Extensionality c₂ c₂ |
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13 |
| 606 | 14 ≡cong = Relation.Binary.PropositionalEquality.cong |
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15 |
| 604 | 16 lemma1 : { c₂ : Level } {a b : Obj (Sets { c₂})} {f g : Hom Sets a b} → |
| 524 | 17 Sets [ f ≈ g ] → (x : a ) → f x ≡ g x |
| 604 | 18 lemma1 refl x = refl |
| 503 | 19 |
| 504 | 20 record Σ {a} (A : Set a) (B : Set a) : Set a where |
| 503 | 21 constructor _,_ |
| 22 field | |
| 23 proj₁ : A | |
| 606 | 24 proj₂ : B |
| 503 | 25 |
| 26 open Σ public | |
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27 |
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28 |
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29 SetsProduct : { c₂ : Level} → CreateProduct ( Sets { c₂} ) |
| 606 | 30 SetsProduct { c₂ } = record { |
| 504 | 31 product = λ a b → Σ a b |
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32 ; π1 = λ a b → λ ab → (proj₁ ab) |
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33 ; π2 = λ a b → λ ab → (proj₂ ab) |
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34 ; isProduct = λ a b → record { |
| 606 | 35 _×_ = λ f g x → record { proj₁ = f x ; proj₂ = g x } -- ( f x , g x ) |
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36 ; π1fxg=f = refl |
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37 ; π2fxg=g = refl |
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38 ; uniqueness = refl |
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39 ; ×-cong = λ {c} {f} {f'} {g} {g'} f=f g=g → prod-cong a b f=f g=g |
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40 } |
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41 } where |
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42 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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43 → Sets [ f ≈ f' ] → Sets [ g ≈ g' ] |
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44 → Sets [ (λ x → f x , g x) ≈ (λ x → f' x , g' x) ] |
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45 prod-cong a b {c} {f} {.f} {g} {.g} refl refl = refl |
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46 |
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47 |
| 604 | 48 record iproduct {a} (I : Set a) ( pi0 : I → Set a ) : Set a where |
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49 field |
| 604 | 50 pi1 : ( i : I ) → pi0 i |
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51 |
| 604 | 52 open iproduct |
| 574 | 53 |
| 606 | 54 SetsIProduct : { c₂ : Level} → (I : Obj Sets) (ai : I → Obj Sets ) |
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55 → IProduct ( Sets { c₂} ) I |
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56 SetsIProduct I fi = record { |
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57 ai = fi |
| 604 | 58 ; iprod = iproduct I fi |
| 59 ; pi = λ i prod → pi1 prod i | |
| 509 | 60 ; isIProduct = record { |
| 604 | 61 iproduct = iproduct1 |
| 509 | 62 ; pif=q = pif=q |
| 63 ; ip-uniqueness = ip-uniqueness | |
| 604 | 64 ; ip-cong = ip-cong |
| 509 | 65 } |
| 66 } where | |
| 604 | 67 iproduct1 : {q : Obj Sets} → ((i : I) → Hom Sets q (fi i)) → Hom Sets q (iproduct I fi) |
| 68 iproduct1 {q} qi x = record { pi1 = λ i → (qi i) x } | |
| 69 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 ] | |
| 509 | 70 pif=q {q} qi {i} = refl |
| 604 | 71 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 ] |
| 509 | 72 ip-uniqueness = refl |
| 604 | 73 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 |
| 606 | 74 ipcx {q} {qi} {qi'} qi=qi x = |
| 604 | 75 begin |
| 76 record { pi1 = λ i → (qi i) x } | |
| 77 ≡⟨ ≡cong ( λ QIX → record { pi1 = QIX } ) ( extensionality Sets (λ i → ≡cong ( λ f → f x ) (qi=qi i) )) ⟩ | |
| 78 record { pi1 = λ i → (qi' i) x } | |
| 79 ∎ where | |
