Mercurial > hg > Members > kono > Proof > category
annotate pullback.agda @ 481:65e6906782bb
Completeness of Comma Category begin
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Fri, 10 Mar 2017 23:57:49 +0900 |
| parents | 4c0a955b651d |
| children | fcae3025d900 |
| 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 |
| 282 | 18 -- ^ ^ |
| 260 | 19 -- π1 | |g |
| 20 -- | | | |
| 300 | 21 -- ab ------→ 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 |
| 261 | 31 open Product |
| 32 open Pullback | |
| 33 | |
| 282 | 34 pullback-from : (a b c ab d : Obj A) |
| 260 | 35 ( f : Hom A a c ) ( g : Hom A b c ) |
| 261 | 36 ( π1 : Hom A ab a ) ( π2 : Hom A ab b ) ( e : Hom A d ab ) |
| 443 | 37 ( eqa : {a b c : Obj A} → (f g : Hom A a b) → {e : Hom A c a } → IsEqualizer A e f g ) |
| 282 | 38 ( prod : Product A a b ab π1 π2 ) → Pullback A a b c d f g |
| 443 | 39 ( A [ π1 o equalizer1 ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) ) ] ) |
| 40 ( A [ π2 o equalizer1 ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) ) ] ) | |
| 261 | 41 pullback-from a b c ab d f g π1 π2 e eqa prod = record { |
| 260 | 42 commute = commute1 ; |
| 282 | 43 p = p1 ; |
| 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 |
| 282 | 47 } where |
| 443 | 48 commute1 : A [ A [ f o A [ π1 o equalizer1 (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] ] |
| 49 ≈ A [ g o A [ π2 o equalizer1 (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] ] ] | |
| 262 | 50 commute1 = let open ≈-Reasoning (A) in |
| 51 begin | |
| 443 | 52 f o ( π1 o equalizer1 (eqa ( f o π1 ) ( g o π2 )) ) |
| 262 | 53 ≈⟨ assoc ⟩ |
| 443 | 54 ( f o π1 ) o equalizer1 (eqa ( f o π1 ) ( g o π2 )) |
| 262 | 55 ≈⟨ fe=ge (eqa (A [ f o π1 ]) (A [ g o π2 ])) ⟩ |
| 443 | 56 ( g o π2 ) o equalizer1 (eqa ( f o π1 ) ( g o π2 )) |
| 262 | 57 ≈↑⟨ assoc ⟩ |
| 443 | 58 g o ( π2 o equalizer1 (eqa ( f o π1 ) ( g o π2 )) ) |
| 262 | 59 ∎ |
| 282 | 60 lemma1 : {d' : Obj A} {π1' : Hom A d' a} {π2' : Hom A d' b} → A [ A [ f o π1' ] ≈ A [ g o π2' ] ] → |
| 262 | 61 A [ A [ A [ f o π1 ] o (prod × π1') π2' ] ≈ A [ A [ g o π2 ] o (prod × π1') π2' ] ] |
| 282 | 62 lemma1 {d'} { π1' } { π2' } eq = let open ≈-Reasoning (A) in |
| 262 | 63 begin |
| 64 ( f o π1 ) o (prod × π1') π2' | |
| 65 ≈↑⟨ assoc ⟩ | |
| 66 f o ( π1 o (prod × π1') π2' ) | |
| 67 ≈⟨ cdr (π1fxg=f prod) ⟩ | |
| 68 f o π1' | |
| 69 ≈⟨ eq ⟩ | |
| 70 g o π2' | |
| 71 ≈↑⟨ cdr (π2fxg=g prod) ⟩ | |
| 72 g o ( π2 o (prod × π1') π2' ) | |
| 73 ≈⟨ assoc ⟩ | |
| 74 ( g o π2 ) o (prod × π1') π2' | |
| 75 ∎ | |
| 261 | 76 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 | 77 p1 {d'} { π1' } { π2' } eq = |
| 262 | 78 let open ≈-Reasoning (A) in k ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} ) (_×_ prod π1' π2' ) ( lemma1 eq ) |
| 282 | 79 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' ] ]} → |
| 443 | 80 A [ A [ A [ π1 o equalizer1 (eqa (A [ f o π1 ]) (A [ g o π2 ]) {e} ) ] o p1 eq ] ≈ π1' ] |
| 262 | 81 π1p=π11 {d'} {π1'} {π2'} {eq} = let open ≈-Reasoning (A) in |
| 82 begin | |
| 443 | 83 ( π1 o equalizer1 (eqa (A [ f o π1 ]) (A [ g o π2 ]) {e} ) ) o p1 eq |
| 262 | 84 ≈⟨⟩ |
| 85 ( π1 o e) o k ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} ) (_×_ prod π1' π2' ) (lemma1 eq) | |
| 86 ≈↑⟨ assoc ⟩ | |
| 87 π1 o ( e o k ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} ) (_×_ prod π1' π2' ) (lemma1 eq) ) | |
| 88 ≈⟨ cdr ( ek=h ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} )) ⟩ | |
| 282 | 89 π1 o (_×_ prod π1' π2' ) |
| 262 | 90 ≈⟨ π1fxg=f prod ⟩ |
| 91 π1' | |
| 92 ∎ | |
| 282 | 93 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' ] ]} → |
| 443 | 94 A [ A [ A [ π2 o equalizer1 (eqa (A [ f o π1 ]) (A [ g o π2 ]) {e} ) ] o p1 eq ] ≈ π2' ] |
| 262 | 95 π2p=π21 {d'} {π1'} {π2'} {eq} = let open ≈-Reasoning (A) in |
| 96 begin | |
| 443 | 97 ( π2 o equalizer1 (eqa (A [ f o π1 ]) (A [ g o π2 ]) {e} ) ) o p1 eq |
| 262 | 98 ≈⟨⟩ |
| 99 ( π2 o e) o k ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} ) (_×_ prod π1' π2' ) (lemma1 eq) | |
| 100 ≈↑⟨ assoc ⟩ | |
| 101 π2 o ( e o k ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} ) (_×_ prod π1' π2' ) (lemma1 eq) ) | |
| 102 ≈⟨ cdr ( ek=h ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} )) ⟩ | |
