Mercurial > hg > Members > kono > Proof > category
annotate pullback.agda @ 790:1e7319868d77
Sets is CCC
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Fri, 19 Apr 2019 23:42:19 +0900 |
| parents | 6b4bd02efd80 |
| children |
| rev | line source |
|---|---|
| 260 | 1 -- Pullback from product and equalizer |
| 2 -- | |
| 3 -- | |
| 4 -- Shinji KONO <kono@ie.u-ryukyu.ac.jp> | |
| 5 ---- | |
| 6 | |
| 7 open import Category -- https://github.com/konn/category-agda | |
| 8 open import Level | |
| 266 | 9 module pullback { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) { c₁' c₂' ℓ' : Level} ( I : Category c₁' c₂' ℓ') ( Γ : Functor I A ) where |
| 260 | 10 |
| 11 open import HomReasoning | |
| 12 open import cat-utility | |
| 13 | |
| 282 | 14 -- |
| 264 | 15 -- Pullback from equalizer and product |
| 260 | 16 -- f |
| 300 | 17 -- a ------→ c |
| 682 | 18 -- ^ ^ |
| 19 -- π1 | |g | |
| 20 -- | | | |
| 21 -- axb -----→ b | |
| 260 | 22 -- ^ π2 |
| 23 -- | | |
| 282 | 24 -- | e = equalizer (f π1) (g π1) |
| 264 | 25 -- | |
| 26 -- d <------------------ d' | |
| 27 -- k (π1' × π2' ) | |
| 260 | 28 |
| 261 | 29 open Equalizer |
| 443 | 30 open IsEqualizer |
| 672 | 31 open IsProduct |
| 261 | 32 |
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33 pullback-from : {a b c : Obj A} |
| 260 | 34 ( f : Hom A a c ) ( g : Hom A b c ) |
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35 ( eqa : {a b : Obj A} → (f g : Hom A a b) → Equalizer A f g ) |
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36 ( prod : ( a b : Obj A ) → Product A a b ) → Pullback A f g |
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37 pullback-from {a} {b} {c} f g eqa prod0 = record { |
| 682 | 38 ab = d ; |
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39 π1 = A [ π1 o e ] ; |
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40 π2 = A [ π2 o e ] ; |
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41 isPullback = record { |
| 260 | 42 commute = commute1 ; |
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43 pullback = p1 ; |
| 282 | 44 π1p=π1 = λ {d} {π1'} {π2'} {eq} → π1p=π11 {d} {π1'} {π2'} {eq} ; |
| 45 π2p=π2 = λ {d} {π1'} {π2'} {eq} → π2p=π21 {d} {π1'} {π2'} {eq} ; | |
| 260 | 46 uniqueness = uniqueness1 |
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47 } |
| 282 | 48 } where |
| 686 | 49 axb : Obj A |
| 50 axb = Product.product (prod0 a b) | |
| 51 π1 : Hom A axb a | |
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52 π1 = Product.π1 (prod0 a b ) |
| 686 | 53 π2 : Hom A axb b |
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54 π2 = Product.π2 (prod0 a b ) |
| 682 | 55 d : Obj A |
| 56 d = equalizer-c (eqa (A [ f o π1 ]) (A [ g o π2 ])) | |
| 686 | 57 e : Hom A d axb |
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58 e = equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) |
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59 prod = Product.isProduct (prod0 a b) |
| 682 | 60 commute1 : A [ A [ f o A [ π1 o e ] ] |
| 61 ≈ A [ g o A [ π2 o e ] ] ] | |
| 262 | 62 commute1 = let open ≈-Reasoning (A) in |
| 63 begin | |
| 682 | 64 f o ( π1 o e ) |
| 262 | 65 ≈⟨ assoc ⟩ |
| 682 | 66 ( f o π1 ) o e |
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67 ≈⟨ fe=ge (isEqualizer (eqa (A [ f o π1 ]) (A [ g o π2 ]))) ⟩ |
| 682 | 68 ( g o π2 ) o e |
| 262 | 69 ≈↑⟨ assoc ⟩ |
| 682 | 70 g o ( π2 o e ) |
| 262 | 71 ∎ |
| 282 | 72 lemma1 : {d' : Obj A} {π1' : Hom A d' a} {π2' : Hom A d' b} → A [ A [ f o π1' ] ≈ A [ g o π2' ] ] → |
| 262 | 73 A [ A [ A [ f o π1 ] o (prod × π1') π2' ] ≈ A [ A [ g o π2 ] o (prod × π1') π2' ] ] |
| 282 | 74 lemma1 {d'} { π1' } { π2' } eq = let open ≈-Reasoning (A) in |
| 262 | 75 begin |
| 76 ( f o π1 ) o (prod × π1') π2' | |
| 77 ≈↑⟨ assoc ⟩ | |
| 78 f o ( π1 o (prod × π1') π2' ) | |
| 79 ≈⟨ cdr (π1fxg=f prod) ⟩ | |
| 80 f o π1' | |
| 81 ≈⟨ eq ⟩ | |
| 82 g o π2' | |
| 83 ≈↑⟨ cdr (π2fxg=g prod) ⟩ | |
| 84 g o ( π2 o (prod × π1') π2' ) | |
| 85 ≈⟨ assoc ⟩ | |
| 86 ( g o π2 ) o (prod × π1') π2' | |
| 87 ∎ | |
| 682 | 88 p1 : {d' : Obj A} {π1' : Hom A d' a} {π2' : Hom A d' b} → A [ A [ f o π1' ] ≈ A [ g o π2' ] ] → Hom A d' d |
| 282 | 89 p1 {d'} { π1' } { π2' } eq = |
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90 let open ≈-Reasoning (A) in k (isEqualizer (eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ))) (_×_ prod π1' π2' ) ( lemma1 eq ) |
| 282 | 91 π1p=π11 : {d₁ : Obj A} {π1' : Hom A d₁ a} {π2' : Hom A d₁ b} {eq : A [ A [ f o π1' ] ≈ A [ g o π2' ] ]} → |
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92 A [ A [ A [ π1 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ]) ) ] o p1 eq ] ≈ π1' ] |
| 262 | 93 π1p=π11 {d'} {π1'} {π2'} {eq} = let open ≈-Reasoning (A) in |
| 94 begin | |
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95 ( π1 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ]) ) ) o p1 eq |
| 262 | 96 ≈⟨⟩ |
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97 ( π1 o e) o k (isEqualizer ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) )) (_×_ prod π1' π2' ) (lemma1 eq) |
| 262 | 98 ≈↑⟨ assoc ⟩ |
