Mercurial > hg > Members > kono > Proof > category
annotate pullback.agda @ 272:5f2b8a5cc115
adjunction to limit
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 22 Sep 2013 21:27:03 +0900 |
| parents | 28278175d696 |
| children | fae4bb967d76 |
| 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 | |
| 14 -- | |
| 264 | 15 -- Pullback from equalizer and product |
| 260 | 16 -- f |
| 17 -- a -------> c | |
| 18 -- ^ ^ | |
| 19 -- π1 | |g | |
| 20 -- | | | |
| 21 -- ab -------> b | |
| 22 -- ^ π2 | |
| 23 -- | | |
| 264 | 24 -- | e = equalizer (f π1) (g π1) |
| 25 -- | | |
| 26 -- d <------------------ d' | |
| 27 -- k (π1' × π2' ) | |
| 260 | 28 |
| 261 | 29 open Equalizer |
| 30 open Product | |
| 31 open Pullback | |
| 32 | |
| 33 pullback-from : (a b c ab d : Obj A) | |
| 260 | 34 ( f : Hom A a c ) ( g : Hom A b c ) |
| 261 | 35 ( π1 : Hom A ab a ) ( π2 : Hom A ab b ) ( e : Hom A d ab ) |
| 260 | 36 ( eqa : {a b c : Obj A} → (f g : Hom A a b) → {e : Hom A c a } → Equalizer A e f g ) |
| 261 | 37 ( prod : Product A a b ab π1 π2 ) → Pullback A a b c d f g |
| 38 ( A [ π1 o equalizer ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ){e} ) ] ) | |
| 39 ( A [ π2 o equalizer ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ){e} ) ] ) | |
| 40 pullback-from a b c ab d f g π1 π2 e eqa prod = record { | |
| 260 | 41 commute = commute1 ; |
| 42 p = p1 ; | |
| 261 | 43 π1p=π1 = λ {d} {π1'} {π2'} {eq} → π1p=π11 {d} {π1'} {π2'} {eq} ; |
| 44 π2p=π2 = λ {d} {π1'} {π2'} {eq} → π2p=π21 {d} {π1'} {π2'} {eq} ; | |
| 260 | 45 uniqueness = uniqueness1 |
| 46 } where | |
| 261 | 47 commute1 : A [ A [ f o A [ π1 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] ] ≈ A [ g o A [ π2 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] ] ] |
| 262 | 48 commute1 = let open ≈-Reasoning (A) in |
| 49 begin | |
| 50 f o ( π1 o equalizer (eqa ( f o π1 ) ( g o π2 )) ) | |
| 51 ≈⟨ assoc ⟩ | |
| 52 ( f o π1 ) o equalizer (eqa ( f o π1 ) ( g o π2 )) | |
| 53 ≈⟨ fe=ge (eqa (A [ f o π1 ]) (A [ g o π2 ])) ⟩ | |
| 54 ( g o π2 ) o equalizer (eqa ( f o π1 ) ( g o π2 )) | |
| 55 ≈↑⟨ assoc ⟩ | |
| 56 g o ( π2 o equalizer (eqa ( f o π1 ) ( g o π2 )) ) | |
| 57 ∎ | |
| 58 lemma1 : {d' : Obj A} {π1' : Hom A d' a} {π2' : Hom A d' b} → A [ A [ f o π1' ] ≈ A [ g o π2' ] ] → | |
| 59 A [ A [ A [ f o π1 ] o (prod × π1') π2' ] ≈ A [ A [ g o π2 ] o (prod × π1') π2' ] ] | |
| 60 lemma1 {d'} { π1' } { π2' } eq = let open ≈-Reasoning (A) in | |
| 61 begin | |
| 62 ( f o π1 ) o (prod × π1') π2' | |
| 63 ≈↑⟨ assoc ⟩ | |
| 64 f o ( π1 o (prod × π1') π2' ) | |
| 65 ≈⟨ cdr (π1fxg=f prod) ⟩ | |
| 66 f o π1' | |
| 67 ≈⟨ eq ⟩ | |
| 68 g o π2' | |
| 69 ≈↑⟨ cdr (π2fxg=g prod) ⟩ | |
| 70 g o ( π2 o (prod × π1') π2' ) | |
| 71 ≈⟨ assoc ⟩ | |
| 72 ( g o π2 ) o (prod × π1') π2' | |
| 73 ∎ | |
| 261 | 74 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 |
| 262 | 75 p1 {d'} { π1' } { π2' } eq = |