| 282 | 103 π2 o (_×_ prod π1' π2' ) |
| 262 | 104 ≈⟨ π2fxg=g prod ⟩ |
| 105 π2' | |
| 106 ∎ | |
| 302 | 107 uniqueness1 : {d₁ : Obj A} (p' : Hom A d₁ d) {π1' : Hom A d₁ a} {π2' : Hom A d₁ b} |
| 108 {eq : A [ A [ f o π1' ] ≈ A [ g o π2' ] ]} → | |
| 443 | 109 {eq1 : A [ A [ A [ π1 o equalizer1 (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] o p' ] ≈ π1' ]} → |
| 110 {eq2 : A [ A [ A [ π2 o equalizer1 (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] o p' ] ≈ π2' ]} → | |
| 261 | 111 A [ p1 eq ≈ p' ] |
| 264 | 112 uniqueness1 {d'} p' {π1'} {π2'} {eq} {eq1} {eq2} = let open ≈-Reasoning (A) in |
| 263 | 113 begin |
| 114 p1 eq | |
| 115 ≈⟨⟩ | |
| 116 k ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} ) (_×_ prod π1' π2' ) (lemma1 eq) | |
| 443 | 117 ≈⟨ IsEqualizer.uniqueness (eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e}) ( begin |
| 264 | 118 e o p' |
| 119 ≈⟨⟩ | |
| 443 | 120 equalizer1 (eqa (A [ f o π1 ]) (A [ g o π2 ])) o p' |
| 264 | 121 ≈↑⟨ Product.uniqueness prod ⟩ |
| 443 | 122 (prod × ( π1 o equalizer1 (eqa (A [ f o π1 ]) (A [ g o π2 ])) o p') ) ( π2 o (equalizer1 (eqa (A [ f o π1 ]) (A [ g o π2 ])) o p')) |
| 264 | 123 ≈⟨ ×-cong prod (assoc) (assoc) ⟩ |
| 443 | 124 (prod × (A [ A [ π1 o equalizer1 (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] o p' ])) |
| 125 (A [ A [ π2 o equalizer1 (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] o p' ]) | |
| 264 | 126 ≈⟨ ×-cong prod eq1 eq2 ⟩ |
| 127 ((prod × π1') π2') | |
| 128 ∎ ) ⟩ | |
| 263 | 129 p' |
| 130 ∎ | |
| 131 | |
| 266 | 132 -------------------------------- |
| 133 -- | |
| 134 -- If we have two limits on c and c', there are isomorphic pair h, h' | |
| 135 | |
| 136 open Limit | |
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137 open NTrans |
| 266 | 138 |
| 139 iso-l : { c₁' c₂' ℓ' : Level} ( I : Category c₁' c₂' ℓ' ) ( Γ : Functor I A ) | |
| 291 | 140 ( a0 a0' : Obj A ) ( t0 : NTrans I A ( K A I a0 ) Γ ) ( t0' : NTrans I A ( K A I a0' ) Γ ) |
| 141 ( lim : Limit A I Γ a0 t0 ) → ( lim' : Limit A I Γ a0' t0' ) | |
| 266 | 142 → Hom A a0 a0' |
| 143 iso-l I Γ a0 a0' t0 t0' lim lim' = limit lim' a0 t0 | |
| 144 | |
| 145 iso-r : { c₁' c₂' ℓ' : Level} ( I : Category c₁' c₂' ℓ' ) ( Γ : Functor I A ) | |
| 291 | 146 ( a0 a0' : Obj A ) ( t0 : NTrans I A ( K A I a0 ) Γ ) ( t0' : NTrans I A ( K A I a0' ) Γ ) |
| 147 ( lim : Limit A I Γ a0 t0 ) → ( lim' : Limit A I Γ a0' t0' ) | |
| 266 | 148 → Hom A a0' a0 |
| 149 iso-r I Γ a0 a0' t0 t0' lim lim' = limit lim a0' t0' | |
| 150 | |
| 151 | |
| 152 iso-lr : { c₁' c₂' ℓ' : Level} ( I : Category c₁' c₂' ℓ' ) ( Γ : Functor I A ) | |
| 291 | 153 ( a0 a0' : Obj A ) ( t0 : NTrans I A ( K A I a0 ) Γ ) ( t0' : NTrans I A ( K A I a0' ) Γ ) |
| 154 ( lim : Limit A I Γ a0 t0 ) → ( lim' : Limit A I Γ a0' t0' ) → ∀{ i : Obj I } → | |
| 266 | 155 A [ A [ iso-l I Γ a0 a0' t0 t0' lim lim' o iso-r I Γ a0 a0' t0 t0' lim lim' ] ≈ id1 A a0' ] |
| 156 iso-lr I Γ a0 a0' t0 t0' lim lim' {i} = let open ≈-Reasoning (A) in begin | |
| 157 limit lim' a0 t0 o limit lim a0' t0' | |
| 440 | 158 ≈↑⟨ limit-uniqueness lim' ( limit lim' a0 t0 o limit lim a0' t0' )( λ {i} → ( begin |
| 266 | 159 TMap t0' i o ( limit lim' a0 t0 o limit lim a0' t0' ) |
| 160 ≈⟨ assoc ⟩ | |
| 282 | 161 ( TMap t0' i o limit lim' a0 t0 ) o limit lim a0' t0' |
| 266 | 162 ≈⟨ car ( t0f=t lim' ) ⟩ |
| 282 | 163 TMap t0 i o limit lim a0' t0' |
| 266 | 164 ≈⟨ t0f=t lim ⟩ |
| 282 | 165 TMap t0' i |
| 271 | 166 ∎) ) ⟩ |
| 266 | 167 limit lim' a0' t0' |
| 440 | 168 ≈⟨ limit-uniqueness lim' (id a0') idR ⟩ |
| 266 | 169 id a0' |
| 170 ∎ | |
| 171 | |
| 172 | |
| 282 | 173 open import CatExponetial |
| 267 | 174 |
| 175 open Functor | |
| 176 | |
| 177 -------------------------------- | |
| 178 -- | |
| 363 | 179 -- Constancy Functor |
| 266 | 180 |
| 268 | 181 KI : { c₁' c₂' ℓ' : Level} ( I : Category c₁' c₂' ℓ' ) → Functor A ( A ^ I ) |
| 182 KI { c₁'} {c₂'} {ℓ'} I = record { | |
| 291 | 183 FObj = λ a → K A I a ; |
| 184 FMap = λ f → record { -- NTrans I A (K A I a) (K A I b) | |
| 267 | 185 TMap = λ a → f ; |
| 282 | 186 isNTrans = record { |
| 267 | 187 commute = λ {a b f₁} → commute1 {a} {b} {f₁} f |
| 188 } | |
| 282 | 189 } ; |
| 266 | 190 isFunctor = let open ≈-Reasoning (A) in record { |
| 267 | 191 ≈-cong = λ f=g {x} → f=g |