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99 π1 o ( e o k (isEqualizer ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) )) (_×_ prod π1' π2' ) (lemma1 eq) ) |
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100 ≈⟨ cdr ( ek=h (isEqualizer ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) ))) ⟩ |
| 282 | 101 π1 o (_×_ prod π1' π2' ) |
| 262 | 102 ≈⟨ π1fxg=f prod ⟩ |
| 103 π1' | |
| 104 ∎ | |
| 282 | 105 π2p=π21 : {d₁ : Obj A} {π1' : Hom A d₁ a} {π2' : Hom A d₁ b} {eq : A [ A [ f o π1' ] ≈ A [ g o π2' ] ]} → |
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106 A [ A [ A [ π2 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ]) ) ] o p1 eq ] ≈ π2' ] |
| 262 | 107 π2p=π21 {d'} {π1'} {π2'} {eq} = let open ≈-Reasoning (A) in |
| 108 begin | |
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109 ( π2 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ]) ) ) o p1 eq |
| 262 | 110 ≈⟨⟩ |
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111 ( π2 o e) o k (isEqualizer ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) )) (_×_ prod π1' π2' ) (lemma1 eq) |
| 262 | 112 ≈↑⟨ assoc ⟩ |
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113 π2 o ( e o k (isEqualizer ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) )) (_×_ prod π1' π2' ) (lemma1 eq) ) |
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114 ≈⟨ cdr ( ek=h (isEqualizer ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) ))) ⟩ |
| 282 | 115 π2 o (_×_ prod π1' π2' ) |
| 262 | 116 ≈⟨ π2fxg=g prod ⟩ |
| 117 π2' | |
| 118 ∎ | |
| 682 | 119 uniqueness1 : {d₁ : Obj A} (p' : Hom A d₁ d) {π1' : Hom A d₁ a} {π2' : Hom A d₁ b} |
| 302 | 120 {eq : A [ A [ f o π1' ] ≈ A [ g o π2' ] ]} → |
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121 {eq1 : A [ A [ A [ π1 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] o p' ] ≈ π1' ]} → |
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122 {eq2 : A [ A [ A [ π2 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] o p' ] ≈ π2' ]} → |
| 261 | 123 A [ p1 eq ≈ p' ] |
| 264 | 124 uniqueness1 {d'} p' {π1'} {π2'} {eq} {eq1} {eq2} = let open ≈-Reasoning (A) in |
| 263 | 125 begin |
| 126 p1 eq | |
| 127 ≈⟨⟩ | |
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128 k (isEqualizer ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) )) (_×_ prod π1' π2' ) (lemma1 eq) |
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129 ≈⟨ IsEqualizer.uniqueness (isEqualizer (eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) )) ( begin |
| 264 | 130 e o p' |
| 131 ≈⟨⟩ | |
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132 equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) o p' |
| 672 | 133 ≈↑⟨ IsProduct.uniqueness prod ⟩ |
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134 (prod × ( π1 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) o p') ) ( π2 o (equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) o p')) |
| 264 | 135 ≈⟨ ×-cong prod (assoc) (assoc) ⟩ |
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136 (prod × (A [ A [ π1 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] o p' ])) |
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137 (A [ A [ π2 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] o p' ]) |
| 264 | 138 ≈⟨ ×-cong prod eq1 eq2 ⟩ |
| 139 ((prod × π1') π2') | |
| 140 ∎ ) ⟩ | |
| 263 | 141 p' |
| 142 ∎ | |
| 143 | |
| 266 | 144 -------------------------------- |
| 145 -- | |
| 146 -- If we have two limits on c and c', there are isomorphic pair h, h' | |
| 147 | |
| 148 open Limit | |
| 487 | 149 open IsLimit |
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150 open NTrans |
| 266 | 151 |
| 152 iso-l : { c₁' c₂' ℓ' : Level} ( I : Category c₁' c₂' ℓ' ) ( Γ : Functor I A ) | |
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153 ( lim : Limit I A Γ ) → ( lim' : Limit I A Γ ) |
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154 → Hom A (a0 lim )(a0 lim') |
| 487 | 155 iso-l I Γ lim lim' = limit (isLimit lim') (a0 lim) ( t0 lim) |
| 266 | 156 |
| 157 iso-r : { c₁' c₂' ℓ' : Level} ( I : Category c₁' c₂' ℓ' ) ( Γ : Functor I A ) | |
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158 ( lim : Limit I A Γ ) → ( lim' : Limit I A Γ ) |
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159 → Hom A (a0 lim') (a0 lim) |
| 487 | 160 iso-r I Γ lim lim' = limit (isLimit lim) (a0 lim') (t0 lim') |
| 266 | 161 |
| 162 | |
| 163 iso-lr : { c₁' c₂' ℓ' : Level} ( I : Category c₁' c₂' ℓ' ) ( Γ : Functor I A ) | |
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164 ( lim : Limit I A Γ ) → ( lim' : Limit I A Γ ) → |
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165 ∀{ i : Obj I } → A [ A [ iso-l I Γ lim lim' o iso-r I Γ lim lim' ] ≈ id1 A (a0 lim') ] |
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166 iso-lr I Γ lim lim' {i} = let open ≈-Reasoning (A) in begin |
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167 iso-l I Γ lim lim' o iso-r I Γ lim lim' |
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168 ≈⟨⟩ |
| 487 | 169 limit (isLimit lim') (a0 lim) ( t0 lim) o limit (isLimit lim) (a0 lim') (t0 lim') |
| 495 | 170 ≈↑⟨ limit-uniqueness (isLimit lim') ( λ {i} → ( begin |
| 487 | 171 TMap (t0 lim') i o ( limit (isLimit lim') (a0 lim) (t0 lim) o limit (isLimit lim) (a0 lim') (t0 lim') ) |
