Mercurial > hg > Members > kono > Proof > category
annotate SetsCompleteness.agda @ 593:a158ebb391f2
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Sun, 21 May 2017 19:53:58 +0900 |
| parents | 0448a74c0a03 |
| children | db76b73b526c |
| 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 |
| 573 | 18 ≈-to-≡ : { c₂ : Level } {a b : Obj (Sets { c₂})} {f g : Hom Sets a b} → |
| 524 | 19 Sets [ f ≈ g ] → (x : a ) → f x ≡ g x |
| 573 | 20 ≈-to-≡ 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 |
| 578 | 50 record sproduct {a} (I : Set a) ( Product : I → Set a ) : Set a where |
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51 field |
| 573 | 52 proj : ( i : I ) → Product i |
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53 |
| 578 | 54 open sproduct |
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55 |
| 578 | 56 iproduct1 : { c₂ : Level} → (I : Obj (Sets { c₂}) ) (fi : I → Obj Sets ) {q : Obj Sets} → ((i : I) → Hom Sets q (fi i)) → Hom Sets q (sproduct I fi) |
| 574 | 57 iproduct1 I fi {q} qi x = record { proj = λ i → (qi i) x } |
| 58 ipcx : { c₂ : Level} → (I : Obj (Sets { c₂})) (fi : I → Obj Sets ) {q : Obj Sets} {qi qi' : (i : I) → Hom Sets q (fi i)} → ((i : I) → Sets [ qi i ≈ qi' i ]) → (x : q) → iproduct1 I fi qi x ≡ iproduct1 I fi qi' x | |
| 59 ipcx I fi {q} {qi} {qi'} qi=qi x = | |
| 60 begin | |
| 61 record { proj = λ i → (qi i) x } | |
| 62 ≡⟨ ≡cong ( λ qi → record { proj = qi } ) ( extensionality Sets (λ i → ≈-to-≡ (qi=qi i) x )) ⟩ | |
| 63 record { proj = λ i → (qi' i) x } | |
| 64 ∎ where | |
| 65 open import Relation.Binary.PropositionalEquality | |
| 66 open ≡-Reasoning | |
| 67 ip-cong : { c₂ : Level} → (I : Obj (Sets { c₂}) ) (fi : I → Obj Sets ) {q : Obj Sets} {qi qi' : (i : I) → Hom Sets q (fi i)} → ((i : I) → Sets [ qi i ≈ qi' i ]) → Sets [ iproduct1 I fi qi ≈ iproduct1 I fi qi' ] | |
| 68 ip-cong I fi {q} {qi} {qi'} qi=qi = extensionality Sets ( ipcx I fi qi=qi ) | |
| 69 | |
| 570 | 70 SetsIProduct : { c₂ : Level} → (I : Obj Sets) (fi : I → Obj Sets ) |
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71 → IProduct ( Sets { c₂} ) I |
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72 SetsIProduct I fi = record { |
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73 ai = fi |
| 578 | 74 ; iprod = sproduct I fi |
| 573 | 75 ; pi = λ i prod → proj prod i |
| 509 | 76 ; isIProduct = record { |
| 574 | 77 iproduct = iproduct1 I fi |
| 509 | 78 ; pif=q = pif=q |
| 79 ; ip-uniqueness = ip-uniqueness | |
| 574 | 80 ; ip-cong = ip-cong I fi |
| 509 | 81 } |
| 82 } where | |
| 574 | 83 pif=q : {q : Obj Sets} (qi : (i : I) → Hom Sets q (fi i)) {i : I} → Sets [ Sets [ (λ prod → proj prod i) o iproduct1 I fi qi ] ≈ qi i ] |
| 509 | 84 pif=q {q} qi {i} = refl |
| 578 | 85 ip-uniqueness : {q : Obj Sets} {h : Hom Sets q (sproduct I fi)} → Sets [ iproduct1 I fi (λ i → Sets [ (λ prod → proj prod i) o h ]) ≈ h ] |
| 509 | 86 ip-uniqueness = refl |
| 87 | |
| 88 | |
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89 -- |
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90 -- e f |