| 266 | 192 ; identity = refl-hom |
| 267 | 193 ; distr = refl-hom |
| 266 | 194 } |
| 267 | 195 } where |
| 196 commute1 : {a b : Obj I} {f₁ : Hom I a b} → {a' b' : Obj A} → (f : Hom A a' b' ) → | |
| 291 | 197 A [ A [ FMap (K A I b') f₁ o f ] ≈ A [ f o FMap (K A I a') f₁ ] ] |
| 282 | 198 commute1 {a} {b} {f₁} {a'} {b'} f = let open ≈-Reasoning (A) in begin |
| 291 | 199 FMap (K A I b') f₁ o f |
| 267 | 200 ≈⟨ idL ⟩ |
| 201 f | |
| 202 ≈↑⟨ idR ⟩ | |
| 291 | 203 f o FMap (K A I a') f₁ |
| 267 | 204 ∎ |
| 205 | |
| 206 | |
| 272 | 207 --------- |
| 208 -- | |
| 298 | 209 -- Limit Constancy Functor F : A → A^I has right adjoint |
| 210 -- | |
| 211 -- we are going to prove universal mapping | |
| 212 | |
| 213 --------- | |
| 214 -- | |
| 272 | 215 -- limit gives co universal mapping ( i.e. adjunction ) |
| 216 -- | |
| 217 -- F = KI I : Functor A (A ^ I) | |
| 282 | 218 -- U = λ b → A0 (lim b {a0 b} {t0 b} |
| 219 -- ε = λ b → T0 ( lim b {a0 b} {t0 b} ) | |
| 475 | 220 -- |
| 221 -- a0 : Obj A and t0 : NTrans K Γ come from the limit | |
| 272 | 222 |
| 282 | 223 limit2couniv : |
| 291 | 224 ( lim : ( Γ : Functor I A ) → { a0 : Obj A } { t0 : NTrans I A ( K A I a0 ) Γ } → Limit A I Γ a0 t0 ) |
| 475 | 225 → ( a0 : ( Γ : Functor I A ) → Obj A ) ( t0 : ( Γ : Functor I A ) → NTrans I A ( K A I (a0 Γ) ) Γ ) |
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226 → coUniversalMapping A ( A ^ I ) (KI I) (λ b → A0 (lim b {a0 b} {t0 b} ) ) ( λ b → T0 ( lim b {a0 b} {t0 b} ) ) |
| 277 | 227 limit2couniv lim a0 t0 = record { -- F U ε |
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228 _*' = λ {b} {a} k → limit (lim b {a0 b} {t0 b} ) a k ; -- η |
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229 iscoUniversalMapping = record { |
| 282 | 230 couniversalMapping = λ{ b a f} → couniversalMapping1 {b} {a} {f} ; |
| 271 | 231 couniquness = couniquness2 |
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232 } |
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233 } where |
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234 couniversalMapping1 : {b : Obj (A ^ I)} {a : Obj A} {f : Hom (A ^ I) (FObj (KI I) a) b} → |
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235 A ^ I [ A ^ I [ T0 (lim b {a0 b} {t0 b}) o FMap (KI I) (limit (lim b {a0 b} {t0 b}) a f) ] ≈ f ] |
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236 couniversalMapping1 {b} {a} {f} {i} = let open ≈-Reasoning (A) in begin |
| 282 | 237 TMap (T0 (lim b {a0 b} {t0 b})) i o TMap ( FMap (KI I) (limit (lim b {a0 b} {t0 b}) a f) ) i |
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238 ≈⟨⟩ |
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239 TMap (t0 b) i o (limit (lim b) a f) |
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240 ≈⟨ t0f=t (lim b) ⟩ |
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241 TMap f i -- i comes from ∀{i} → B [ TMap f i ≈ TMap g i ] |
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242 ∎ |
| 271 | 243 couniquness2 : {b : Obj (A ^ I)} {a : Obj A} {f : Hom (A ^ I) (FObj (KI I) a) b} {g : Hom A a (A0 (lim b {a0 b} {t0 b} ))} → |
| 282 | 244 ( ∀ { i : Obj I } → A [ A [ TMap (T0 (lim b {a0 b} {t0 b} )) i o TMap ( FMap (KI I) g) i ] ≈ TMap f i ] ) |
| 272 | 245 → A [ limit (lim b {a0 b} {t0 b} ) a f ≈ g ] |
| 271 | 246 couniquness2 {b} {a} {f} {g} lim-g=f = let open ≈-Reasoning (A) in begin |
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247 limit (lim b {a0 b} {t0 b} ) a f |
| 440 | 248 ≈⟨ limit-uniqueness ( lim b {a0 b} {t0 b} ) g lim-g=f ⟩ |
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249 g |
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250 ∎ |
| 268 | 251 |
| 272 | 252 open import Category.Cat |
| 275 | 253 |
| 254 | |
| 278 | 255 open coUniversalMapping |
| 282 | 256 |
| 257 univ2limit : | |
| 258 ( U : Obj (A ^ I ) → Obj A ) | |
| 291 | 259 ( ε : ( b : Obj (A ^ I ) ) → NTrans I A (K A I (U b)) b ) |
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260 ( univ : coUniversalMapping A (A ^ I) (KI I) U (ε) ) → |
| 291 | 261 ( Γ : Functor I A ) → Limit A I Γ (U Γ) (ε Γ) |
| 278 | 262 univ2limit U ε univ Γ = record { |
| 272 | 263 limit = λ a t → limit1 a t ; |
| 282 | 264 t0f=t = λ {a t i } → t0f=t1 {a} {t} {i} ; |
| 440 | 265 limit-uniqueness = λ {a} {t} f t=f → limit-uniqueness1 {a} {t} {f} t=f |