| 266 | 172 ≈⟨ assoc ⟩ |
| 487 | 173 ( TMap (t0 lim') i o limit (isLimit lim') (a0 lim) (t0 lim) ) o limit (isLimit lim) (a0 lim') (t0 lim') |
| 174 ≈⟨ car ( t0f=t (isLimit lim') ) ⟩ | |
| 175 TMap (t0 lim) i o limit (isLimit lim) (a0 lim') (t0 lim') | |
| 176 ≈⟨ t0f=t (isLimit lim) ⟩ | |
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177 TMap (t0 lim') i |
| 271 | 178 ∎) ) ⟩ |
| 487 | 179 limit (isLimit lim') (a0 lim') (t0 lim') |
| 495 | 180 ≈⟨ limit-uniqueness (isLimit lim') idR ⟩ |
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181 id (a0 lim' ) |
| 266 | 182 ∎ |
| 183 | |
| 184 | |
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185 |
| 282 | 186 open import CatExponetial |
| 267 | 187 |
| 188 open Functor | |
| 189 | |
| 190 -------------------------------- | |
| 191 -- | |
| 363 | 192 -- Constancy Functor |
| 266 | 193 |
| 268 | 194 KI : { c₁' c₂' ℓ' : Level} ( I : Category c₁' c₂' ℓ' ) → Functor A ( A ^ I ) |
| 195 KI { c₁'} {c₂'} {ℓ'} I = record { | |
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196 FObj = λ a → K I A a ; |
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197 FMap = λ f → record { -- NTrans I A (K I A a) (K I A b) |
| 267 | 198 TMap = λ a → f ; |
| 282 | 199 isNTrans = record { |
| 267 | 200 commute = λ {a b f₁} → commute1 {a} {b} {f₁} f |
| 201 } | |
| 282 | 202 } ; |
| 266 | 203 isFunctor = let open ≈-Reasoning (A) in record { |
| 267 | 204 ≈-cong = λ f=g {x} → f=g |
| 266 | 205 ; identity = refl-hom |
| 267 | 206 ; distr = refl-hom |
| 266 | 207 } |
| 267 | 208 } where |
| 209 commute1 : {a b : Obj I} {f₁ : Hom I a b} → {a' b' : Obj A} → (f : Hom A a' b' ) → | |
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210 A [ A [ FMap (K I A b') f₁ o f ] ≈ A [ f o FMap (K I A a') f₁ ] ] |
| 282 | 211 commute1 {a} {b} {f₁} {a'} {b'} f = let open ≈-Reasoning (A) in begin |
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212 FMap (K I A b') f₁ o f |
| 267 | 213 ≈⟨ idL ⟩ |
| 214 f | |
| 215 ≈↑⟨ idR ⟩ | |
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216 f o FMap (K I A a') f₁ |
| 267 | 217 ∎ |
| 218 | |
| 219 | |
| 272 | 220 --------- |
| 221 -- | |
| 298 | 222 -- Limit Constancy Functor F : A → A^I has right adjoint |
| 223 -- | |
| 224 -- we are going to prove universal mapping | |
| 225 | |
| 226 --------- | |
| 227 -- | |
| 272 | 228 -- limit gives co universal mapping ( i.e. adjunction ) |
| 229 -- | |
| 230 -- F = KI I : Functor A (A ^ I) | |
| 282 | 231 -- U = λ b → A0 (lim b {a0 b} {t0 b} |
| 232 -- ε = λ b → T0 ( lim b {a0 b} {t0 b} ) | |
| 475 | 233 -- |
| 234 -- a0 : Obj A and t0 : NTrans K Γ come from the limit | |
| 272 | 235 |
| 282 | 236 limit2couniv : |
|
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|
237 ( lim : ( Γ : Functor I A ) → Limit I A Γ ) |
|
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|
238 → coUniversalMapping A ( A ^ I ) (KI I) |
|
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parents:
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|
239 limit2couniv lim = record { |
|
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|
240 U = λ b → a0 ( lim b) ; |
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241 ε = λ b → t0 (lim b) ; |
|
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|
242 _*' = λ {b} {a} k → limit (isLimit (lim b )) a k ; |
| 693 | 243 iscoUniversalMapping = record { couniversalMapping = λ{ b a f} → couniversalMapping1 {b} {a} {f} ; |
| 271 | 244 couniquness = couniquness2 |
|
270
8ba03259a177
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|
245 } |
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246 } where |
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269
diff
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|
247 couniversalMapping1 : {b : Obj (A ^ I)} {a : Obj A} {f : Hom (A ^ I) (FObj (KI I) a) b} → |
| 487 | 248 A ^ I [ A ^ I [ t0 (lim b) o FMap (KI I) (limit (isLimit (lim b)) a f) ] ≈ f ] |
|
270
8ba03259a177
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269
diff
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|
249 couniversalMapping1 {b} {a} {f} {i} = let open ≈-Reasoning (A) in begin |
| 487 | 250 TMap (t0 (lim b )) i o TMap ( FMap (KI I) (limit (isLimit (lim b )) a f) ) i |
|
270
8ba03259a177
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diff
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|
251 ≈⟨⟩ |
| 487 | 252 TMap (t0 (lim b)) i o (limit (isLimit (lim b)) a f) |
| 253 ≈⟨ t0f=t (isLimit (lim b)) ⟩ | |
|
270
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254 TMap f i -- i comes from ∀{i} → B [ TMap f i ≈ TMap g i ] |
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|
255 ∎ |
|
484
fcae3025d900
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|
256 couniquness2 : {b : Obj (A ^ I)} {a : Obj A} {f : Hom (A ^ I) (FObj (KI I) a) b} {g : Hom A a (a0 (lim b ))} → |
|
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257 ( ∀ { i : Obj I } → A [ A [ TMap (t0 (lim b )) i o TMap ( FMap (KI I) g) i ] ≈ TMap f i ] ) |
| 487 | 258 → A [ limit (isLimit (lim b )) a f ≈ g ] |
| 271 | 259 couniquness2 {b} {a} {f} {g} lim-g=f = let open ≈-Reasoning (A) in begin |