| 606 | 80 open import Relation.Binary.PropositionalEquality |
| 81 open ≡-Reasoning | |
| 604 | 82 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' ] |
| 83 ip-cong {q} {qi} {qi'} qi=qi = extensionality Sets ( ipcx qi=qi ) | |
| 509 | 84 |
| 85 | |
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86 -- |
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87 -- e f |
| 606 | 88 -- c -------→ a ---------→ b f ( f' |
| 604 | 89 -- ^ . ---------→ |
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90 -- | . g |
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91 -- |k . |
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92 -- | . h |
| 604 | 93 --y : d |
| 509 | 94 |
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95 -- cf. https://github.com/danr/Agda-projects/blob/master/Category-Theory/Equalizer.agda |
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96 |
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97 data sequ {c : Level} (A B : Set c) ( f g : A → B ) : Set c where |
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98 elem : (x : A ) → (eq : f x ≡ g x) → sequ A B f g |
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99 |
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100 equ : { c₂ : Level} {a b : Obj (Sets {c₂}) } { f g : Hom (Sets {c₂}) a b } → ( sequ a b f g ) → a |
| 606 | 101 equ (elem x eq) = x |
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102 |
| 606 | 103 fe=ge0 : { c₂ : Level} {a b : Obj (Sets {c₂}) } { f g : Hom (Sets {c₂}) a b } → |
| 533 | 104 (x : sequ a b f g) → (Sets [ f o (λ e → equ e) ]) x ≡ (Sets [ g o (λ e → equ e) ]) x |
| 105 fe=ge0 (elem x eq ) = eq | |
| 106 | |
| 541 | 107 irr : { c₂ : Level} {d : Set c₂ } { x y : d } ( eq eq' : x ≡ y ) → eq ≡ eq' |
| 108 irr refl refl = refl | |
| 109 | |
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110 open sequ |
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111 |
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112 -- equalizer-c = sequ a b f g |
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113 -- ; equalizer = λ e → equ e |
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114 |
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115 SetsIsEqualizer : { c₂ : Level} → (a b : Obj (Sets {c₂}) ) (f g : Hom (Sets {c₂}) a b) → IsEqualizer Sets (λ e → equ e )f g |
| 606 | 116 SetsIsEqualizer {c₂} a b f g = record { |
| 604 | 117 fe=ge = fe=ge |
| 118 ; k = k | |
| 119 ; ek=h = λ {d} {h} {eq} → ek=h {d} {h} {eq} | |
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120 ; uniqueness = uniqueness |
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121 } where |
| 604 | 122 fe=ge : Sets [ Sets [ f o (λ e → equ e ) ] ≈ Sets [ g o (λ e → equ e ) ] ] |
| 606 | 123 fe=ge = extensionality Sets (fe=ge0 ) |
| 604 | 124 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) |
| 125 k {d} h eq = λ x → elem (h x) ( ≡cong ( λ y → y x ) eq ) | |
| 126 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 ] | |
| 606 | 127 ek=h {d} {h} {eq} = refl |
| 523 | 128 injection : { c₂ : Level } {a b : Obj (Sets { c₂})} (f : Hom Sets a b) → Set c₂ |
| 129 injection f = ∀ x y → f x ≡ f y → x ≡ y | |
| 604 | 130 elm-cong : (x y : sequ a b f g) → equ x ≡ equ y → x ≡ y |
| 131 elm-cong ( elem x eq ) (elem .x eq' ) refl = ≡cong ( λ ee → elem x ee ) ( irr eq eq' ) | |
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132 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)} → |
| 604 | 133 Sets [ Sets [ (λ e → equ e) o k' ] ≈ h ] → (x : d ) → equ (k h fh=gh x) ≡ equ (k' x) |
| 134 lemma5 refl x = refl -- somehow this is not equal to lemma1 | |
| 512 | 135 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)} → |
| 136 Sets [ Sets [ (λ e → equ e) o k' ] ≈ h ] → Sets [ k h fh=gh ≈ k' ] | |
| 525 | 137 uniqueness {d} {h} {fh=gh} {k'} ek'=h = extensionality Sets ( λ ( x : d ) → begin |