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91 -- c -------→ a ---------→ b f ( f' |
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92 -- ^ . ---------→ |
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93 -- | . g |
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94 -- |k . |
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95 -- | . h |
| 514 | 96 --y : d |
| 509 | 97 |
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98 -- cf. https://github.com/danr/Agda-projects/blob/master/Category-Theory/Equalizer.agda |
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99 |
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100 data sequ {c : Level} (A B : Set c) ( f g : A → B ) : Set c where |
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101 elem : (x : A ) → (eq : f x ≡ g x) → sequ A B f g |
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102 |
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103 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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104 equ (elem x eq) = x |
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105 |
| 533 | 106 fe=ge0 : { c₂ : Level} {a b : Obj (Sets {c₂}) } { f g : Hom (Sets {c₂}) a b } → |
| 107 (x : sequ a b f g) → (Sets [ f o (λ e → equ e) ]) x ≡ (Sets [ g o (λ e → equ e) ]) x | |
| 108 fe=ge0 (elem x eq ) = eq | |
| 109 | |
| 541 | 110 irr : { c₂ : Level} {d : Set c₂ } { x y : d } ( eq eq' : x ≡ y ) → eq ≡ eq' |
| 111 irr refl refl = refl | |
| 112 | |
| 555 | 113 elm-cong : { c₂ : Level} → {a b : Obj (Sets {c₂}) } {f g : Hom (Sets {c₂}) a b} → (x y : sequ a b f g) → equ x ≡ equ y → x ≡ y |
| 114 elm-cong ( elem x eq ) (elem .x eq' ) refl = ≡cong ( λ ee → elem x ee ) ( irr eq eq' ) | |
| 115 | |
| 563 | 116 fe=ge : { c₂ : Level} → {a b : Obj (Sets {c₂}) } {f g : Hom (Sets {c₂}) a b} |
| 117 → Sets [ Sets [ f o (λ e → equ {_} {a} {b} {f} {g} e ) ] ≈ Sets [ g o (λ e → equ e ) ] ] | |
| 558 | 118 fe=ge = extensionality Sets (fe=ge0 ) |
| 119 k : { c₂ : Level} → {a b : Obj (Sets {c₂}) } {f g : Hom (Sets {c₂}) a b} → {d : Obj Sets} (h : Hom Sets d a) | |
| 120 → Sets [ Sets [ f o h ] ≈ Sets [ g o h ] ] → Hom Sets d (sequ a b f g) | |
| 573 | 121 k {_} {_} {_} {_} {_} {d} h eq = λ x → elem (h x) ( ≈-to-≡ eq x ) |
| 563 | 122 ek=h : { c₂ : Level} → {a b : Obj (Sets {c₂}) } {f g : Hom (Sets {c₂}) a b} → {d : Obj Sets} {h : Hom Sets d a} {eq : Sets [ Sets [ f o h ] ≈ Sets [ g o h ] ]} → Sets [ Sets [ (λ e → equ {_} {a} {b} {f} {g} e ) o k h eq ] ≈ h ] |
| 558 | 123 ek=h {_} {_} {_} {_} {_} {d} {h} {eq} = refl |
| 124 | |
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125 open sequ |
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126 |
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127 -- equalizer-c = sequ a b f g |
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128 -- ; equalizer = λ e → equ e |
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129 |