| 487 | 260 limit (isLimit (lim b )) a f |
| 495 | 261 ≈⟨ limit-uniqueness (isLimit ( lim b )) lim-g=f ⟩ |
|
270
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262 g |
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263 ∎ |
| 268 | 264 |
| 282 | 265 univ2limit : |
|
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266 ( univ : coUniversalMapping A (A ^ I) (KI I) ) → |
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267 ( Γ : Functor I A ) → Limit I A Γ |
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268 univ2limit univ Γ = record { |
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269 a0 = (coUniversalMapping.U univ ) Γ ; |
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270 t0 = (coUniversalMapping.ε univ ) Γ ; |
| 487 | 271 isLimit = record { |
| 272 limit = λ a t → limit1 a t ; | |
| 273 t0f=t = λ {a t i } → t0f=t1 {a} {t} {i} ; | |
| 495 | 274 limit-uniqueness = λ {a} {t} {f} t=f → limit-uniqueness1 {a} {t} {f} t=f |
| 487 | 275 } |
| 272 | 276 } where |
|
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277 U : Obj (A ^ I) → Obj A |
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278 U b = coUniversalMapping.U univ b |
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279 ε : ( b : Obj (A ^ I ) ) → NTrans I A (FObj (KI I) (U b)) b |
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280 ε b = coUniversalMapping.ε univ b |
| 689 | 281 limit1 : (a : Obj A) → NTrans I A (FObj (KI I) a) Γ → Hom A a (U Γ) |
| 282 limit1 a t = coUniversalMapping._*' univ {_} {a} t | |
|
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283 t0f=t1 : {a : Obj A} {t : NTrans I A (K I A a) Γ} {i : Obj I} → |
| 689 | 284 A [ A [ TMap (ε Γ) i o limit1 a t ] ≈ TMap t i ] |
|
274
1b651faa2391
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273
diff
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|
285 t0f=t1 {a} {t} {i} = let open ≈-Reasoning (A) in begin |
| 689 | 286 TMap (ε Γ) i o limit1 a t |
|
274
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273
diff
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|
287 ≈⟨⟩ |
| 689 | 288 TMap (ε Γ) i o coUniversalMapping._*' univ t |
| 289 ≈⟨ coIsUniversalMapping.couniversalMapping ( coUniversalMapping.iscoUniversalMapping univ) {Γ} {a} {t} ⟩ | |
|
274
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diff
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|
290 TMap t i |
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291 ∎ |
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292 limit-uniqueness1 : { a : Obj A } → { t : NTrans I A ( K I A a ) Γ } → { f : Hom A a (U Γ)} |
| 689 | 293 → ( ∀ { i : Obj I } → A [ A [ TMap (ε Γ) i o f ] ≈ TMap t i ] ) → A [ limit1 a t ≈ f ] |
|
274
1b651faa2391
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273
diff
changeset
|
294 limit-uniqueness1 {a} {t} {f} εf=t = let open ≈-Reasoning (A) in begin |
| 689 | 295 coUniversalMapping._*' univ t |
| 296 ≈⟨ ( coIsUniversalMapping.couniquness ( coUniversalMapping.iscoUniversalMapping univ) ) εf=t ⟩ | |
|
274
1b651faa2391
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273
diff
changeset
|
297 f |
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diff
changeset
|
298 ∎ |
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diff
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|
299 |
|
303
7f40d6a51c72
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parents:
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diff
changeset
|
300 |
|
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changeset
|
301 |
| 686 | 302 -- another form of uniqueness of a product |
| 672 | 303 lemma-p0 : (a b ab : Obj A) ( π1 : Hom A ab a ) ( π2 : Hom A ab b ) ( prod : IsProduct A a b ab π1 π2 ) → |
|
303
7f40d6a51c72
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parents:
302
diff
changeset
|
304 A [ _×_ prod π1 π2 ≈ id1 A ab ] |
|
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302
diff
changeset
|
305 lemma-p0 a b ab π1 π2 prod = let open ≈-Reasoning (A) in begin |
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parents:
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diff
changeset
|
306 _×_ prod π1 π2 |
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parents:
302
diff
changeset
|
307 ≈↑⟨ ×-cong prod idR idR ⟩ |
|
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parents:
302
diff
changeset
|
308 _×_ prod (A [ π1 o id1 A ab ]) (A [ π2 o id1 A ab ]) |
| 672 | 309 ≈⟨ IsProduct.uniqueness prod ⟩ |
|
303
7f40d6a51c72
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parents:
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diff
changeset
|
310 id1 A ab |
|
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diff
changeset
|
311 ∎ |
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parents:
302
diff
changeset
|
312 |
|
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diff
changeset
|
313 |
| 282 | 314 open IProduct |
|
508
3ce21b2a671a
IProduct is written in Sets
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parents:
495
diff
changeset
|
315 open IsIProduct |
| 283 | 316 open Equalizer |
|
281
dbd2044add2a
limit from product and equalizer continue...