| 272 | 266 } where |
| 291 | 267 limit1 : (a : Obj A) → NTrans I A (K A I a) Γ → Hom A a (U Γ) |
| 282 | 268 limit1 a t = _*' univ {_} {a} t |
| 291 | 269 t0f=t1 : {a : Obj A} {t : NTrans I A (K A I a) Γ} {i : Obj I} → |
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270 A [ A [ TMap (ε Γ) i o limit1 a t ] ≈ TMap t i ] |
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|
271 t0f=t1 {a} {t} {i} = let open ≈-Reasoning (A) in begin |
|
279
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|
272 TMap (ε Γ) i o limit1 a t |
|
274
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273
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|
273 ≈⟨⟩ |
| 280 | 274 TMap (ε Γ) i o _*' univ {Γ} {a} t |
| 275 ≈⟨ coIsUniversalMapping.couniversalMapping ( iscoUniversalMapping univ) {Γ} {a} {t} ⟩ | |
|
274
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273
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changeset
|
276 TMap t i |
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273
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|
277 ∎ |
| 291 | 278 limit-uniqueness1 : { a : Obj A } → { t : NTrans I A ( K A I a ) Γ } → { f : Hom A a (U Γ)} |
|
279
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278
diff
changeset
|
279 → ( ∀ { i : Obj I } → A [ A [ TMap (ε Γ) i o f ] ≈ TMap t i ] ) → A [ limit1 a t ≈ f ] |
|
274
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273
diff
changeset
|
280 limit-uniqueness1 {a} {t} {f} εf=t = let open ≈-Reasoning (A) in begin |
| 278 | 281 _*' univ t |
| 282 ≈⟨ ( coIsUniversalMapping.couniquness ( iscoUniversalMapping univ) ) εf=t ⟩ | |
|
274
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273
diff
changeset
|
283 f |
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273
diff
changeset
|
284 ∎ |
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changeset
|
285 |
|
303
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changeset
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286 |
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|
287 lemma-p0 : (a b ab : Obj A) ( π1 : Hom A ab a ) ( π2 : Hom A ab b ) ( prod : Product A a b ab π1 π2 ) → |
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diff
changeset
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288 A [ _×_ prod π1 π2 ≈ id1 A ab ] |
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289 lemma-p0 a b ab π1 π2 prod = let open ≈-Reasoning (A) in begin |
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|
290 _×_ prod π1 π2 |
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302
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changeset
|
291 ≈↑⟨ ×-cong prod idR idR ⟩ |
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diff
changeset
|
292 _×_ prod (A [ π1 o id1 A ab ]) (A [ π2 o id1 A ab ]) |
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diff
changeset
|
293 ≈⟨ Product.uniqueness prod ⟩ |
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changeset
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294 id1 A ab |
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changeset
|
295 ∎ |
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diff
changeset
|
296 |
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diff
changeset
|
297 |
| 282 | 298 open IProduct |
| 283 | 299 open Equalizer |
|
281
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280
diff
changeset
|
300 |
| 282 | 301 -- |
| 302 -- limit from equalizer and product | |
| 303 -- | |
| 304 -- | |
| 283 | 305 -- ai |
| 306 -- ^ K f = id lim | |
| 300 | 307 -- | pi lim = K i -----------→ K j = lim |
| 283 | 308 -- | | | |
| 309 -- p | | | |
|
303
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changeset
|
310 -- ^ proj i o e = ε i | | ε j = proj j o e |
| 283 | 311 -- | | | |
|
285
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|
312 -- | e = equalizer (id p) (id p) | | |
| 283 | 313 -- | v v |
| 300 | 314 -- lim <------------------ d' a i = Γ i -----------→ Γ j = a j |
| 283 | 315 -- k ( product pi ) Γ f |
| 316 -- Γ f o ε i = ε j | |
| 317 -- | |
| 291 | 318 |
| 283 | 319 limit-ε : |
| 443 | 320 ( eqa : {a b c : Obj A} → (e : Hom A c a ) → (f g : Hom A a b) → IsEqualizer A e f g ) |
| 282 | 321 ( lim p : Obj A ) ( e : Hom A lim p ) |