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130 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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131 SetsIsEqualizer {c₂} a b f g = record { |
| 560 | 132 fe=ge = fe=ge { c₂ } {a} {b} {f} {g} |
| 133 ; k = λ {d} h eq → k { c₂ } {a} {b} {f} {g} {d} h eq | |
| 558 | 134 ; ek=h = λ {d} {h} {eq} → ek=h {c₂} {a} {b} {f} {g} {d} {h} {eq} |
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135 ; uniqueness = uniqueness |
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136 } where |
| 523 | 137 injection : { c₂ : Level } {a b : Obj (Sets { c₂})} (f : Hom Sets a b) → Set c₂ |
| 138 injection f = ∀ x y → f x ≡ f y → x ≡ y | |
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139 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)} → |
| 563 | 140 Sets [ Sets [ (λ e → equ e) o k' ] ≈ h ] → (x : d ) → equ {_} {a} {b} {f} {g} (k h fh=gh x) ≡ equ (k' x) |
| 573 | 141 lemma5 refl x = refl -- somehow this is not equal to ≈-to-≡ |
| 512 | 142 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)} → |
| 143 Sets [ Sets [ (λ e → equ e) o k' ] ≈ h ] → Sets [ k h fh=gh ≈ k' ] | |
| 525 | 144 uniqueness {d} {h} {fh=gh} {k'} ek'=h = extensionality Sets ( λ ( x : d ) → begin |
| 145 k h fh=gh x | |
| 146 ≡⟨ elm-cong ( k h fh=gh x) ( k' x ) (lemma5 {d} {h} {fh=gh} {k'} ek'=h x ) ⟩ | |
| 147 k' x | |
| 148 ∎ ) where | |
| 149 open import Relation.Binary.PropositionalEquality | |
| 150 open ≡-Reasoning | |
| 151 | |
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152 |
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153 open Functor |
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154 |
| 538 | 155 ---- |
| 156 -- C is locally small i.e. Hom C i j is a set c₁ | |
| 157 -- | |
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158 record Small { c₁ c₂ ℓ : Level} ( C : Category c₁ c₂ ℓ ) ( I : Set c₁ ) |
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159 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ )) where |
| 507 | 160 field |
| 552 | 161 hom→ : {i j : Obj C } → Hom C i j → I → I |
| 162 hom← : {i j : Obj C } → ( f : I → I ) → Hom C i j | |
| 540 | 163 hom-iso : {i j : Obj C } → { f : Hom C i j } → hom← ( hom→ f ) ≡ f |
| 536 | 164 -- ≈-≡ : {a b : Obj C } { x y : Hom C a b } → (x≈y : C [ x ≈ y ] ) → x ≡ y |
| 507 | 165 |
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166 open Small |
| 507 | 167 |
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168 ΓObj : { c₁ c₂ ℓ : Level} { C : Category c₁ c₂ ℓ } { I : Set c₁ } ( s : Small C I ) ( Γ : Functor C ( Sets { c₁} )) |
| 538 | 169 (i : Obj C ) → Set c₁ |
| 170 ΓObj s Γ i = FObj Γ i | |
| 507 | 171 |
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172 ΓMap : { c₁ c₂ ℓ : Level} { C : Category c₁ c₂ ℓ } { I : Set c₁ } ( s : Small C I ) ( Γ : Functor C ( Sets { c₁} )) |
| 552 | 173 {i j : Obj C } → ( f : I → I ) → ΓObj s Γ i → ΓObj s Γ j |
| 540 | 174 ΓMap s Γ {i} {j} f = FMap Γ ( hom← s f ) |
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175 |
| 591 | 176 slid : { c₁ c₂ ℓ : Level} ( C : Category c₁ c₂ ℓ ) ( I : Set c₁ ) ( s : Small C I ) → (x : Obj C) → I → I |