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parents:
280
diff
changeset
|
317 |
| 282 | 318 -- |
| 319 -- limit from equalizer and product | |
| 320 -- | |
| 685 | 321 -- Γu |
| 692 | 322 -- → Γj → Γk ← |
| 779 | 323 -- / ↑ ↑ \ |
| 324 -- proj j / | | \ proj k | |
| 325 -- / μu| |μu \ Equalizer have to be independent from j and k | |
| 692 | 326 -- | | | | so we need products of Obj I and Hom I |
| 686 | 327 -- |product of Hom I | |
| 328 -- | ↑ ↑ | K u = id lim | |
| 779 | 329 -- | } | | |
| 330 -- | f(id)| |g(Γ) | lim = K j -----------→ K k = lim | |
| 686 | 331 -- | | | | | u | |
| 779 | 332 -- \ | | / proj j o e = ε j | | ε k = proj k o e |
| 686 | 333 -- product of Obj I μ u o g o e | | μ u o f o e |
| 334 -- ↑ | | | |
|
678
867ea41b331c
extensionality remains
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parents:
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diff
changeset
|
335 -- | e = equalizer f g | | |
| 682 | 336 -- | ↓ ↓ |
| 686 | 337 -- lim ←---------------- d' Γ j ----------→ Γ k |
| 682 | 338 -- k ( product pi ) Γ u |
| 339 -- Γ u o ε j = ε k | |
| 283 | 340 -- |
| 291 | 341 |
|
678
867ea41b331c
extensionality remains
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parents:
677
diff
changeset
|
342 -- homprod should be written by IProduct |
| 686 | 343 -- If I is locally small, this is iso to a set |
|
678
867ea41b331c
extensionality remains
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parents:
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diff
changeset
|
344 record homprod {c : Level } : Set (suc c₁' ⊔ suc c₂' ) where |
|
867ea41b331c
extensionality remains
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parents:
677
diff
changeset
|
345 field |
|
867ea41b331c
extensionality remains
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parents:
677
diff
changeset
|
346 hom-j : Obj I |
|
867ea41b331c
extensionality remains
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parents:
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diff
changeset
|
347 hom-k : Obj I |
|
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diff
changeset
|
348 hom : Hom I hom-j hom-k |
|
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diff
changeset
|
349 open homprod |
|
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extensionality remains
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diff
changeset
|
350 |
|
680
5d894993477f
fh=gh done. limit from product and equalizer done
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parents:
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diff
changeset
|
351 Homprod : {j k : Obj I} (u : Hom I j k) → homprod {c₁} |
|
5d894993477f
fh=gh done. limit from product and equalizer done
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parents:
679
diff
changeset
|
352 Homprod {j} {k} u = record {hom-j = j ; hom-k = k ; hom = u} |
|
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parents:
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diff
changeset
|
353 |
| 282 | 354 limit-from : |
|
691
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changeset
|
355 ( prod : {c : Level} { I : Set c } → ( ai : I → Obj A ) → IProduct I A ai ) |
|
673
0007f9a25e9c
fix limit from product and equalizer (not yet finished )
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parents:
672
diff
changeset
|
356 ( eqa : {a b : Obj A} → (f g : Hom A a b) → Equalizer A f g ) |
|
691
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parents:
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changeset
|
357 → Limit I A Γ |
|
673
0007f9a25e9c
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parents:
672
diff
changeset
|
358 limit-from prod eqa = record { |
| 682 | 359 a0 = d ; |
| 685 | 360 t0 = cone-ε ; |
| 487 | 361 isLimit = record { |
| 685 | 362 limit = λ a t → cone1 a t ; |
|
673
0007f9a25e9c
fix limit from product and equalizer (not yet finished )
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parents:
672
diff
changeset
|
363 t0f=t = λ {a t i } → t0f=t1 {a} {t} {i} ; |
| 495 | 364 limit-uniqueness = λ {a} {t} {f} t=f → limit-uniqueness1 {a} {t} {f} t=f |
| 487 | 365 } |
| 282 | 366 } where |
|
680
5d894993477f
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diff
changeset
|
367 p0 : Obj A |
|
673
0007f9a25e9c
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diff
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|
368 p0 = iprod (prod (FObj Γ)) |
| 682 | 369 Fcod : homprod {c₁} → Obj A |
| 370 Fcod = λ u → FObj Γ ( hom-k u ) | |
| 679 | 371 f : Hom A p0 (iprod (prod Fcod)) |
|
680
5d894993477f
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parents:
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diff
changeset
|
372 f = iproduct (isIProduct (prod Fcod)) (λ u → pi (prod (FObj Γ)) (hom-k u )) |
| 679 | 373 g : Hom A p0 (iprod (prod Fcod)) |
|
680
5d894993477f
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parents:
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diff
changeset
|
374 g = iproduct (isIProduct (prod Fcod)) (λ u → A [ FMap Γ (hom u) o pi (prod (FObj Γ)) (hom-j u ) ] ) |
| 682 | 375 equ-ε : Equalizer A g f |
| 376 equ-ε = eqa g f | |
| 377 d : Obj A | |
| 378 d = equalizer-c equ-ε | |
| 379 e : Hom A d p0 | |
