Mercurial > hg > Members > kono > Proof > category
annotate pullback.agda @ 333:26f44a4fa494
factorial still have a problem
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 22 Mar 2014 14:55:51 +0700 |
| parents | 702adc45704f |
| children | cf9ee72f9b0e |
| 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 |
| 30 open Product | |
| 31 open Pullback | |
| 32 | |
| 282 | 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 ) |
| 282 | 36 ( eqa : {a b c : Obj A} → (f g : Hom A a b) → {e : Hom A c a } → Equalizer A e f g ) |
| 37 ( prod : Product A a b ab π1 π2 ) → Pullback A a b c d f g | |
| 261 | 38 ( A [ π1 o equalizer ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ){e} ) ] ) |
| 282 | 39 ( A [ π2 o equalizer ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ){e} ) ] ) |
| 261 | 40 pullback-from a b c ab d f g π1 π2 e eqa prod = record { |
| 260 | 41 commute = commute1 ; |
| 282 | 42 p = p1 ; |
| 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 |
| 282 | 46 } where |
| 302 | 47 commute1 : A [ A [ f o A [ π1 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] ] |
| 48 ≈ A [ g o A [ π2 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] ] ] | |
| 262 | 49 commute1 = let open ≈-Reasoning (A) in |
| 50 begin | |
| 282 | 51 f o ( π1 o equalizer (eqa ( f o π1 ) ( g o π2 )) ) |
| 262 | 52 ≈⟨ assoc ⟩ |
| 282 | 53 ( f o π1 ) o equalizer (eqa ( f o π1 ) ( g o π2 )) |
| 262 | 54 ≈⟨ fe=ge (eqa (A [ f o π1 ]) (A [ g o π2 ])) ⟩ |
| 282 | 55 ( g o π2 ) o equalizer (eqa ( f o π1 ) ( g o π2 )) |
| 262 | 56 ≈↑⟨ assoc ⟩ |
| 282 | 57 g o ( π2 o equalizer (eqa ( f o π1 ) ( g o π2 )) ) |
| 262 | 58 ∎ |
| 282 | 59 lemma1 : {d' : Obj A} {π1' : Hom A d' a} {π2' : Hom A d' b} → A [ A [ f o π1' ] ≈ A [ g o π2' ] ] → |
| 262 | 60 A [ A [ A [ f o π1 ] o (prod × π1') π2' ] ≈ A [ A [ g o π2 ] o (prod × π1') π2' ] ] |
| 282 | 61 lemma1 {d'} { π1' } { π2' } eq = let open ≈-Reasoning (A) in |
| 262 | 62 begin |
| 63 ( f o π1 ) o (prod × π1') π2' | |
| 64 ≈↑⟨ assoc ⟩ | |
| 65 f o ( π1 o (prod × π1') π2' ) | |
| 66 ≈⟨ cdr (π1fxg=f prod) ⟩ | |
| 67 f o π1' | |
| 68 ≈⟨ eq ⟩ | |
| 69 g o π2' | |
| 70 ≈↑⟨ cdr (π2fxg=g prod) ⟩ | |
| 71 g o ( π2 o (prod × π1') π2' ) | |
| 72 ≈⟨ assoc ⟩ | |
| 73 ( g o π2 ) o (prod × π1') π2' | |
| 74 ∎ | |
| 261 | 75 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 | 76 p1 {d'} { π1' } { π2' } eq = |
| 262 | 77 let open ≈-Reasoning (A) in k ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} ) (_×_ prod π1' π2' ) ( lemma1 eq ) |
| 282 | 78 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' ] ]} → |
| 262 | 79 A [ A [ A [ π1 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ]) {e} ) ] o p1 eq ] ≈ π1' ] |
| 80 π1p=π11 {d'} {π1'} {π2'} {eq} = let open ≈-Reasoning (A) in | |
| 81 begin | |
| 82 ( π1 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ]) {e} ) ) o p1 eq | |
| 83 ≈⟨⟩ | |
| 84 ( π1 o e) o k ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} ) (_×_ prod π1' π2' ) (lemma1 eq) | |
| 85 ≈↑⟨ assoc ⟩ | |
| 86 π1 o ( e o k ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} ) (_×_ prod π1' π2' ) (lemma1 eq) ) | |
| 87 ≈⟨ cdr ( ek=h ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} )) ⟩ | |
| 282 | 88 π1 o (_×_ prod π1' π2' ) |
| 262 | 89 ≈⟨ π1fxg=f prod ⟩ |
| 90 π1' | |
| 91 ∎ | |
| 282 | 92 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' ] ]} → |
| 263 | 93 A [ A [ A [ π2 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ]) {e} ) ] o p1 eq ] ≈ π2' ] |
| 262 | 94 π2p=π21 {d'} {π1'} {π2'} {eq} = let open ≈-Reasoning (A) in |
| 95 begin | |
| 96 ( π2 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ]) {e} ) ) o p1 eq | |
| 97 ≈⟨⟩ | |
| 98 ( π2 o e) o k ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} ) (_×_ prod π1' π2' ) (lemma1 eq) | |
| 99 ≈↑⟨ assoc ⟩ | |
| 100 π2 o ( e o k ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} ) (_×_ prod π1' π2' ) (lemma1 eq) ) | |
| 101 ≈⟨ cdr ( ek=h ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} )) ⟩ | |
| 282 | 102 π2 o (_×_ prod π1' π2' ) |
| 262 | 103 ≈⟨ π2fxg=g prod ⟩ |
| 104 π2' | |
| 105 ∎ | |
| 302 | 106 uniqueness1 : {d₁ : Obj A} (p' : Hom A d₁ d) {π1' : Hom A d₁ a} {π2' : Hom A d₁ b} |
| 107 {eq : A [ A [ f o π1' ] ≈ A [ g o π2' ] ]} → | |
| 261 | 108 {eq1 : A [ A [ A [ π1 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] o p' ] ≈ π1' ]} → |