| 322 ( proj : (i : Obj I ) → Hom A p (FObj Γ i) ) → | |
| 291 | 323 NTrans I A (K A I lim) Γ |
|
303
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diff
changeset
|
324 limit-ε eqa lim p e proj = record { |
| 282 | 325 TMap = tmap ; |
|
303
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parents:
302
diff
changeset
|
326 isNTrans = record { commute = commute1 } |
|
281
dbd2044add2a
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parents:
280
diff
changeset
|
327 } where |
| 291 | 328 tmap : (i : Obj I) → Hom A (FObj (K A I lim) i) (FObj Γ i) |
|
285
46d4ad55b948
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diff
changeset
|
329 tmap i = A [ proj i o e ] |
| 283 | 330 commute1 : {i j : Obj I} {f : Hom I i j} → |
| 291 | 331 A [ A [ FMap Γ f o tmap i ] ≈ A [ tmap j o FMap (K A I lim) f ] ] |
|
285
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parents:
284
diff
changeset
|
332 commute1 {i} {j} {f} = let open ≈-Reasoning (A) in begin |
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284
diff
changeset
|
333 FMap Γ f o tmap i |
|
46d4ad55b948
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284
diff
changeset
|
334 ≈⟨⟩ |
|
46d4ad55b948
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diff
changeset
|
335 FMap Γ f o ( proj i o e ) |
|
46d4ad55b948
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284
diff
changeset
|
336 ≈⟨ assoc ⟩ |
|
46d4ad55b948
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diff
changeset
|
337 ( FMap Γ f o proj i ) o e |
|
303
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Limit form equalizer and product done.
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parents:
302
diff
changeset
|
338 ≈⟨ fe=ge ( eqa e (FMap Γ f o proj i) ( proj j )) ⟩ |
|
285
46d4ad55b948
commutativity continue...
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parents:
284
diff
changeset
|
339 proj j o e |
|
46d4ad55b948
commutativity continue...
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parents:
284
diff
changeset
|
340 ≈↑⟨ idR ⟩ |
|
46d4ad55b948
commutativity continue...
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parents:
284
diff
changeset
|
341 (proj j o e ) o id1 A lim |
|
46d4ad55b948
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parents:
284
diff
changeset
|
342 ≈⟨⟩ |
| 291 | 343 tmap j o FMap (K A I lim) f |
| 288 | 344 ∎ |
|
281
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parents:
280
diff
changeset
|
345 |
| 282 | 346 limit-from : |
|
285
46d4ad55b948
commutativity continue...
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parents:
284
diff
changeset
|
347 ( prod : (p : Obj A) ( ai : Obj I → Obj A ) ( pi : (i : Obj I) → Hom A p ( ai i ) ) |
|
281
dbd2044add2a
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parents:
280
diff
changeset
|
348 → IProduct {c₁'} A (Obj I) p ai pi ) |
| 443 | 349 ( eqa : {a b c : Obj A} → (e : Hom A c a ) → (f g : Hom A a b) → IsEqualizer A e f g ) |
| 290 | 350 ( lim p : Obj A ) -- limit to be made |
|
285
46d4ad55b948
commutativity continue...
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parents:
284
diff
changeset
|
351 ( e : Hom A lim p ) -- existing of equalizer |
|
46d4ad55b948
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parents:
284
diff
changeset
|
352 ( proj : (i : Obj I ) → Hom A p (FObj Γ i) ) -- existing of product ( projection actually ) |
|
303
7f40d6a51c72
Limit form equalizer and product done.
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parents:
302
diff
changeset
|
353 → Limit A I Γ lim ( limit-ε eqa lim p e proj ) |
| 290 | 354 limit-from prod eqa lim p e proj = record { |
| 282 | 355 limit = λ a t → limit1 a t ; |
| 356 t0f=t = λ {a t i } → t0f=t1 {a} {t} {i} ; | |
| 440 | 357 limit-uniqueness = λ {a} {t} f t=f → limit-uniqueness1 {a} {t} {f} t=f |
| 282 | 358 } where |
| 291 | 359 limit1 : (a : Obj A) → NTrans I A (K A I a) Γ → Hom A a lim |
|
468
c375d8f93a2c
discrete category and product from a limit
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parents:
443
diff
changeset
|
360 limit1 a t = let open ≈-Reasoning (A) in k (eqa e (id1 A p) (id1 A p )) (iproduct ( prod p (FObj Γ) proj ) (TMap t) ) refl-hom |
| 291 | 361 t0f=t1 : {a : Obj A} {t : NTrans I A (K A I a) Γ} {i : Obj I} → |
|
303
7f40d6a51c72
Limit form equalizer and product done.