| 380 e = equalizer equ-ε | |
| 381 equ = isEqualizer equ-ε | |
| 683 | 382 -- projection of the product of Obj I |
| 682 | 383 proj : (i : Obj I) → Hom A p0 (FObj Γ i) |
| 384 proj = pi ( prod (FObj Γ) ) | |
| 385 prodΓ = isIProduct ( prod (FObj Γ) ) | |
| 683 | 386 -- projection of the product of Hom I |
| 686 | 387 μ : {j k : Obj I} → (u : Hom I j k ) → Hom A (iprod (prod Fcod)) (Fcod (Homprod u)) |
| 388 μ u = pi (prod Fcod ) (Homprod u) | |
|
691
917e51be9bbf
change argument of Limit and K
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parents:
690
diff
changeset
|
389 cone-ε : NTrans I A (K I A (equalizer-c equ-ε ) ) Γ |
| 685 | 390 cone-ε = record { |
| 682 | 391 TMap = λ i → tmap i ; |
| 392 isNTrans = record { commute = commute1 } | |
| 393 } where | |
|
691
917e51be9bbf
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394 tmap : (i : Obj I) → Hom A (FObj (K I A d) i) (FObj Γ i) |
| 682 | 395 tmap i = A [ proj i o e ] |
| 396 commute1 : {j k : Obj I} {u : Hom I j k} → | |
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397 A [ A [ FMap Γ u o tmap j ] ≈ A [ tmap k o FMap (K I A d) u ] ] |
| 682 | 398 commute1 {j} {k} {u} = let open ≈-Reasoning (A) in begin |
| 399 FMap Γ u o tmap j | |
| 400 ≈⟨⟩ | |
| 401 FMap Γ u o ( proj j o e ) | |
| 402 ≈⟨ assoc ⟩ | |
| 403 ( FMap Γ u o pi (prod (FObj Γ)) j ) o e | |
| 404 ≈↑⟨ car ( pif=q (isIProduct (prod Fcod )) ) ⟩ | |
| 686 | 405 ( μ u o g ) o e |
| 682 | 406 ≈↑⟨ assoc ⟩ |
| 686 | 407 μ u o (g o e ) |
| 682 | 408 ≈⟨ cdr ( fe=ge (isEqualizer equ-ε )) ⟩ |
| 686 | 409 μ u o (f o e ) |
| 682 | 410 ≈⟨ assoc ⟩ |
| 686 | 411 (μ u o f ) o e |
| 682 | 412 ≈⟨ car ( pif=q (isIProduct (prod Fcod ))) ⟩ |
| 413 pi (prod (FObj Γ)) k o e | |
| 414 ≈⟨⟩ | |
| 415 proj k o e | |
| 416 ≈↑⟨ idR ⟩ | |
| 417 (proj k o e ) o id1 A d | |
| 418 ≈⟨⟩ | |
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419 tmap k o FMap (K I A d) u |
| 682 | 420 ∎ |
| 683 | 421 -- an arrow to our product of Obj I ( is an equalizer because of commutativity of t ) |
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422 h : {a : Obj A} → (t : NTrans I A (K I A a) Γ ) → Hom A a p0 |
| 683 | 423 h t = iproduct prodΓ (TMap t) |
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424 fh=gh : (a : Obj A) → (t : NTrans I A (K I A a) Γ ) → |
| 683 | 425 A [ A [ g o h t ] ≈ A [ f o h t ] ] |
| 679 | 426 fh=gh a t = let open ≈-Reasoning (A) in begin |
| 683 | 427 g o h t |
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428 ≈↑⟨ ip-uniqueness (isIProduct (prod Fcod)) ⟩ |
| 683 | 429 iproduct (isIProduct (prod Fcod)) (λ u → pi (prod Fcod) u o ( g o h t )) |
| 682 | 430 ≈⟨ ip-cong (isIProduct (prod Fcod)) ( λ u → begin |
| 683 | 431 pi (prod Fcod) u o ( g o h t ) |
| 682 | 432 ≈⟨ assoc ⟩ |
| 683 | 433 ( pi (prod Fcod) u o g ) o h t |
| 682 | 434 ≈⟨ car (pif=q (isIProduct (prod Fcod ))) ⟩ |
| 683 | 435 (FMap Γ (hom u) o pi (prod (FObj Γ)) (hom-j u) ) o h t |
| 682 | 436 ≈↑⟨ assoc ⟩ |
| 683 | 437 FMap Γ (hom u) o (pi (prod (FObj Γ)) (hom-j u) o h t ) |
| 682 | 438 ≈⟨ cdr ( pif=q prodΓ ) ⟩ |
| 439 FMap Γ (hom u) o TMap t (hom-j u) | |
| 440 ≈⟨ IsNTrans.commute (isNTrans t) ⟩ | |
| 441 TMap t (hom-k u) o id1 A a | |
| 442 ≈⟨ idR ⟩ | |
| 443 TMap t (hom-k u) | |
| 444 ≈↑⟨ pif=q prodΓ ⟩ | |
| 683 | 445 pi (prod (FObj Γ)) (hom-k u) o h t |
| 682 | 446 ≈↑⟨ car (pif=q (isIProduct (prod Fcod ))) ⟩ |
| 683 | 447 (pi (prod Fcod) u o f ) o h t |
| 682 | 448 ≈↑⟨ assoc ⟩ |
| 683 | 449 pi (prod Fcod) u o ( f o h t ) |
| 682 | 450 ∎ |
| 451 ) ⟩ | |
| 683 | 452 iproduct (isIProduct (prod Fcod)) (λ u → pi (prod Fcod) u o ( f o h t )) |
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453 ≈⟨ ip-uniqueness (isIProduct (prod Fcod)) ⟩ |
| 683 | 454 f o h t |
|
678
867ea41b331c
extensionality remains
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455 ∎ |
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456 cone1 : (a : Obj A) → NTrans I A (K I A a) Γ → Hom A a d |
| 685 | 457 cone1 a t = k equ (h t) ( fh=gh a t ) |
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458 t0f=t1 : {a : Obj A} {t : NTrans I A (K I A a) Γ} {i : Obj I} → |
| 685 | 459 A [ A [ TMap cone-ε i o cone1 a t ] ≈ TMap t i ] |
| 283 | 460 t0f=t1 {a} {t} {i} = let open ≈-Reasoning (A) in begin |
| 685 | 461 TMap cone-ε i o cone1 a t |
| 283 | 462 ≈⟨⟩ |
| 683 | 463 ( ( proj i ) o e ) o k equ (h t) (fh=gh a t) |
| 283 | 464 ≈↑⟨ assoc ⟩ |
| 683 | 465 proj i o ( e o k equ (h t) (fh=gh a t ) ) |
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466 ≈⟨ cdr ( ek=h equ) ⟩ |
| 683 | 467 proj i o h t |
| 676 | 468 ≈⟨ pif=q prodΓ ⟩ |
| 283 | 469 TMap t i |
| 470 ∎ | |
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471 limit-uniqueness1 : {a : Obj A} {t : NTrans I A (K I A a) Γ} {f : Hom A a d} |
| 685 | 472 → ({i : Obj I} → A [ A [ TMap cone-ε i o f ] ≈ TMap t i ]) → |
| 473 A [ cone1 a t ≈ f ] | |
| 283 | 474 limit-uniqueness1 {a} {t} {f} lim=t = let open ≈-Reasoning (A) in begin |
| 685 | 475 cone1 a t |
| 283 | 476 ≈⟨⟩ |
| 683 | 477 k equ (h t) (fh=gh a t) |
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478 ≈⟨ IsEqualizer.uniqueness equ ( begin |
| 283 | 479 e o f |
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480 ≈↑⟨ ip-uniqueness prodΓ ⟩ |