| 76 let open ≈-Reasoning (A) in k ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} ) (_×_ prod π1' π2' ) ( lemma1 eq ) | |
| 77 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' ] ]} → | |
| 78 A [ A [ A [ π1 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ]) {e} ) ] o p1 eq ] ≈ π1' ] | |
| 79 π1p=π11 {d'} {π1'} {π2'} {eq} = let open ≈-Reasoning (A) in | |
| 80 begin | |
| 81 ( π1 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ]) {e} ) ) o p1 eq | |
| 82 ≈⟨⟩ | |
| 83 ( π1 o e) o k ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} ) (_×_ prod π1' π2' ) (lemma1 eq) | |
| 84 ≈↑⟨ assoc ⟩ | |
| 85 π1 o ( e o k ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} ) (_×_ prod π1' π2' ) (lemma1 eq) ) | |
| 86 ≈⟨ cdr ( ek=h ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} )) ⟩ | |
| 87 π1 o (_×_ prod π1' π2' ) | |
| 88 ≈⟨ π1fxg=f prod ⟩ | |
| 89 π1' | |
| 90 ∎ | |
| 263 | 91 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' ] ]} → |
| 92 A [ A [ A [ π2 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ]) {e} ) ] o p1 eq ] ≈ π2' ] | |
| 262 | 93 π2p=π21 {d'} {π1'} {π2'} {eq} = let open ≈-Reasoning (A) in |
| 94 begin | |
| 95 ( π2 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ]) {e} ) ) o p1 eq | |
| 96 ≈⟨⟩ | |
| 97 ( π2 o e) o k ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} ) (_×_ prod π1' π2' ) (lemma1 eq) | |
| 98 ≈↑⟨ assoc ⟩ | |
| 99 π2 o ( e o k ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} ) (_×_ prod π1' π2' ) (lemma1 eq) ) | |
| 100 ≈⟨ cdr ( ek=h ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} )) ⟩ | |
| 101 π2 o (_×_ prod π1' π2' ) | |
| 102 ≈⟨ π2fxg=g prod ⟩ | |
| 103 π2' | |
| 104 ∎ | |
| 261 | 105 uniqueness1 : {d₁ : Obj A} (p' : Hom A d₁ d) {π1' : Hom A d₁ a} {π2' : Hom A d₁ b} {eq : A [ A [ f o π1' ] ≈ A [ g o π2' ] ]} → |
| 106 {eq1 : A [ A [ A [ π1 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] o p' ] ≈ π1' ]} → | |
| 107 {eq2 : A [ A [ A [ π2 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] o p' ] ≈ π2' ]} → | |
| 108 A [ p1 eq ≈ p' ] | |
| 264 | 109 uniqueness1 {d'} p' {π1'} {π2'} {eq} {eq1} {eq2} = let open ≈-Reasoning (A) in |
| 263 | 110 begin |
| 111 p1 eq | |
| 112 ≈⟨⟩ | |
| 113 k ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} ) (_×_ prod π1' π2' ) (lemma1 eq) | |
| 264 | 114 ≈⟨ Equalizer.uniqueness (eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e}) ( begin |
| 115 e o p' | |
| 116 ≈⟨⟩ | |
| 117 equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) o p' | |
| 118 ≈↑⟨ Product.uniqueness prod ⟩ | |
| 119 (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')) | |
| 120 ≈⟨ ×-cong prod (assoc) (assoc) ⟩ | |
| 121 (prod × (A [ A [ π1 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] o p' ])) | |
| 122 (A [ A [ π2 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] o p' ]) | |
| 123 ≈⟨ ×-cong prod eq1 eq2 ⟩ | |
| 124 ((prod × π1') π2') | |
| 125 ∎ ) ⟩ | |
| 263 | 126 p' |
| 127 ∎ | |
| 128 | |
| 266 | 129 ------ |
| 130 -- | |