| 109 {eq2 : A [ A [ A [ π2 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] o p' ] ≈ π2' ]} → | |
| 110 A [ p1 eq ≈ p' ] | |
| 264 | 111 uniqueness1 {d'} p' {π1'} {π2'} {eq} {eq1} {eq2} = let open ≈-Reasoning (A) in |
| 263 | 112 begin |
| 113 p1 eq | |
| 114 ≈⟨⟩ | |
| 115 k ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} ) (_×_ prod π1' π2' ) (lemma1 eq) | |
| 264 | 116 ≈⟨ Equalizer.uniqueness (eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e}) ( begin |
| 117 e o p' | |
| 118 ≈⟨⟩ | |
| 119 equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) o p' | |
| 120 ≈↑⟨ Product.uniqueness prod ⟩ | |
| 121 (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')) | |
| 122 ≈⟨ ×-cong prod (assoc) (assoc) ⟩ | |
| 123 (prod × (A [ A [ π1 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] o p' ])) | |
| 282 | 124 (A [ A [ π2 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] o p' ]) |
| 264 | 125 ≈⟨ ×-cong prod eq1 eq2 ⟩ |
| 126 ((prod × π1') π2') | |
| 127 ∎ ) ⟩ | |
| 263 | 128 p' |
| 129 ∎ | |
| 130 | |
| 266 | 131 -------------------------------- |
| 132 -- | |
| 133 -- If we have two limits on c and c', there are isomorphic pair h, h' | |
| 134 | |
| 135 open Limit | |
|
312
702adc45704f
is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
303
diff
changeset
|
136 open NTrans |
| 266 | 137 |
| 138 iso-l : { c₁' c₂' ℓ' : Level} ( I : Category c₁' c₂' ℓ' ) ( Γ : Functor I A ) | |
| 291 | 139 ( a0 a0' : Obj A ) ( t0 : NTrans I A ( K A I a0 ) Γ ) ( t0' : NTrans I A ( K A I a0' ) Γ ) |
| 140 ( lim : Limit A I Γ a0 t0 ) → ( lim' : Limit A I Γ a0' t0' ) | |
| 266 | 141 → Hom A a0 a0' |
| 142 iso-l I Γ a0 a0' t0 t0' lim lim' = limit lim' a0 t0 | |
| 143 | |
| 144 iso-r : { c₁' c₂' ℓ' : Level} ( I : Category c₁' c₂' ℓ' ) ( Γ : Functor I A ) | |
| 291 | 145 ( a0 a0' : Obj A ) ( t0 : NTrans I A ( K A I a0 ) Γ ) ( t0' : NTrans I A ( K A I a0' ) Γ ) |
| 146 ( lim : Limit A I Γ a0 t0 ) → ( lim' : Limit A I Γ a0' t0' ) | |
| 266 | 147 → Hom A a0' a0 |
| 148 iso-r I Γ a0 a0' t0 t0' lim lim' = limit lim a0' t0' | |
| 149 | |
| 150 | |
| 151 iso-lr : { c₁' c₂' ℓ' : Level} ( I : Category c₁' c₂' ℓ' ) ( Γ : Functor I A ) | |
| 291 | 152 ( a0 a0' : Obj A ) ( t0 : NTrans I A ( K A I a0 ) Γ ) ( t0' : NTrans I A ( K A I a0' ) Γ ) |
| 153 ( lim : Limit A I Γ a0 t0 ) → ( lim' : Limit A I Γ a0' t0' ) → ∀{ i : Obj I } → | |
| 266 | 154 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' ] |
| 155 iso-lr I Γ a0 a0' t0 t0' lim lim' {i} = let open ≈-Reasoning (A) in begin | |
| 156 limit lim' a0 t0 o limit lim a0' t0' | |
| 271 | 157 ≈↑⟨ limit-uniqueness lim' ( λ {i} → ( begin |
| 266 | 158 TMap t0' i o ( limit lim' a0 t0 o limit lim a0' t0' ) |
| 159 ≈⟨ assoc ⟩ | |
| 282 | 160 ( TMap t0' i o limit lim' a0 t0 ) o limit lim a0' t0' |
| 266 | 161 ≈⟨ car ( t0f=t lim' ) ⟩ |
| 282 | 162 TMap t0 i o limit lim a0' t0' |
| 266 | 163 ≈⟨ t0f=t lim ⟩ |
| 282 | 164 TMap t0' i |
| 271 | 165 ∎) ) ⟩ |
| 266 | 166 limit lim' a0' t0' |
| 271 | 167 ≈⟨ limit-uniqueness lim' idR ⟩ |
| 266 | 168 id a0' |
| 169 ∎ | |
| 170 | |
| 171 | |
| 282 | 172 open import CatExponetial |
| 267 | 173 |
| 174 open Functor | |
| 175 | |
| 176 -------------------------------- | |
| 177 -- | |
| 178 -- Contancy Functor | |
| 266 | 179 |
| 268 | 180 KI : { c₁' c₂' ℓ' : Level} ( I : Category c₁' c₂' ℓ' ) → Functor A ( A ^ I ) |
| 181 KI { c₁'} {c₂'} {ℓ'} I = record { | |
| 291 | 182 FObj = λ a → K A I a ; |
| 183 FMap = λ f → record { -- NTrans I A (K A I a) (K A I b) | |
| 267 | 184 TMap = λ a → f ; |
| 282 | 185 isNTrans = record { |
| 267 | 186 commute = λ {a b f₁} → commute1 {a} {b} {f₁} f |
| 187 } | |
| 282 | 188 } ; |
| 266 | 189 isFunctor = let open ≈-Reasoning (A) in record { |
| 267 | 190 ≈-cong = λ f=g {x} → f=g |
| 266 | 191 ; identity = refl-hom |
| 267 | 192 ; distr = refl-hom |
| 266 | 193 } |
| 267 | 194 } where |
| 195 commute1 : {a b : Obj I} {f₁ : Hom I a b} → {a' b' : Obj A} → (f : Hom A a' b' ) → | |
| 291 | 196 A [ A [ FMap (K A I b') f₁ o f ] ≈ A [ f o FMap (K A I a') f₁ ] ] |
| 282 | 197 commute1 {a} {b} {f₁} {a'} {b'} f = let open ≈-Reasoning (A) in begin |
| 291 | 198 FMap (K A I b') f₁ o f |
| 267 | 199 ≈⟨ idL ⟩ |
| 200 f | |
| 201 ≈↑⟨ idR ⟩ | |
| 291 | 202 f o FMap (K A I a') f₁ |
| 267 | 203 ∎ |
| 204 | |
| 205 | |
| 272 | 206 --------- |