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481 iproduct prodΓ (λ i → ( proj i o ( e o f ) ) ) |
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482 ≈⟨ ip-cong prodΓ ( λ i → begin |
|
285
46d4ad55b948
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483 proj i o ( e o f ) |
| 284 | 484 ≈⟨ assoc ⟩ |
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285
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485 ( proj i o e ) o f |
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486 ≈⟨ lim=t {i} ⟩ |
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487 TMap t i |
| 284 | 488 ∎ ) ⟩ |
| 683 | 489 h t |
| 283 | 490 ∎ ) ⟩ |
| 491 f | |
| 492 ∎ | |
| 493 | |
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494 |
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495 -- |
| 291 | 496 -- |
| 497 -- Adjoint functor preserves limits | |
| 498 -- | |
| 499 -- | |
| 500 | |
| 501 open import Category.Cat | |
| 502 | |
| 503 ta1 : { c₁' c₂' ℓ' : Level} (B : Category c₁' c₂' ℓ') ( Γ : Functor I B ) | |
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504 ( lim : Obj B ) ( tb : NTrans I B ( K I B lim ) Γ ) → |
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505 ( U : Functor B A) → NTrans I A ( K I A (FObj U lim) ) (U ○ Γ) |
|
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506 ta1 B Γ lim tb U = record { |
| 291 | 507 TMap = TMap (Functor*Nat I A U tb) ; |
| 508 isNTrans = record { commute = λ {a} {b} {f} → let open ≈-Reasoning (A) in begin | |
| 509 FMap (U ○ Γ) f o TMap (Functor*Nat I A U tb) a | |
| 510 ≈⟨ nat ( Functor*Nat I A U tb ) ⟩ | |
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511 TMap (Functor*Nat I A U tb) b o FMap (U ○ K I B lim) f |
| 291 | 512 ≈⟨ cdr (IsFunctor.identity (isFunctor U) ) ⟩ |
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513 TMap (Functor*Nat I A U tb) b o FMap (K I A (FObj U lim)) f |
| 291 | 514 ∎ |
| 515 } } | |
| 516 | |
| 517 adjoint-preseve-limit : | |
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518 { c₁' c₂' ℓ' : Level} (B : Category c₁' c₂' ℓ') ( Γ : Functor I B ) ( limitb : Limit I B Γ ) → |
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519 ( adj : Adjunction A B ) → Limit I A (Adjunction.U adj ○ Γ) |
|
688
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|
520 adjoint-preseve-limit B Γ limitb adj = record { |
|
484
fcae3025d900
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|
521 a0 = FObj U lim ; |
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|
522 t0 = ta1 B Γ lim tb U ; |
| 487 | 523 isLimit = record { |
| 524 limit = λ a t → limit1 a t ; | |
| 525 t0f=t = λ {a t i } → t0f=t1 {a} {t} {i} ; | |
| 495 | 526 limit-uniqueness = λ {a} {t} {f} t=f → limit-uniqueness1 {a} {t} {f} t=f |
| 487 | 527 } |
| 291 | 528 } where |
|
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|
529 U : Functor B A |
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|
530 U = Adjunction.U adj |
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|
531 F : Functor A B |
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|
532 F = Adjunction.F adj |
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diff
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|
533 η : NTrans A A identityFunctor ( U ○ F ) |
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|
534 η = Adjunction.η adj |
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|
535 ε : NTrans B B ( F ○ U ) identityFunctor |
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|
536 ε = Adjunction.ε adj |
|
484
fcae3025d900
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|
537 ta = ta1 B Γ (a0 limitb) (t0 limitb) U |
|
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|
538 tb = t0 limitb |
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|
539 lim = a0 limitb |
|
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|
540 tfmap : (a : Obj A) → NTrans I A (K I A a) (U ○ Γ) → (i : Obj I) → Hom B (FObj (K I B (FObj F a)) i) (FObj Γ i) |
| 293 | 541 tfmap a t i = B [ TMap ε (FObj Γ i) o FMap F (TMap t i) ] |
|
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diff
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|
542 tF : (a : Obj A) → NTrans I A (K I A a) (U ○ Γ) → NTrans I B (K I B (FObj F a)) Γ |
| 293 | 543 tF a t = record { |
| 544 TMap = tfmap a t ; | |
| 545 isNTrans = record { commute = λ {a'} {b} {f} → let open ≈-Reasoning (B) in begin | |
| 546 FMap Γ f o tfmap a t a' | |
| 294 | 547 ≈⟨⟩ |
| 548 FMap Γ f o ( TMap ε (FObj Γ a') o FMap F (TMap t a')) | |
| 549 ≈⟨ assoc ⟩ | |
| 550 (FMap Γ f o TMap ε (FObj Γ a') ) o FMap F (TMap t a') | |
| 551 ≈⟨ car (nat ε) ⟩ | |
| 552 (TMap ε (FObj Γ b) o FMap (F ○ U) (FMap Γ f) ) o FMap F (TMap t a') | |
| 553 ≈↑⟨ assoc ⟩ | |
| 554 TMap ε (FObj Γ b) o ( FMap (F ○ U) (FMap Γ f) o FMap F (TMap t a') ) | |
| 555 ≈↑⟨ cdr ( distr F ) ⟩ | |
| 556 TMap ε (FObj Γ b) o ( FMap F (A [ FMap U (FMap Γ f) o TMap t a' ] ) ) | |
| 557 ≈⟨ cdr ( fcong F (nat t) ) ⟩ | |
|
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|
558 TMap ε (FObj Γ b) o FMap F (A [ TMap t b o FMap (K I A a) f ]) |
| 294 | 559 ≈⟨⟩ |
|
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|
560 TMap ε (FObj Γ b) o FMap F (A [ TMap t b o id1 A (FObj (K I A a) b) ]) |
|
299
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|