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parents:
302
diff
changeset
|
362 A [ A [ TMap (limit-ε eqa lim p e proj ) i o limit1 a t ] ≈ TMap t i ] |
| 283 | 363 t0f=t1 {a} {t} {i} = let open ≈-Reasoning (A) in begin |
|
303
7f40d6a51c72
Limit form equalizer and product done.
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parents:
302
diff
changeset
|
364 TMap (limit-ε eqa lim p e proj ) i o limit1 a t |
| 283 | 365 ≈⟨⟩ |
|
468
c375d8f93a2c
discrete category and product from a limit
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parents:
443
diff
changeset
|
366 ( ( proj i ) o e ) o k (eqa e (id1 A p) (id1 A p )) (iproduct ( prod p (FObj Γ) proj ) (TMap t) ) refl-hom |
| 283 | 367 ≈↑⟨ assoc ⟩ |
|
468
c375d8f93a2c
discrete category and product from a limit
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parents:
443
diff
changeset
|
368 proj i o ( e o k (eqa e (id1 A p) (id1 A p )) (iproduct ( prod p (FObj Γ) proj ) (TMap t) ) refl-hom ) |
| 283 | 369 ≈⟨ cdr ( ek=h ( eqa e (id1 A p) (id1 A p ) ) ) ⟩ |
|
468
c375d8f93a2c
discrete category and product from a limit
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parents:
443
diff
changeset
|
370 proj i o iproduct (prod p (FObj Γ) proj) (TMap t) |
| 283 | 371 ≈⟨ pif=q (prod p (FObj Γ) proj) (TMap t) ⟩ |
| 372 TMap t i | |
| 373 ∎ | |
| 291 | 374 limit-uniqueness1 : {a : Obj A} {t : NTrans I A (K A I a) Γ} {f : Hom A a lim} |
|
303
7f40d6a51c72
Limit form equalizer and product done.
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parents:
302
diff
changeset
|
375 → ({i : Obj I} → A [ A [ TMap (limit-ε eqa lim p e proj ) i o f ] ≈ TMap t i ]) → |
| 282 | 376 A [ limit1 a t ≈ f ] |
| 283 | 377 limit-uniqueness1 {a} {t} {f} lim=t = let open ≈-Reasoning (A) in begin |
| 378 limit1 a t | |
| 379 ≈⟨⟩ | |
|
468
c375d8f93a2c
discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
443
diff
changeset
|
380 k (eqa e (id1 A p) (id1 A p )) (iproduct ( prod p (FObj Γ) proj ) (TMap t) ) refl-hom |
| 443 | 381 ≈⟨ IsEqualizer.uniqueness (eqa e (id1 A p) (id1 A p )) ( begin |
| 283 | 382 e o f |
| 383 ≈↑⟨ ip-uniqueness (prod p (FObj Γ) proj) ⟩ | |
|
468
c375d8f93a2c
discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
443
diff
changeset
|
384 iproduct (prod p (FObj Γ) proj) (λ i → ( proj i o ( e o f ) ) ) |
| 284 | 385 ≈⟨ ip-cong (prod p (FObj Γ) proj) ( λ i → begin |
|
285
46d4ad55b948
commutativity continue...
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parents:
284
diff
changeset
|
386 proj i o ( e o f ) |
| 284 | 387 ≈⟨ assoc ⟩ |
|
285
46d4ad55b948
commutativity continue...
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parents:
284
diff
changeset
|
388 ( proj i o e ) o f |
|
46d4ad55b948
commutativity continue...
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parents:
284
diff
changeset
|
389 ≈⟨ lim=t {i} ⟩ |
|
46d4ad55b948
commutativity continue...
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parents:
284
diff
changeset
|
390 TMap t i |
| 284 | 391 ∎ ) ⟩ |
|
468
c375d8f93a2c
discrete category and product from a limit
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parents:
443
diff
changeset
|
392 iproduct (prod p (FObj Γ) proj) (TMap t) |
| 283 | 393 ∎ ) ⟩ |
| 394 f | |
| 395 ∎ | |
| 396 | |
| 291 | 397 ---- |
| 398 -- | |
| 399 -- Adjoint functor preserves limits | |
| 400 -- | |
| 401 -- | |
| 402 | |
| 403 open import Category.Cat | |
| 404 | |
| 405 ta1 : { c₁' c₂' ℓ' : Level} (B : Category c₁' c₂' ℓ') ( Γ : Functor I B ) | |
|
299
8c72f5284bc8
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parents:
298
diff
changeset
|
406 ( lim : Obj B ) ( tb : NTrans I B ( K B I lim ) Γ ) → |
| 291 | 407 ( U : Functor B A) → NTrans I A ( K A I (FObj U lim) ) (U ○ Γ) |
|
299
8c72f5284bc8
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parents:
298
diff
changeset
|
408 ta1 B Γ lim tb U = record { |
| 291 | 409 TMap = TMap (Functor*Nat I A U tb) ; |
| 410 isNTrans = record { commute = λ {a} {b} {f} → let open ≈-Reasoning (A) in begin | |
| 411 FMap (U ○ Γ) f o TMap (Functor*Nat I A U tb) a | |
| 412 ≈⟨ nat ( Functor*Nat I A U tb ) ⟩ | |
| 413 TMap (Functor*Nat I A U tb) b o FMap (U ○ K B I lim) f | |
| 414 ≈⟨ cdr (IsFunctor.identity (isFunctor U) ) ⟩ | |
| 415 TMap (Functor*Nat I A U tb) b o FMap (K A I (FObj U lim)) f | |