| 207 -- | |
| 298 | 208 -- Limit Constancy Functor F : A → A^I has right adjoint |
| 209 -- | |
| 210 -- we are going to prove universal mapping | |
| 211 | |
| 212 --------- | |
| 213 -- | |
| 272 | 214 -- limit gives co universal mapping ( i.e. adjunction ) |
| 215 -- | |
| 216 -- F = KI I : Functor A (A ^ I) | |
| 282 | 217 -- U = λ b → A0 (lim b {a0 b} {t0 b} |
| 218 -- ε = λ b → T0 ( lim b {a0 b} {t0 b} ) | |
| 272 | 219 |
| 282 | 220 limit2couniv : |
| 291 | 221 ( lim : ( Γ : Functor I A ) → { a0 : Obj A } { t0 : NTrans I A ( K A I a0 ) Γ } → Limit A I Γ a0 t0 ) |
| 222 → ( a0 : ( b : Functor I A ) → Obj A ) ( t0 : ( b : Functor I A ) → NTrans I A ( K A I (a0 b) ) b ) | |
|
270
8ba03259a177
one yellow remain on ∀{x} → B [ TMap f x ≈ TMap g x ]
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
269
diff
changeset
|
223 → 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 | 224 limit2couniv lim a0 t0 = record { -- F U ε |
|
274
1b651faa2391
adjoint2limit problems are written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
273
diff
changeset
|
225 _*' = λ {b} {a} k → limit (lim b {a0 b} {t0 b} ) a k ; -- η |
|
270
8ba03259a177
one yellow remain on ∀{x} → B [ TMap f x ≈ TMap g x ]
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
269
diff
changeset
|
226 iscoUniversalMapping = record { |
| 282 | 227 couniversalMapping = λ{ b a f} → couniversalMapping1 {b} {a} {f} ; |
| 271 | 228 couniquness = couniquness2 |
|
270
8ba03259a177
one yellow remain on ∀{x} → B [ TMap f x ≈ TMap g x ]
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
269
diff
changeset
|
229 } |
|
8ba03259a177
one yellow remain on ∀{x} → B [ TMap f x ≈ TMap g x ]
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
269
diff
changeset
|
230 } where |
|
8ba03259a177
one yellow remain on ∀{x} → B [ TMap f x ≈ TMap g x ]
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
269
diff
changeset
|
231 couniversalMapping1 : {b : Obj (A ^ I)} {a : Obj A} {f : Hom (A ^ I) (FObj (KI I) a) b} → |
|
8ba03259a177
one yellow remain on ∀{x} → B [ TMap f x ≈ TMap g x ]
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
269
diff
changeset
|
232 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 ] |
|
8ba03259a177
one yellow remain on ∀{x} → B [ TMap f x ≈ TMap g x ]
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
269
diff
changeset
|
233 couniversalMapping1 {b} {a} {f} {i} = let open ≈-Reasoning (A) in begin |
| 282 | 234 TMap (T0 (lim b {a0 b} {t0 b})) i o TMap ( FMap (KI I) (limit (lim b {a0 b} {t0 b}) a f) ) i |
|
270
8ba03259a177
one yellow remain on ∀{x} → B [ TMap f x ≈ TMap g x ]
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
269
diff
changeset
|
235 ≈⟨⟩ |
|
8ba03259a177
one yellow remain on ∀{x} → B [ TMap f x ≈ TMap g x ]
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
269
diff
changeset
|
236 TMap (t0 b) i o (limit (lim b) a f) |
|
8ba03259a177
one yellow remain on ∀{x} → B [ TMap f x ≈ TMap g x ]
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
269
diff
changeset
|
237 ≈⟨ t0f=t (lim b) ⟩ |
|
8ba03259a177
one yellow remain on ∀{x} → B [ TMap f x ≈ TMap g x ]
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
269
diff
changeset
|
238 TMap f i -- i comes from ∀{i} → B [ TMap f i ≈ TMap g i ] |
|
8ba03259a177
one yellow remain on ∀{x} → B [ TMap f x ≈ TMap g x ]
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
269
diff
changeset
|
239 ∎ |
| 271 | 240 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 | 241 ( ∀ { 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 | 242 → A [ limit (lim b {a0 b} {t0 b} ) a f ≈ g ] |
| 271 | 243 couniquness2 {b} {a} {f} {g} lim-g=f = let open ≈-Reasoning (A) in begin |
|
270
8ba03259a177
one yellow remain on ∀{x} → B [ TMap f x ≈ TMap g x ]
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
269
diff
changeset
|
244 limit (lim b {a0 b} {t0 b} ) a f |
| 271 | 245 ≈⟨ limit-uniqueness ( lim b {a0 b} {t0 b} ) lim-g=f ⟩ |
|
270
8ba03259a177
one yellow remain on ∀{x} → B [ TMap f x ≈ TMap g x ]
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
269
diff
changeset
|
246 g |
|
8ba03259a177
one yellow remain on ∀{x} → B [ TMap f x ≈ TMap g x ]
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
269
diff
changeset
|
247 ∎ |
| 268 | 248 |
| 272 | 249 open import Category.Cat |
| 275 | 250 |