| 131 -- Limit | |
| 132 -- | |
| 133 ----- | |
| 134 | |
| 135 -- Constancy Functor | |
| 136 | |
| 137 K : { c₁' c₂' ℓ' : Level} ( I : Category c₁' c₂' ℓ' ) → ( a : Obj A ) → Functor I A | |
| 138 K I a = record { | |
| 265 | 139 FObj = λ i → a ; |
| 140 FMap = λ f → id1 A a ; | |
| 141 isFunctor = let open ≈-Reasoning (A) in record { | |
| 142 ≈-cong = λ f=g → refl-hom | |
| 143 ; identity = refl-hom | |
| 144 ; distr = sym idL | |
| 145 } | |
| 146 } | |
| 147 | |
| 148 open NTrans | |
| 149 | |
| 150 record Limit { c₁' c₂' ℓ' : Level} ( I : Category c₁' c₂' ℓ' ) ( Γ : Functor I A ) | |
| 266 | 151 ( a0 : Obj A ) ( t0 : NTrans I A ( K I a0 ) Γ ) : Set (suc (c₁' ⊔ c₂' ⊔ ℓ' ⊔ c₁ ⊔ c₂ ⊔ ℓ )) where |
| 265 | 152 field |
| 266 | 153 limit : ( a : Obj A ) → ( t : NTrans I A ( K I a ) Γ ) → Hom A a a0 |
| 154 t0f=t : { a : Obj A } → { t : NTrans I A ( K I a ) Γ } → ∀ { i : Obj I } → | |
| 265 | 155 A [ A [ TMap t0 i o limit a t ] ≈ TMap t i ] |
| 271 | 156 limit-uniqueness : { a : Obj A } → { t : NTrans I A ( K I a ) Γ } → { f : Hom A a a0 } → ( ∀ { i : Obj I } → |
| 157 A [ A [ TMap t0 i o f ] ≈ TMap t i ] ) → A [ limit a t ≈ f ] | |
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158 A0 : Obj A |
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159 A0 = a0 |
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160 T0 : NTrans I A ( K I a0 ) Γ |
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161 T0 = t0 |
| 265 | 162 |
| 266 | 163 -------------------------------- |
| 164 -- | |
| 165 -- If we have two limits on c and c', there are isomorphic pair h, h' | |
| 166 | |
| 167 open Limit | |
| 168 | |
| 169 iso-l : { c₁' c₂' ℓ' : Level} ( I : Category c₁' c₂' ℓ' ) ( Γ : Functor I A ) | |
| 170 ( a0 a0' : Obj A ) ( t0 : NTrans I A ( K I a0 ) Γ ) ( t0' : NTrans I A ( K I a0' ) Γ ) | |
| 171 ( lim : Limit I Γ a0 t0 ) → ( lim' : Limit I Γ a0' t0' ) | |
| 172 → Hom A a0 a0' | |
| 173 iso-l I Γ a0 a0' t0 t0' lim lim' = limit lim' a0 t0 | |
| 174 | |
| 175 iso-r : { c₁' c₂' ℓ' : Level} ( I : Category c₁' c₂' ℓ' ) ( Γ : Functor I A ) | |
| 176 ( a0 a0' : Obj A ) ( t0 : NTrans I A ( K I a0 ) Γ ) ( t0' : NTrans I A ( K I a0' ) Γ ) | |
| 177 ( lim : Limit I Γ a0 t0 ) → ( lim' : Limit I Γ a0' t0' ) | |
| 178 → Hom A a0' a0 | |
| 179 iso-r I Γ a0 a0' t0 t0' lim lim' = limit lim a0' t0' | |
| 180 | |
| 181 | |
| 182 iso-lr : { c₁' c₂' ℓ' : Level} ( I : Category c₁' c₂' ℓ' ) ( Γ : Functor I A ) | |
| 183 ( a0 a0' : Obj A ) ( t0 : NTrans I A ( K I a0 ) Γ ) ( t0' : NTrans I A ( K I a0' ) Γ ) | |
| 184 ( lim : Limit I Γ a0 t0 ) → ( lim' : Limit I Γ a0' t0' ) → ∀{ i : Obj I } → | |
| 185 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' ] | |
| 186 iso-lr I Γ a0 a0' t0 t0' lim lim' {i} = let open ≈-Reasoning (A) in begin | |
| 187 limit lim' a0 t0 o limit lim a0' t0' | |
| 271 | 188 ≈↑⟨ limit-uniqueness lim' ( λ {i} → ( begin |
| 266 | 189 TMap t0' i o ( limit lim' a0 t0 o limit lim a0' t0' ) |
| 190 ≈⟨ assoc ⟩ | |