| 138 k h fh=gh x | |
| 139 ≡⟨ elm-cong ( k h fh=gh x) ( k' x ) (lemma5 {d} {h} {fh=gh} {k'} ek'=h x ) ⟩ | |
| 140 k' x | |
| 141 ∎ ) where | |
| 142 open import Relation.Binary.PropositionalEquality | |
| 143 open ≡-Reasoning | |
| 144 | |
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145 |
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146 open Functor |
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147 |
| 538 | 148 ---- |
| 149 -- C is locally small i.e. Hom C i j is a set c₁ | |
| 150 -- | |
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151 record Small { c₁ c₂ ℓ : Level} ( C : Category c₁ c₂ ℓ ) ( I : Set c₁ ) |
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152 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ )) where |
| 507 | 153 field |
| 606 | 154 hom→ : {i j : Obj C } → Hom C i j → I |
| 155 hom← : {i j : Obj C } → ( f : I ) → Hom C i j | |
| 156 hom-iso : {i j : Obj C } → { f : Hom C i j } → hom← ( hom→ f ) ≡ f | |
| 507 | 157 |
| 606 | 158 open Small |
| 507 | 159 |
| 606 | 160 ΓObj : { c₁ c₂ ℓ : Level} { C : Category c₁ c₂ ℓ } { I : Set c₁ } ( s : Small C I ) ( Γ : Functor C ( Sets { c₁} )) |
| 538 | 161 (i : Obj C ) → Set c₁ |
| 162 ΓObj s Γ i = FObj Γ i | |
| 507 | 163 |
| 606 | 164 ΓMap : { c₁ c₂ ℓ : Level} { C : Category c₁ c₂ ℓ } { I : Set c₁ } ( s : Small C I ) ( Γ : Functor C ( Sets { c₁} )) |
| 165 {i j : Obj C } → ( f : I ) → ΓObj s Γ i → ΓObj s Γ j | |
| 166 ΓMap s Γ {i} {j} f = FMap Γ ( hom← s f ) | |
| 598 | 167 |
| 606 | 168 record snat { c₂ } { I OC : Set c₂ } ( sobj : OC → Set c₂ ) |
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169 ( smap : { i j : OC } → (f : I ) → sobj i → sobj j ) : Set c₂ where |
| 606 | 170 field |
| 171 snmap : ( i : OC ) → sobj i | |
| 172 sncommute : ( i j : OC ) → ( f : I ) → smap f ( snmap i ) ≡ snmap j | |
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173 smap0 : { i j : OC } → (f : I ) → sobj i → sobj j |
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174 smap0 {i} {j} f x = smap f x |
| 598 | 175 |
| 604 | 176 open snat |
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177 |
| 608 | 178 ≡cong2 : { c c' : Level } { A B : Set c } { C : Set c' } { a a' : A } { b b' : B } ( f : A → B → C ) |
| 179 → a ≡ a' | |
| 180 → b ≡ b' | |
| 181 → f a b ≡ f a' b' | |
| 182 ≡cong2 _ refl refl = refl | |
| 183 | |
| 184 open import Relation.Binary.HeterogeneousEquality as HE renaming ( cong to cong' ; sym to sym' ; subst₂ to subst₂' ; Extensionality to Extensionality' ) | |
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185 using (_≅_;refl) |
| 664 | 186 postulate ≅extensionality : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → HE.Extensionality c₂ c₂ |
| 608 | 187 |
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188 snat-cong : {c : Level} |
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189 {I OC : Set c} |
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190 {sobj : OC → Set c} |
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191 {smap : {i j : OC} → (f : I) → sobj i → sobj j} |
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192 → (s t : snat sobj smap) |
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193 → (snmap-≡ : snmap s ≡ snmap t) |
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194 → (sncommute-≅ : sncommute s ≅ sncommute t) |
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195 → s ≡ t |
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196 snat-cong _ _ refl refl = refl |
| 590 | 197 |
| 598 | 198 open import HomReasoning |
| 199 open NTrans | |
| 590 | 200 |
| 606 | 201 Cone : { c₁ c₂ ℓ : Level} ( C : Category c₁ c₂ ℓ ) ( I : Set c₁ ) ( s : Small C I ) ( Γ : Functor C (Sets {c₁} ) ) |
| 604 | 202 → NTrans C Sets (K Sets C (snat (ΓObj s Γ) (ΓMap s Γ) ) ) Γ |
| 203 Cone C I s Γ = record { | |