| 251 | |
| 278 | 252 open coUniversalMapping |
| 282 | 253 |
| 254 univ2limit : | |
| 255 ( U : Obj (A ^ I ) → Obj A ) | |
| 291 | 256 ( ε : ( b : Obj (A ^ I ) ) → NTrans I A (K A I (U b)) b ) |
|
279
8df8e74e6316
limit and prod/equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
278
diff
changeset
|
257 ( univ : coUniversalMapping A (A ^ I) (KI I) U (ε) ) → |
| 291 | 258 ( Γ : Functor I A ) → Limit A I Γ (U Γ) (ε Γ) |
| 278 | 259 univ2limit U ε univ Γ = record { |
| 272 | 260 limit = λ a t → limit1 a t ; |
| 282 | 261 t0f=t = λ {a t i } → t0f=t1 {a} {t} {i} ; |
| 262 limit-uniqueness = λ {a} {t} {f} t=f → limit-uniqueness1 {a} {t} {f} t=f | |
| 272 | 263 } where |
| 291 | 264 limit1 : (a : Obj A) → NTrans I A (K A I a) Γ → Hom A a (U Γ) |
| 282 | 265 limit1 a t = _*' univ {_} {a} t |
| 291 | 266 t0f=t1 : {a : Obj A} {t : NTrans I A (K A I a) Γ} {i : Obj I} → |
|
279
8df8e74e6316
limit and prod/equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
278
diff
changeset
|
267 A [ A [ TMap (ε Γ) i o limit1 a t ] ≈ TMap t i ] |
|
274
1b651faa2391
adjoint2limit problems are written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
273
diff
changeset
|
268 t0f=t1 {a} {t} {i} = let open ≈-Reasoning (A) in begin |
|
279
8df8e74e6316
limit and prod/equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
278
diff
changeset
|
269 TMap (ε Γ) i o limit1 a t |
|
274
1b651faa2391
adjoint2limit problems are written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
273
diff
changeset
|
270 ≈⟨⟩ |
| 280 | 271 TMap (ε Γ) i o _*' univ {Γ} {a} t |
| 272 ≈⟨ coIsUniversalMapping.couniversalMapping ( iscoUniversalMapping univ) {Γ} {a} {t} ⟩ | |
|
274
1b651faa2391
adjoint2limit problems are written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
273
diff
changeset
|
273 TMap t i |
|
1b651faa2391
adjoint2limit problems are written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
273
diff
changeset
|
274 ∎ |
| 291 | 275 limit-uniqueness1 : { a : Obj A } → { t : NTrans I A ( K A I a ) Γ } → { f : Hom A a (U Γ)} |
|
279
8df8e74e6316
limit and prod/equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
278
diff
changeset
|
276 → ( ∀ { i : Obj I } → A [ A [ TMap (ε Γ) i o f ] ≈ TMap t i ] ) → A [ limit1 a t ≈ f ] |
|
274
1b651faa2391
adjoint2limit problems are written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
273
diff
changeset
|
277 limit-uniqueness1 {a} {t} {f} εf=t = let open ≈-Reasoning (A) in begin |
| 278 | 278 _*' univ t |
| 279 ≈⟨ ( coIsUniversalMapping.couniquness ( iscoUniversalMapping univ) ) εf=t ⟩ | |
|
274
1b651faa2391
adjoint2limit problems are written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
273
diff
changeset
|
280 f |
|
1b651faa2391
adjoint2limit problems are written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
273
diff
changeset
|
281 ∎ |
|
1b651faa2391
adjoint2limit problems are written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
273
diff
changeset
|
282 |
|
303
7f40d6a51c72
Limit form equalizer and product done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
302
diff
changeset
|
283 |
|
7f40d6a51c72
Limit form equalizer and product done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
302
diff
changeset
|
284 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 ) → |
|
7f40d6a51c72
Limit form equalizer and product done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
302
diff
changeset
|
285 A [ _×_ prod π1 π2 ≈ id1 A ab ] |
|
7f40d6a51c72
Limit form equalizer and product done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
302
diff
changeset
|
286 lemma-p0 a b ab π1 π2 prod = let open ≈-Reasoning (A) in begin |
|
7f40d6a51c72
Limit form equalizer and product done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
302
diff
changeset
|
287 _×_ prod π1 π2 |
|
7f40d6a51c72
Limit form equalizer and product done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
302
diff
changeset
|
288 ≈↑⟨ ×-cong prod idR idR ⟩ |
|
7f40d6a51c72
Limit form equalizer and product done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
302
diff
changeset
|
289 _×_ prod (A [ π1 o id1 A ab ]) (A [ π2 o id1 A ab ]) |
|
7f40d6a51c72
Limit form equalizer and product done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
302
diff
changeset
|
290 ≈⟨ Product.uniqueness prod ⟩ |
|
7f40d6a51c72