| 177 slid C I s x = hom→ s ( id1 C x ) | |
| 507 | 178 |
| 583 | 179 record slim { c₂ } { I OC : Set c₂ } |
| 180 ( sobj : OC → Set c₂ ) ( smap : { i j : OC } → (f : I → I ) → sobj i → sobj j ) | |
| 558 | 181 : Set c₂ where |
| 182 field | |
| 593 | 183 slproj : ( i : OC ) → sobj i |
| 184 slequ : (i j : OC) (f : I → I ) → sequ OC (sobj j) (λ x → smap f (slproj i) ) (λ x → slproj j ) | |
| 185 slprod : sproduct OC sobj | |
| 186 slprod = record { proj = slproj } | |
| 590 | 187 slmap : { i j : OC } → (f : I → I ) → sobj i → sobj j |
| 188 slmap f x = smap f x | |
| 591 | 189 open slim |
| 558 | 190 |
| 530 | 191 open import HomReasoning |
| 192 open NTrans | |
| 569 | 193 |
| 590 | 194 |
| 593 | 195 Cone : { c₁ c₂ ℓ : Level} ( C : Category c₁ c₂ ℓ ) ( I : Set c₁ ) ( s : Small C I ) ( Γ : Functor C (Sets {c₁} ) ) |
| 196 → NTrans C Sets (K Sets C ( slim (ΓObj s Γ) (ΓMap s Γ) )) Γ | |
| 197 Cone {c₁} C I s Γ = record { | |
| 198 TMap = λ i → λ se → proj ( slprod se ) i | |
| 199 ; isNTrans = record { commute = commute1 } | |
| 200 } where | |
| 201 commute1 : {a b : Obj C} {f : Hom C a b} → Sets [ Sets [ FMap Γ f o (λ se → proj (slprod se) a) ] ≈ | |
| 202 Sets [ (λ se → proj (slprod se) b) o FMap (K Sets C (slim (ΓObj s Γ) (ΓMap s Γ))) f ] ] | |
| 203 commute1 {a} {b} {f} = extensionality Sets ( λ se → begin | |
| 204 FMap Γ f (proj (slprod se ) a ) | |
| 205 ≡⟨ ≡cong ( λ g → FMap Γ g (proj (slprod se) a)) (sym ( hom-iso s ) ) ⟩ | |
| 206 FMap Γ (hom← s ( hom→ s f)) (proj (slprod se) a) | |
| 207 ≡⟨ fe=ge0 (slequ se a b ( hom→ s f) ) ⟩ | |
| 208 proj (slprod se) b | |
| 592 | 209 ∎ ) where |
| 210 open import Relation.Binary.PropositionalEquality | |
| 211 open ≡-Reasoning | |
| 212 | |
| 585 | 213 SetsLimit : { c₁ c₂ ℓ : Level} ( C : Category c₁ c₂ ℓ ) ( I : Set c₁ ) ( small : Small C I ) ( Γ : Functor C (Sets {c₁} ) ) |
| 214 → Limit Sets C Γ | |
| 215 SetsLimit { c₂} C I s Γ = record { | |
| 216 a0 = slim (ΓObj s Γ) (ΓMap s Γ) | |
| 587 | 217 ; t0 = Cone C I s Γ |
| 585 | 218 ; isLimit = record { |
| 219 limit = limit1 | |
| 593 | 220 ; t0f=t = λ {a t i } → refl |
| 221 ; limit-uniqueness = λ {a t f } → limit-uniqueness {a} {t} {f} | |
| 585 | 222 } |
| 223 } where | |
| 224 limit1 : (a : Obj Sets) → NTrans C Sets (K Sets C a) Γ → Hom Sets a (slim (ΓObj s Γ) (ΓMap s Γ)) | |
| 593 | 225 limit1 a t = λ x → record { |
| 226 slequ = λ i j f → elem {!!} ( ≡cong ( λ f → f x ) ( IsNTrans.commute ( isNTrans t ) )) | |
| 227 ; slproj = λ i → ( TMap t i ) x | |
| 228 } | |
| 229 limit-uniqueness : {a : Obj Sets} {t : NTrans C Sets (K Sets C a) Γ} {f : Hom Sets a (slim (ΓObj s Γ) (ΓMap s Γ))} → ({i : Obj C} → Sets [ Sets [ TMap (Cone C I s Γ) i o f ] ≈ TMap t i ]) → Sets [ limit1 a t ≈ f ] | |
| 230 limit-uniqueness {a} {t} {f} cif=t = extensionality Sets ( λ x → begin | |
| 231 limit1 a t x | |
| 232 ≡⟨ {!!} ⟩ | |
| 233 f x | |
| 234 ∎ ) where | |
| 235 open import Relation.Binary.PropositionalEquality | |
| 236 open ≡-Reasoning | |
| 237 | |
| 238 | |
| 239 | |
| 240 | |
| 241 | |
| 242 | |
| 243 | |
| 244 |