| 191 ( TMap t0' i o limit lim' a0 t0 ) o limit lim a0' t0' | |
| 192 ≈⟨ car ( t0f=t lim' ) ⟩ | |
| 193 TMap t0 i o limit lim a0' t0' | |
| 194 ≈⟨ t0f=t lim ⟩ | |
| 195 TMap t0' i | |
| 271 | 196 ∎) ) ⟩ |
| 266 | 197 limit lim' a0' t0' |
| 271 | 198 ≈⟨ limit-uniqueness lim' idR ⟩ |
| 266 | 199 id a0' |
| 200 ∎ | |
| 201 | |
| 202 | |
| 267 | 203 open import CatExponetial |
| 204 | |
| 205 open Functor | |
| 206 | |
| 207 -------------------------------- | |
| 208 -- | |
| 209 -- Contancy Functor | |
| 266 | 210 |
| 268 | 211 KI : { c₁' c₂' ℓ' : Level} ( I : Category c₁' c₂' ℓ' ) → Functor A ( A ^ I ) |
| 212 KI { c₁'} {c₂'} {ℓ'} I = record { | |
| 267 | 213 FObj = λ a → K I a ; |
| 214 FMap = λ f → record { -- NTrans I A (K I a) (K I b) | |
| 215 TMap = λ a → f ; | |
| 216 isNTrans = record { | |
| 217 commute = λ {a b f₁} → commute1 {a} {b} {f₁} f | |
| 218 } | |
| 219 } ; | |
| 266 | 220 isFunctor = let open ≈-Reasoning (A) in record { |
| 267 | 221 ≈-cong = λ f=g {x} → f=g |
| 266 | 222 ; identity = refl-hom |
| 267 | 223 ; distr = refl-hom |
| 266 | 224 } |
| 267 | 225 } where |
| 226 commute1 : {a b : Obj I} {f₁ : Hom I a b} → {a' b' : Obj A} → (f : Hom A a' b' ) → | |
| 227 A [ A [ FMap (K I b') f₁ o f ] ≈ A [ f o FMap (K I a') f₁ ] ] | |
| 228 commute1 {a} {b} {f₁} {a'} {b'} f = let open ≈-Reasoning (A) in begin | |
| 229 FMap (K I b') f₁ o f | |
| 230 ≈⟨ idL ⟩ | |
| 231 f | |
| 232 ≈↑⟨ idR ⟩ | |
| 233 f o FMap (K I a') f₁ | |
| 234 ∎ | |
| 235 | |
| 236 | |
| 268 | 237 open import Function |
| 267 | 238 |
| 272 | 239 --------- |
| 240 -- | |
| 241 -- limit gives co universal mapping ( i.e. adjunction ) | |
| 242 -- | |
| 243 -- F = KI I : Functor A (A ^ I) | |
| 244 -- U = λ b → A0 (lim b {a0 b} {t0 b} | |
| 245 -- ε = λ b → T0 ( lim b {a0 b} {t0 b} ) | |
| 246 | |
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247 limit2adjoint : |
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248 ( lim : ( Γ : Functor I A ) → { a0 : Obj A } { t0 : NTrans I A ( K I a0 ) Γ } → Limit I Γ a0 t0 ) |
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249 → ( a0 : ( b : Functor I A ) → Obj A ) ( t0 : ( b : Functor I A ) → NTrans I A ( K I (a0 b) ) b ) |
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250 → coUniversalMapping A ( A ^ I ) (KI I) (λ b → A0 (lim b {a0 b} {t0 b} ) ) ( λ b → T0 ( lim b {a0 b} {t0 b} ) ) |
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251 limit2adjoint lim a0 t0 = record { |
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252 _*' = λ {b} {a} k → limit (lim b {a0 b} {t0 b} ) a k ; |
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253 iscoUniversalMapping = record { |
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254 couniversalMapping = λ{ b a f} → couniversalMapping1 {b} {a} {f} ; |
| 271 | 255 couniquness = couniquness2 |
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256 } |
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257 } where |