Limit form equalizer and product done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
302
diff
changeset
|
291 id1 A ab |
|
7f40d6a51c72
Limit form equalizer and product done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
302
diff
changeset
|
292 ∎ |
|
7f40d6a51c72
Limit form equalizer and product done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
302
diff
changeset
|
293 |
|
7f40d6a51c72
Limit form equalizer and product done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
302
diff
changeset
|
294 |
|
281
dbd2044add2a
limit from product and equalizer continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
280
diff
changeset
|
295 ----- |
|
dbd2044add2a
limit from product and equalizer continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
280
diff
changeset
|
296 -- |
|
dbd2044add2a
limit from product and equalizer continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
280
diff
changeset
|
297 -- product on arbitrary index |
|
dbd2044add2a
limit from product and equalizer continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
280
diff
changeset
|
298 -- |
|
dbd2044add2a
limit from product and equalizer continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
280
diff
changeset
|
299 |
|
dbd2044add2a
limit from product and equalizer continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
280
diff
changeset
|
300 record IProduct { c c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) ( I : Set c) |
|
dbd2044add2a
limit from product and equalizer continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
280
diff
changeset
|
301 ( p : Obj A ) -- product |
|
dbd2044add2a
limit from product and equalizer continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
280
diff
changeset
|
302 ( ai : I → Obj A ) -- families |
|
dbd2044add2a
limit from product and equalizer continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
280
diff
changeset
|
303 ( pi : (i : I ) → Hom A p ( ai i ) ) -- projections |
|
dbd2044add2a
limit from product and equalizer continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
280
diff
changeset
|
304 : Set (c ⊔ ℓ ⊔ (c₁ ⊔ c₂)) where |
|
dbd2044add2a
limit from product and equalizer continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
280
diff
changeset
|
305 field |
|
dbd2044add2a
limit from product and equalizer continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
280
diff
changeset
|
306 product : {q : Obj A} → ( qi : (i : I) → Hom A q (ai i) ) → Hom A q p |
|
dbd2044add2a
limit from product and equalizer continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
280
diff
changeset
|
307 pif=q : {q : Obj A} → ( qi : (i : I) → Hom A q (ai i) ) → ∀ { i : I } → A [ A [ ( pi i ) o ( product qi ) ] ≈ (qi i) ] |
|
dbd2044add2a
limit from product and equalizer continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
280
diff
changeset
|
308 ip-uniqueness : {q : Obj A} { h : Hom A q p } → A [ product ( λ (i : I) → A [ (pi i) o h ] ) ≈ h ] |
| 283 | 309 ip-cong : {q : Obj A} → { qi : (i : I) → Hom A q (ai i) } → { qi' : (i : I) → Hom A q (ai i) } |
| 282 | 310 → ( ∀ (i : I ) → A [ qi i ≈ qi' i ] ) → A [ product qi ≈ product qi' ] |
| 302 | 311 -- another form of uniquness |
|
303
7f40d6a51c72
Limit form equalizer and product done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
302
diff
changeset
|
312 ip-uniqueness1 : {q : Obj A} → ( qi : (i : I) → Hom A q (ai i) ) → ( product' : Hom A q p ) |
| 302 | 313 → ( ∀ { i : I } → A [ A [ ( pi i ) o product' ] ≈ (qi i) ] ) |
|
299
8c72f5284bc8
remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
298
diff
changeset
|
314 → A [ product' ≈ product qi ] |
|
303
7f40d6a51c72
Limit form equalizer and product done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
302
diff
changeset
|
315 ip-uniqueness1 {a} qi product' eq = let open ≈-Reasoning (A) in begin |
| 302 | 316 product' |
| 317 ≈↑⟨ ip-uniqueness ⟩ | |
| 318 product (λ i₁ → A [ pi i₁ o product' ]) | |
| 319 ≈⟨ ip-cong ( λ i → begin | |
| 320 pi i o product' | |
| 321 ≈⟨ eq {i} ⟩ | |
| 322 qi i | |
| 323 ∎ ) ⟩ | |
| 324 product qi | |
| 325 ∎ | |
| 282 | 326 |
| 327 open IProduct | |
| 283 | 328 open Equalizer |
|
281
dbd2044add2a
limit from product and equalizer continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
280
diff
changeset
|
329 |
| 282 | 330 -- |