| 416 ∎ | |
| 417 } } | |
| 418 | |
| 419 adjoint-preseve-limit : | |
| 420 { c₁' c₂' ℓ' : Level} (B : Category c₁' c₂' ℓ') ( Γ : Functor I B ) | |
|
299
8c72f5284bc8
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parents:
298
diff
changeset
|
421 ( lim : Obj B ) ( tb : NTrans I B ( K B I lim ) Γ ) → ( limitb : Limit B I Γ lim tb ) → |
| 291 | 422 { U : Functor B A } { F : Functor A B } |
| 293 | 423 { η : NTrans A A identityFunctor ( U ○ F ) } |
| 291 | 424 { ε : NTrans B B ( F ○ U ) identityFunctor } → |
|
299
8c72f5284bc8
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parents:
298
diff
changeset
|
425 ( adj : Adjunction A B U F η ε ) → Limit A I (U ○ Γ) (FObj U lim) (ta1 B Γ lim tb U ) |
| 292 | 426 adjoint-preseve-limit B Γ lim tb limitb {U} {F} {η} {ε} adj = record { |
| 291 | 427 limit = λ a t → limit1 a t ; |
| 428 t0f=t = λ {a t i } → t0f=t1 {a} {t} {i} ; | |
| 440 | 429 limit-uniqueness = λ {a} {t} f t=f → limit-uniqueness1 {a} {t} {f} t=f |
| 291 | 430 } where |
|
299
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parents:
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diff
changeset
|
431 ta = ta1 B Γ lim tb U |
| 293 | 432 tfmap : (a : Obj A) → NTrans I A (K A I a) (U ○ Γ) → (i : Obj I) → Hom B (FObj (K B I (FObj F a)) i) (FObj Γ i) |
| 433 tfmap a t i = B [ TMap ε (FObj Γ i) o FMap F (TMap t i) ] | |
| 434 tF : (a : Obj A) → NTrans I A (K A I a) (U ○ Γ) → NTrans I B (K B I (FObj F a)) Γ | |
| 435 tF a t = record { | |
| 436 TMap = tfmap a t ; | |
| 437 isNTrans = record { commute = λ {a'} {b} {f} → let open ≈-Reasoning (B) in begin | |
| 438 FMap Γ f o tfmap a t a' | |
| 294 | 439 ≈⟨⟩ |
| 440 FMap Γ f o ( TMap ε (FObj Γ a') o FMap F (TMap t a')) | |
| 441 ≈⟨ assoc ⟩ | |
| 442 (FMap Γ f o TMap ε (FObj Γ a') ) o FMap F (TMap t a') | |
| 443 ≈⟨ car (nat ε) ⟩ | |
| 444 (TMap ε (FObj Γ b) o FMap (F ○ U) (FMap Γ f) ) o FMap F (TMap t a') | |
| 445 ≈↑⟨ assoc ⟩ | |
| 446 TMap ε (FObj Γ b) o ( FMap (F ○ U) (FMap Γ f) o FMap F (TMap t a') ) | |
| 447 ≈↑⟨ cdr ( distr F ) ⟩ | |
| 448 TMap ε (FObj Γ b) o ( FMap F (A [ FMap U (FMap Γ f) o TMap t a' ] ) ) | |
| 449 ≈⟨ cdr ( fcong F (nat t) ) ⟩ | |
| 450 TMap ε (FObj Γ b) o FMap F (A [ TMap t b o FMap (K A I a) f ]) | |
| 451 ≈⟨⟩ | |
| 452 TMap ε (FObj Γ b) o FMap F (A [ TMap t b o id1 A (FObj (K A I a) b) ]) | |
|
299
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453 ≈⟨ cdr ( fcong F (idR1 A)) ⟩ |
| 294 | 454 TMap ε (FObj Γ b) o FMap F (TMap t b ) |
| 455 ≈↑⟨ idR ⟩ | |
| 456 ( TMap ε (FObj Γ b) o FMap F (TMap t b)) o id1 B (FObj F (FObj (K A I a) b)) | |
| 457 ≈⟨⟩ | |
| 293 | 458 tfmap a t b o FMap (K B I (FObj F a)) f |
| 459 ∎ | |
| 460 } } | |
| 461 limit1 : (a : Obj A) → NTrans I A (K A I a) (U ○ Γ) → Hom A a (FObj U lim) | |
| 462 limit1 a t = A [ FMap U (limit limitb (FObj F a) (tF a t )) o TMap η a ] | |
| 463 t0f=t1 : {a : Obj A} {t : NTrans I A (K A I a) (U ○ Γ)} {i : Obj I} → | |
| 291 | 464 A [ A [ TMap ta i o limit1 a t ] ≈ TMap t i ] |
| 295 | 465 t0f=t1 {a} {t} {i} = let open ≈-Reasoning (A) in begin |
| 466 TMap ta i o limit1 a t | |
| 467 ≈⟨⟩ | |
| 468 FMap U ( TMap tb i ) o ( FMap U (limit limitb (FObj F a) (tF a t )) o TMap η a ) | |
| 469 ≈⟨ assoc ⟩ | |
| 470 ( FMap U ( TMap tb i ) o FMap U (limit limitb (FObj F a) (tF a t ))) o TMap η a | |
| 471 ≈↑⟨ car ( distr U ) ⟩ | |
| 472 FMap U ( B [ TMap tb i o limit limitb (FObj F a) (tF a t ) ] ) o TMap η a | |
| 473 ≈⟨ car ( fcong U ( t0f=t limitb ) ) ⟩ | |
| 474 FMap U (TMap (tF a t) i) o TMap η a | |
| 475 ≈⟨⟩ | |
| 476 FMap U ( B [ TMap ε (FObj Γ i) o FMap F (TMap t i) ] ) o TMap η a | |
| 477 ≈⟨ car ( distr U ) ⟩ | |
| 478 ( FMap U ( TMap ε (FObj Γ i)) o FMap U ( FMap F (TMap t i) )) o TMap η a | |
| 479 ≈↑⟨ assoc ⟩ | |
| 480 FMap U ( TMap ε (FObj Γ i) ) o ( FMap U ( FMap F (TMap t i) ) o TMap η a ) | |
| 481 ≈⟨ cdr ( nat η ) ⟩ | |
| 482 FMap U (TMap ε (FObj Γ i)) o ( TMap η (FObj U (FObj Γ i)) o FMap (identityFunctor {_} {_} {_} {A}) (TMap t i) ) | |
| 483 ≈⟨ assoc ⟩ | |
| 484 ( FMap U (TMap ε (FObj Γ i)) o TMap η (FObj U (FObj Γ i))) o TMap t i | |
| 485 ≈⟨ car ( IsAdjunction.adjoint1 ( Adjunction.isAdjunction adj ) ) ⟩ | |
| 486 id1 A (FObj (U ○ Γ) i) o TMap t i | |
| 487 ≈⟨ idL ⟩ | |
| 488 TMap t i | |
| 489 ∎ | |