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258 couniversalMapping1 : {b : Obj (A ^ I)} {a : Obj A} {f : Hom (A ^ I) (FObj (KI I) a) b} → |
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259 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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260 couniversalMapping1 {b} {a} {f} {i} = let open ≈-Reasoning (A) in begin |
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261 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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262 ≈⟨⟩ |
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263 TMap (t0 b) i o (limit (lim b) a f) |
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264 ≈⟨ t0f=t (lim b) ⟩ |
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265 TMap f i -- i comes from ∀{i} → B [ TMap f i ≈ TMap g i ] |
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266 ∎ |
| 271 | 267 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} ))} → |
| 272 | 268 ( ∀ { i : Obj I } → A [ A [ TMap (T0 (lim b {a0 b} {t0 b} )) i o TMap ( FMap (KI I) g) i ] ≈ TMap f i ] ) |
| 269 → A [ limit (lim b {a0 b} {t0 b} ) a f ≈ g ] | |
| 271 | 270 couniquness2 {b} {a} {f} {g} lim-g=f = let open ≈-Reasoning (A) in begin |
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270
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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271 limit (lim b {a0 b} {t0 b} ) a f |
| 271 | 272 ≈⟨ limit-uniqueness ( lim b {a0 b} {t0 b} ) lim-g=f ⟩ |
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270
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273 g |
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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274 ∎ |
| 268 | 275 |
| 272 | 276 open import Category.Cat |
| 277 | |
| 278 adjoint2limit : | |
| 279 ( U : Functor (A ^ I) A ) ( η : NTrans A A identityFunctor ( U ○ (KI I) ) ) | |
| 280 ( ε : NTrans (A ^ I) (A ^ I) ( (KI I) ○ U ) identityFunctor ) | |
| 281 ( adj : Adjunction A (A ^ I) U (KI I) η ε ) → | |
| 282 ( Γ : Functor I A ) → { a0 : Obj A } { t0 : NTrans I A ( K I a0 ) Γ } → Limit I Γ a0 t0 | |
| 283 adjoint2limit U η ε adj Γ {a0} {t0} = record { | |
| 284 limit = λ a t → limit1 a t ; | |
| 285 t0f=t = λ {a t i } → t0f=t1 {a} {t} {i} ; | |
| 286 limit-uniqueness = λ {a} {t} {f} t=f → limit-uniqueness1 {a} {t} {f} t=f | |
| 287 } where | |
| 288 limit1 : ( a : Obj A ) → ( t : NTrans I A ( K I a ) Γ ) → Hom A a a0 | |
| 289 limit1 a t = {!!} | |
| 290 t0f=t1 : { a : Obj A } → { t : NTrans I A ( K I a ) Γ } → ∀ { i : Obj I } → | |
| 291 A [ A [ TMap t0 i o limit1 a t ] ≈ TMap t i ] | |
| 292 t0f=t1 = {!!} | |
| 293 limit-uniqueness1 : { a : Obj A } → { t : NTrans I A ( K I a ) Γ } → { f : Hom A a a0 } → ( ∀ { i : Obj I } → | |
| 294 A [ A [ TMap t0 i o f ] ≈ TMap t i ] ) → A [ limit1 a t ≈ f ] | |
| 295 limit-uniqueness1 = {!!} | |
| 269 | 296 |
| 271 | 297 |
| 272 | 298 |