| 331 -- limit from equalizer and product | |
| 332 -- | |
| 333 -- | |
| 283 | 334 -- ai |
| 335 -- ^ K f = id lim | |
| 300 | 336 -- | pi lim = K i -----------→ K j = lim |
| 283 | 337 -- | | | |
| 338 -- p | | | |
|
303
7f40d6a51c72
Limit form equalizer and product done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
302
diff
changeset
|
339 -- ^ proj i o e = ε i | | ε j = proj j o e |
| 283 | 340 -- | | | |
|
285
46d4ad55b948
commutativity continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
284
diff
changeset
|
341 -- | e = equalizer (id p) (id p) | | |
| 283 | 342 -- | v v |
| 300 | 343 -- lim <------------------ d' a i = Γ i -----------→ Γ j = a j |
| 283 | 344 -- k ( product pi ) Γ f |
| 345 -- Γ f o ε i = ε j | |
| 346 -- | |
| 291 | 347 |
| 283 | 348 limit-ε : |
|
303
7f40d6a51c72
Limit form equalizer and product done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
302
diff
changeset
|
349 ( eqa : {a b c : Obj A} → (e : Hom A c a ) → (f g : Hom A a b) → Equalizer A e f g ) |
| 282 | 350 ( lim p : Obj A ) ( e : Hom A lim p ) |
| 351 ( proj : (i : Obj I ) → Hom A p (FObj Γ i) ) → | |
| 291 | 352 NTrans I A (K A I lim) Γ |
|
303
7f40d6a51c72
Limit form equalizer and product done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
302
diff
changeset
|
353 limit-ε eqa lim p e proj = record { |
| 282 | 354 TMap = tmap ; |
|
303
7f40d6a51c72
Limit form equalizer and product done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
302
diff
changeset
|
355 isNTrans = record { commute = commute1 } |
|
281
dbd2044add2a
limit from product and equalizer continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
280
diff
changeset
|
356 } where |
| 291 | 357 tmap : (i : Obj I) → Hom A (FObj (K A I lim) i) (FObj Γ i) |
|
285
46d4ad55b948
commutativity continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
284
diff
changeset
|
358 tmap i = A [ proj i o e ] |
| 283 | 359 commute1 : {i j : Obj I} {f : Hom I i j} → |
| 291 | 360 A [ A [ FMap Γ f o tmap i ] ≈ A [ tmap j o FMap (K A I lim) f ] ] |
|
285
46d4ad55b948
commutativity continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
284
diff
changeset
|
361 commute1 {i} {j} {f} = let open ≈-Reasoning (A) in begin |
|
46d4ad55b948
commutativity continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
284
diff
changeset
|
362 FMap Γ f o tmap i |
|
46d4ad55b948
commutativity continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
284
diff
changeset
|
363 ≈⟨⟩ |
|
46d4ad55b948
commutativity continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
284
diff
changeset
|
364 FMap Γ f o ( proj i o e ) |
|
46d4ad55b948
commutativity continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
284
diff
changeset
|
365 ≈⟨ assoc ⟩ |
|
46d4ad55b948
commutativity continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
284
diff
changeset
|
366 ( FMap Γ f o proj i ) o e |
|
303
7f40d6a51c72
Limit form equalizer and product done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
302
diff
changeset
|
367 ≈⟨ fe=ge ( eqa e (FMap Γ f o proj i) ( proj j )) ⟩ |
|
285
46d4ad55b948
commutativity continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
284
diff
changeset
|
368 proj j o e |
|
46d4ad55b948
commutativity continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
284
diff
changeset
|
369 ≈↑⟨ idR ⟩ |
|
46d4ad55b948
commutativity continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
284
diff
changeset
|
370 (proj j o e ) o id1 A lim |
|
46d4ad55b948
commutativity continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
284
diff
changeset
|
371 ≈⟨⟩ |
| 291 | 372 tmap j o FMap (K A I lim) f |
| 288 | 373 ∎ |
|
281
dbd2044add2a
limit from product and equalizer continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
280
diff
changeset
|
374 |
| 282 | 375 limit-from : |
|
285
46d4ad55b948
commutativity continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
284
diff
changeset
|
376 ( prod : (p : Obj A) ( ai : Obj I → Obj A ) ( pi : (i : Obj I) → Hom A p ( ai i ) ) |
|
281
dbd2044add2a
limit from product and equalizer continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
280
diff
changeset
|
377 → IProduct {c₁'} A (Obj I) p ai pi ) |
|
285
46d4ad55b948
commutativity continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
284
diff
changeset
|