| 296 | 490 -- ta = TMap (Functor*Nat I A U tb) , FMap U ( TMap tb i ) o f ≈ TMap t i |
| 293 | 491 limit-uniqueness1 : {a : Obj A} {t : NTrans I A (K A I a) (U ○ Γ)} {f : Hom A a (FObj U lim)} |
| 291 | 492 → ({i : Obj I} → A [ A [ TMap ta i o f ] ≈ TMap t i ]) → |
| 493 A [ limit1 a t ≈ f ] | |
| 295 | 494 limit-uniqueness1 {a} {t} {f} lim=t = let open ≈-Reasoning (A) in begin |
| 495 limit1 a t | |
| 496 ≈⟨⟩ | |
| 497 FMap U (limit limitb (FObj F a) (tF a t )) o TMap η a | |
| 440 | 498 ≈⟨ car ( fcong U (limit-uniqueness limitb (B [ TMap ε lim o FMap F f ] ) ( λ {i} → lemma1 i) )) ⟩ |
| 298 | 499 FMap U ( B [ TMap ε lim o FMap F f ] ) o TMap η a -- Universal mapping |
|
297
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500 ≈⟨ car (distr U ) ⟩ |
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501 ( (FMap U (TMap ε lim)) o (FMap U ( FMap F f )) ) o TMap η a |
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limit preservation proved.
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502 ≈⟨ sym assoc ⟩ |
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503 (FMap U (TMap ε lim)) o ((FMap U ( FMap F f )) o TMap η a ) |
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parents:
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504 ≈⟨ cdr (nat η) ⟩ |
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parents:
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505 (FMap U (TMap ε lim)) o ((TMap η (FObj U lim )) o f ) |
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537570f6a44f
limit preservation proved.
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parents:
296
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506 ≈⟨ assoc ⟩ |
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parents:
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507 ((FMap U (TMap ε lim)) o (TMap η (FObj U lim))) o f |
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parents:
296
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508 ≈⟨ car ( IsAdjunction.adjoint1 ( Adjunction.isAdjunction adj)) ⟩ |
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parents:
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509 id (FObj U lim) o f |
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parents:
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510 ≈⟨ idL ⟩ |
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parents:
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511 f |
| 296 | 512 ∎ where |
| 513 lemma1 : (i : Obj I) → B [ B [ TMap tb i o B [ TMap ε lim o FMap F f ] ] ≈ TMap (tF a t) i ] | |
| 514 lemma1 i = let open ≈-Reasoning (B) in begin | |
| 515 TMap tb i o (TMap ε lim o FMap F f) | |
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297
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516 ≈⟨ assoc ⟩ |
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517 ( TMap tb i o TMap ε lim ) o FMap F f |
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parents:
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518 ≈⟨ car ( nat ε ) ⟩ |
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296
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519 ( TMap ε (FObj Γ i) o FMap F ( FMap U ( TMap tb i ))) o FMap F f |
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296
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520 ≈↑⟨ assoc ⟩ |
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parents:
296
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521 TMap ε (FObj Γ i) o ( FMap F ( FMap U ( TMap tb i )) o FMap F f ) |
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parents:
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522 ≈↑⟨ cdr ( distr F ) ⟩ |
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limit preservation proved.
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parents:
296
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523 TMap ε (FObj Γ i) o FMap F ( A [ FMap U ( TMap tb i ) o f ] ) |
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limit preservation proved.
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parents:
296
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524 ≈⟨ cdr ( fcong F (lim=t {i}) ) ⟩ |
| 296 | 525 TMap ε (FObj Γ i) o FMap F (TMap t i) |
| 526 ≈⟨⟩ | |
| 527 TMap (tF a t) i | |
| 528 ∎ | |
| 295 | 529 |
| 296 | 530 |
| 531 |