| 606 | 204 TMap = λ i → λ sn → snmap sn i |
| 604 | 205 ; isNTrans = record { commute = comm1 } |
| 598 | 206 } where |
| 604 | 207 comm1 : {a b : Obj C} {f : Hom C a b} → |
| 208 Sets [ Sets [ FMap Γ f o (λ sn → snmap sn a) ] ≈ | |
| 209 Sets [ (λ sn → (snmap sn b)) o FMap (K Sets C (snat (ΓObj s Γ) (ΓMap s Γ))) f ] ] | |
| 210 comm1 {a} {b} {f} = extensionality Sets ( λ sn → begin | |
| 211 FMap Γ f (snmap sn a ) | |
| 212 ≡⟨ ≡cong ( λ f → ( FMap Γ f (snmap sn a ))) (sym ( hom-iso s )) ⟩ | |
| 213 FMap Γ ( hom← s ( hom→ s f)) (snmap sn a ) | |
| 596 | 214 ≡⟨⟩ |
| 606 | 215 ΓMap s Γ (hom→ s f) (snmap sn a ) |
| 216 ≡⟨ sncommute sn a b (hom→ s f) ⟩ | |
| 604 | 217 snmap sn b |
| 596 | 218 ∎ ) where |
| 590 | 219 open import Relation.Binary.PropositionalEquality |
| 220 open ≡-Reasoning | |
| 221 | |
| 222 | |
| 606 | 223 SetsLimit : { c₁ c₂ ℓ : Level} ( C : Category c₁ c₂ ℓ ) ( I : Set c₁ ) ( small : Small C I ) ( Γ : Functor C (Sets {c₁} ) ) |
| 598 | 224 → Limit Sets C Γ |
| 606 | 225 SetsLimit { c₂} C I s Γ = record { |
| 226 a0 = snat (ΓObj s Γ) (ΓMap s Γ) | |
| 604 | 227 ; t0 = Cone C I s Γ |
| 598 | 228 ; isLimit = record { |
| 604 | 229 limit = limit1 |
| 230 ; t0f=t = λ {a t i } → t0f=t {a} {t} {i} | |
| 231 ; limit-uniqueness = λ {a t i } → limit-uniqueness {a} {t} {i} | |
| 598 | 232 } |
| 233 } where | |
| 606 | 234 comm2 : { a : Obj Sets } {x : a } {i j : Obj C} (t : NTrans C Sets (K Sets C a) Γ) (f : I) |
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235 → ΓMap s Γ f (TMap t i x) ≡ TMap t j x |
| 664 | 236 comm2 {a} {x} t f = ≡cong ( λ h → h x ) ( IsNTrans.commute ( isNTrans t ) ) |
| 606 | 237 limit1 : (a : Obj Sets) → NTrans C Sets (K Sets C a) Γ → Hom Sets a (snat (ΓObj s Γ) (ΓMap s Γ)) |
| 604 | 238 limit1 a t = λ x → record { snmap = λ i → ( TMap t i ) x ; |
| 606 | 239 sncommute = λ i j f → comm2 t f } |
| 604 | 240 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 ] |
| 241 t0f=t {a} {t} {i} = extensionality Sets ( λ x → begin | |
| 242 ( Sets [ TMap (Cone C I s Γ) i o limit1 a t ]) x | |
| 243 ≡⟨⟩ | |
| 244 TMap t i x | |
| 245 ∎ ) where | |
| 562 | 246 open import Relation.Binary.PropositionalEquality |
| 247 open ≡-Reasoning | |
| 604 | 248 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))} → |
| 249 ({i : Obj C} → Sets [ Sets [ TMap (Cone C I s Γ) i o f ] ≈ TMap t i ]) → Sets [ limit1 a t ≈ f ] | |
| 250 limit-uniqueness {a} {t} {f} cif=t = extensionality Sets ( λ x → begin | |
| 598 | 251 limit1 a t x |
| 604 | 252 ≡⟨⟩ |
| 606 | 253 record { snmap = λ i → ( TMap t i ) x ; sncommute = λ i j f → comm2 t f } |
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254 ≡⟨ snat-cong (limit1 a t x) (f x ) ( extensionality Sets ( λ i → eq1 x i )) (eq2 x ) ⟩ |
| 664 | 255 record { snmap = λ i → snmap (f x ) i ; sncommute = λ i j g → sncommute (f x ) i j g } |
| 604 | 256 ≡⟨⟩ |
| 598 | 257 f x |
| 604 | 258 ∎ ) where |
| 598 | 259 open import Relation.Binary.PropositionalEquality |
| 260 open ≡-Reasoning | |
| 604 | 261 eq1 : (x : a ) (i : Obj C) → TMap t i x ≡ snmap (f x) i |
| 262 eq1 x i = sym ( ≡cong ( λ f → f x ) cif=t ) | |
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263 irr≅ : { c₂ : Level} {d : Set c₂ } { x y : d } ( eq eq' : x ≡ y ) → eq ≅ eq' |
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264 irr≅ refl refl = refl |
| 664 | 265 eq3 : ( x : a) ( i j : Obj C ) ( g : I ) → ≡cong ( λ h → h x ) ( IsNTrans.commute ( isNTrans t ) {i} {j} {hom← s g } ) |
| 266 ≅ sncommute (f x) i j g | |
| 267 eq3 x i j g = {!!} | |
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268 -- eq2 : ( x : a) → sncommute (limit1 a t x) ≅ sncommute (f x) |
| 664 | 269 eq2 : ( x : a) → ( λ i j g → ≡cong ( λ h → h x ) ( IsNTrans.commute ( isNTrans t ) {i} {j} {hom← s g } )) |
| 270 ≅ ( λ i j g → sncommute (f x) i j g ) | |
| 606 | 271 eq2 x = {!!} |