561 ≈⟨ cdr ( fcong F (idR1 A)) ⟩ |
| 294 | 562 TMap ε (FObj Γ b) o FMap F (TMap t b ) |
| 563 ≈↑⟨ idR ⟩ | |
|
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|
564 ( TMap ε (FObj Γ b) o FMap F (TMap t b)) o id1 B (FObj F (FObj (K I A a) b)) |
| 294 | 565 ≈⟨⟩ |
|
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|
566 tfmap a t b o FMap (K I B (FObj F a)) f |
| 293 | 567 ∎ |
| 568 } } | |
|
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|
569 limit1 : (a : Obj A) → NTrans I A (K I A a) (U ○ Γ) → Hom A a (FObj U (a0 limitb) ) |
| 487 | 570 limit1 a t = A [ FMap U (limit (isLimit limitb) (FObj F a) (tF a t )) o TMap η a ] |
|
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|
571 t0f=t1 : {a : Obj A} {t : NTrans I A (K I A a) (U ○ Γ)} {i : Obj I} → |
| 291 | 572 A [ A [ TMap ta i o limit1 a t ] ≈ TMap t i ] |
| 295 | 573 t0f=t1 {a} {t} {i} = let open ≈-Reasoning (A) in begin |
| 574 TMap ta i o limit1 a t | |
| 575 ≈⟨⟩ | |
| 487 | 576 FMap U ( TMap tb i ) o ( FMap U (limit (isLimit limitb) (FObj F a) (tF a t )) o TMap η a ) |
| 295 | 577 ≈⟨ assoc ⟩ |
| 487 | 578 ( FMap U ( TMap tb i ) o FMap U (limit (isLimit limitb) (FObj F a) (tF a t ))) o TMap η a |
| 295 | 579 ≈↑⟨ car ( distr U ) ⟩ |
| 487 | 580 FMap U ( B [ TMap tb i o limit (isLimit limitb) (FObj F a) (tF a t ) ] ) o TMap η a |
| 581 ≈⟨ car ( fcong U ( t0f=t (isLimit limitb) ) ) ⟩ | |
| 295 | 582 FMap U (TMap (tF a t) i) o TMap η a |
| 583 ≈⟨⟩ | |
| 584 FMap U ( B [ TMap ε (FObj Γ i) o FMap F (TMap t i) ] ) o TMap η a | |
| 585 ≈⟨ car ( distr U ) ⟩ | |
| 586 ( FMap U ( TMap ε (FObj Γ i)) o FMap U ( FMap F (TMap t i) )) o TMap η a | |
| 587 ≈↑⟨ assoc ⟩ | |
| 588 FMap U ( TMap ε (FObj Γ i) ) o ( FMap U ( FMap F (TMap t i) ) o TMap η a ) | |
| 589 ≈⟨ cdr ( nat η ) ⟩ | |
| 590 FMap U (TMap ε (FObj Γ i)) o ( TMap η (FObj U (FObj Γ i)) o FMap (identityFunctor {_} {_} {_} {A}) (TMap t i) ) | |
| 591 ≈⟨ assoc ⟩ | |
| 592 ( FMap U (TMap ε (FObj Γ i)) o TMap η (FObj U (FObj Γ i))) o TMap t i | |
| 593 ≈⟨ car ( IsAdjunction.adjoint1 ( Adjunction.isAdjunction adj ) ) ⟩ | |
| 594 id1 A (FObj (U ○ Γ) i) o TMap t i | |
| 595 ≈⟨ idL ⟩ | |
| 596 TMap t i | |
| 597 ∎ | |
| 296 | 598 -- ta = TMap (Functor*Nat I A U tb) , FMap U ( TMap tb i ) o f ≈ TMap t i |
|
691
917e51be9bbf
change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
690
diff
changeset
|
599 limit-uniqueness1 : {a : Obj A} {t : NTrans I A (K I A a) (U ○ Γ)} {f : Hom A a (FObj U lim)} |
| 291 | 600 → ({i : Obj I} → A [ A [ TMap ta i o f ] ≈ TMap t i ]) → |
| 601 A [ limit1 a t ≈ f ] | |
| 295 | 602 limit-uniqueness1 {a} {t} {f} lim=t = let open ≈-Reasoning (A) in begin |
| 603 limit1 a t | |
| 604 ≈⟨⟩ | |
| 487 | 605 FMap U (limit (isLimit limitb) (FObj F a) (tF a t )) o TMap η a |
| 495 | 606 ≈⟨ car ( fcong U (limit-uniqueness (isLimit limitb) ( λ {i} → lemma1 i) )) ⟩ |
| 298 | 607 FMap U ( B [ TMap ε lim o FMap F f ] ) o TMap η a -- Universal mapping |
|
297
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
608 ≈⟨ car (distr U ) ⟩ |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
609 ( (FMap U (TMap ε lim)) o (FMap U ( FMap F f )) ) o TMap η a |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
610 ≈⟨ sym assoc ⟩ |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
611 (FMap U (TMap ε lim)) o ((FMap U ( FMap F f )) o TMap η a ) |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
612 ≈⟨ cdr (nat η) ⟩ |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
613 (FMap U (TMap ε lim)) o ((TMap η (FObj U lim )) o f ) |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
614 ≈⟨ assoc ⟩ |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
615 ((FMap U (TMap ε lim)) o (TMap η (FObj U lim))) o f |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
616 ≈⟨ car ( IsAdjunction.adjoint1 ( Adjunction.isAdjunction adj)) ⟩ |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
617 id (FObj U lim) o f |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
618 ≈⟨ idL ⟩ |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
619 f |
| 296 | 620 ∎ where |
| 621 lemma1 : (i : Obj I) → B [ B [ TMap tb i o B [ TMap ε lim o FMap F f ] ] ≈ TMap (tF a t) i ] | |
| 622 lemma1 i = let open ≈-Reasoning (B) in begin | |
| 623 TMap tb i o (TMap ε lim o FMap F f) | |
|
297
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
624 ≈⟨ assoc ⟩ |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
625 ( TMap tb i o TMap ε lim ) o FMap F f |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
626 ≈⟨ car ( nat ε ) ⟩ |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
627 ( TMap ε (FObj Γ i) o FMap F ( FMap U ( TMap tb i ))) o FMap F f |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
628 ≈↑⟨ assoc ⟩ |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
629 TMap ε (FObj Γ i) o ( FMap F ( FMap U ( TMap tb i )) o FMap F f ) |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
630 ≈↑⟨ cdr ( distr F ) ⟩ |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
631 TMap ε (FObj Γ i) o FMap F ( A [ FMap U ( TMap tb i ) o f ] ) |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
632 ≈⟨ cdr ( fcong F (lim=t {i}) ) ⟩ |
| 296 | 633 TMap ε (FObj Γ i) o FMap F (TMap t i) |
| 634 ≈⟨⟩ | |
| 635 TMap (tF a t) i | |
| 636 ∎ | |
| 295 | 637 |
| 296 | 638 |
| 639 |