378 ( eqa : {a b c : Obj A} → (e : Hom A c a ) → (f g : Hom A a b) → Equalizer A e f g ) |
| 290 | 379 ( lim p : Obj A ) -- limit to be made |
|
285
46d4ad55b948
commutativity continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
284
diff
changeset
|
380 ( e : Hom A lim p ) -- existing of equalizer |
|
46d4ad55b948
commutativity continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
284
diff
changeset
|
381 ( proj : (i : Obj I ) → Hom A p (FObj Γ i) ) -- existing of product ( projection actually ) |
|
303
7f40d6a51c72
Limit form equalizer and product done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
302
diff
changeset
|
382 → Limit A I Γ lim ( limit-ε eqa lim p e proj ) |
| 290 | 383 limit-from prod eqa lim p e proj = record { |
| 282 | 384 limit = λ a t → limit1 a t ; |
| 385 t0f=t = λ {a t i } → t0f=t1 {a} {t} {i} ; | |
| 386 limit-uniqueness = λ {a} {t} {f} t=f → limit-uniqueness1 {a} {t} {f} t=f | |
| 387 } where | |
| 291 | 388 limit1 : (a : Obj A) → NTrans I A (K A I a) Γ → Hom A a lim |
| 283 | 389 limit1 a t = let open ≈-Reasoning (A) in k (eqa e (id1 A p) (id1 A p )) (product ( prod p (FObj Γ) proj ) (TMap t) ) refl-hom |
| 291 | 390 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.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
302
diff
changeset
|
391 A [ A [ TMap (limit-ε eqa lim p e proj ) i o limit1 a t ] ≈ TMap t i ] |
| 283 | 392 t0f=t1 {a} {t} {i} = let open ≈-Reasoning (A) in begin |
|
303
7f40d6a51c72
Limit form equalizer and product done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
302
diff
changeset
|
393 TMap (limit-ε eqa lim p e proj ) i o limit1 a t |
| 283 | 394 ≈⟨⟩ |
| 395 ( ( proj i ) o e ) o k (eqa e (id1 A p) (id1 A p )) (product ( prod p (FObj Γ) proj ) (TMap t) ) refl-hom | |
| 396 ≈↑⟨ assoc ⟩ | |
| 397 proj i o ( e o k (eqa e (id1 A p) (id1 A p )) (product ( prod p (FObj Γ) proj ) (TMap t) ) refl-hom ) | |
| 398 ≈⟨ cdr ( ek=h ( eqa e (id1 A p) (id1 A p ) ) ) ⟩ | |
| 399 proj i o product (prod p (FObj Γ) proj) (TMap t) | |
| 400 ≈⟨ pif=q (prod p (FObj Γ) proj) (TMap t) ⟩ | |
| 401 TMap t i | |
| 402 ∎ | |
| 291 | 403 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.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
302
diff
changeset
|
404 → ({i : Obj I} → A [ A [ TMap (limit-ε eqa lim p e proj ) i o f ] ≈ TMap t i ]) → |
| 282 | 405 A [ limit1 a t ≈ f ] |
| 283 | 406 limit-uniqueness1 {a} {t} {f} lim=t = let open ≈-Reasoning (A) in begin |
| 407 limit1 a t | |
| 408 ≈⟨⟩ | |
| 409 k (eqa e (id1 A p) (id1 A p )) (product ( prod p (FObj Γ) proj ) (TMap t) ) refl-hom | |
| 410 ≈⟨ Equalizer.uniqueness (eqa e (id1 A p) (id1 A p )) ( begin | |
| 411 e o f | |
| 412 ≈↑⟨ ip-uniqueness (prod p (FObj Γ) proj) ⟩ | |
| 413 product (prod p (FObj Γ) proj) (λ i → ( proj i o ( e o f ) ) ) | |
| 284 | 414 ≈⟨ ip-cong (prod p (FObj Γ) proj) ( λ i → begin |
|
285
46d4ad55b948
commutativity continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
284
diff
changeset
|
415 proj i o ( e o f ) |
| 284 | 416 ≈⟨ assoc ⟩ |
|
285
46d4ad55b948
commutativity continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
284
diff
changeset
|
417 ( proj i o e ) o f |
|
46d4ad55b948
commutativity continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
284
diff
changeset
|
418 ≈⟨ lim=t {i} ⟩ |
|
46d4ad55b948
commutativity continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
284
diff
changeset
|
419 TMap t i |
| 284 | 420 ∎ ) ⟩ |
| 283 | 421 product (prod p (FObj Γ) proj) (TMap t) |
| 422 ∎ ) ⟩ | |
| 423 f | |
| 424 ∎ | |
| 425 | |
| 291 | 426 ---- |
| 427 -- | |
| 428 -- Adjoint functor preserves limits | |
| 429 -- | |
| 430 -- | |
| 431 | |
| 432 open import Category.Cat | |
| 433 | |
| 434 ta1 : { c₁' c₂' ℓ' : Level} (B : Category c₁' c₂' ℓ') ( Γ : Functor I B ) | |
|
299
8c72f5284bc8
remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
298
diff
changeset
|
435 ( lim : Obj B ) ( tb : NTrans I B ( K B I lim ) Γ ) → |
| 291 | 436 ( U : Functor B A) → NTrans I A ( K A I (FObj U lim) ) (U ○ Γ) |
|
299
8c72f5284bc8
remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
298
diff
changeset
|
437 ta1 B Γ lim tb U = record { |
| 291 | 438 TMap = TMap (Functor*Nat I A U tb) ; |
| 439 isNTrans = record { commute = λ {a} {b} {f} → let open ≈-Reasoning (A) in begin | |
| 440 FMap (U ○ Γ) f o TMap (Functor*Nat I A U tb) a | |
| 441 ≈⟨ nat ( Functor*Nat I A U tb ) ⟩ | |
| 442 TMap (Functor*Nat I A U tb) b o FMap (U ○ K B I lim) f | |
| 443 ≈⟨ cdr (IsFunctor.identity (isFunctor U) ) ⟩ | |
| 444 TMap (Functor*Nat I A U tb) b o FMap (K A I (FObj U lim)) f | |
| 445 ∎ | |
| 446 } } | |
| 447 | |
| 448 adjoint-preseve-limit : | |
| 449 { c₁' c₂' ℓ' : Level} (B : Category c₁' c₂' ℓ') ( Γ : Functor I B ) | |
|
299
8c72f5284bc8
remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
298
diff
changeset
|
450 ( lim : Obj B ) ( tb : NTrans I B ( K B I lim ) Γ ) → ( limitb : Limit B I Γ lim tb ) → |
| 291 | 451 { U : Functor B A } { F : Functor A B } |
| 293 | 452 { η : NTrans A A identityFunctor ( U ○ F ) } |
| 291 | 453 { ε : NTrans B B ( F ○ U ) identityFunctor } → |
|
299
8c72f5284bc8
remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
298
diff
changeset
|
454 ( adj : Adjunction A B U F η ε ) → Limit A I (U ○ Γ) (FObj U lim) (ta1 B Γ lim tb U ) |
| 292 | 455 adjoint-preseve-limit B Γ lim tb limitb {U} {F} {η} {ε} adj = record { |
| 291 | 456 limit = λ a t → limit1 a t ; |
| 457 t0f=t = λ {a t i } → t0f=t1 {a} {t} {i} ; | |
| 458 limit-uniqueness = λ {a} {t} {f} t=f → limit-uniqueness1 {a} {t} {f} t=f | |
| 459 } where | |
|
299
8c72f5284bc8
remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
298
diff
changeset
|
460 ta = ta1 B Γ lim tb U |
| 293 | 461 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) |
| 462 tfmap a t i = B [ TMap ε (FObj Γ i) o FMap F (TMap t i) ] | |
| 463 tF : (a : Obj A) → NTrans I A (K A I a) (U ○ Γ) → NTrans I B (K B I (FObj F a)) Γ | |
| 464 tF a t = record { | |
| 465 TMap = tfmap a t ; | |
| 466 isNTrans = record { commute = λ {a'} {b} {f} → let open ≈-Reasoning (B) in begin | |
| 467 FMap Γ f o tfmap a t a' | |
| 294 | 468 ≈⟨⟩ |
| 469 FMap Γ f o ( TMap ε (FObj Γ a') o FMap F (TMap t a')) | |
| 470 ≈⟨ assoc ⟩ | |
| 471 (FMap Γ f o TMap ε (FObj Γ a') ) o FMap F (TMap t a') | |
| 472 ≈⟨ car (nat ε) ⟩ | |
| 473 (TMap ε (FObj Γ b) o FMap (F ○ U) (FMap Γ f) ) o FMap F (TMap t a') | |
| 474 ≈↑⟨ assoc ⟩ | |
| 475 TMap ε (FObj Γ b) o ( FMap (F ○ U) (FMap Γ f) o FMap F (TMap t a') ) | |
| 476 ≈↑⟨ cdr ( distr F ) ⟩ | |
| 477 TMap ε (FObj Γ b) o ( FMap F (A [ FMap U (FMap Γ f) o TMap t a' ] ) ) | |
| 478 ≈⟨ cdr ( fcong F (nat t) ) ⟩ | |
| 479 TMap ε (FObj Γ b) o FMap F (A [ TMap t b o FMap (K A I a) f ]) | |
| 480 ≈⟨⟩ | |
| 481 TMap ε (FObj Γ b) o FMap F (A [ TMap t b o id1 A (FObj (K A I a) b) ]) | |
|
299
8c72f5284bc8
remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
298
diff
changeset
|
482 ≈⟨ cdr ( fcong F (idR1 A)) ⟩ |
| 294 | 483 TMap ε (FObj Γ b) o FMap F (TMap t b ) |
| 484 ≈↑⟨ idR ⟩ | |
| 485 ( TMap ε (FObj Γ b) o FMap F (TMap t b)) o id1 B (FObj F (FObj (K A I a) b)) | |
| 486 ≈⟨⟩ | |
| 293 | 487 tfmap a t b o FMap (K B I (FObj F a)) f |
| 488 ∎ | |
| 489 } } | |
| 490 limit1 : (a : Obj A) → NTrans I A (K A I a) (U ○ Γ) → Hom A a (FObj U lim) | |
| 491 limit1 a t = A [ FMap U (limit limitb (FObj F a) (tF a t )) o TMap η a ] | |
| 492 t0f=t1 : {a : Obj A} {t : NTrans I A (K A I a) (U ○ Γ)} {i : Obj I} → | |
| 291 | 493 A [ A [ TMap ta i o limit1 a t ] ≈ TMap t i ] |
| 295 | 494 t0f=t1 {a} {t} {i} = let open ≈-Reasoning (A) in begin |
| 495 TMap ta i o limit1 a t | |
| 496 ≈⟨⟩ | |
| 497 FMap U ( TMap tb i ) o ( FMap U (limit limitb (FObj F a) (tF a t )) o TMap η a ) | |
| 498 ≈⟨ assoc ⟩ | |
| 499 ( FMap U ( TMap tb i ) o FMap U (limit limitb (FObj F a) (tF a t ))) o TMap η a | |
| 500 ≈↑⟨ car ( distr U ) ⟩ | |
| 501 FMap U ( B [ TMap tb i o limit limitb (FObj F a) (tF a t ) ] ) o TMap η a | |
| 502 ≈⟨ car ( fcong U ( t0f=t limitb ) ) ⟩ | |
| 503 FMap U (TMap (tF a t) i) o TMap η a | |
| 504 ≈⟨⟩ | |
| 505 FMap U ( B [ TMap ε (FObj Γ i) o FMap F (TMap t i) ] ) o TMap η a | |
| 506 ≈⟨ car ( distr U ) ⟩ | |
| 507 ( FMap U ( TMap ε (FObj Γ i)) o FMap U ( FMap F (TMap t i) )) o TMap η a | |
| 508 ≈↑⟨ assoc ⟩ | |
| 509 FMap U ( TMap ε (FObj Γ i) ) o ( FMap U ( FMap F (TMap t i) ) o TMap η a ) | |
| 510 ≈⟨ cdr ( nat η ) ⟩ | |
| 511 FMap U (TMap ε (FObj Γ i)) o ( TMap η (FObj U (FObj Γ i)) o FMap (identityFunctor {_} {_} {_} {A}) (TMap t i) ) | |
| 512 ≈⟨ assoc ⟩ | |
| 513 ( FMap U (TMap ε (FObj Γ i)) o TMap η (FObj U (FObj Γ i))) o TMap t i | |
| 514 ≈⟨ car ( IsAdjunction.adjoint1 ( Adjunction.isAdjunction adj ) ) ⟩ | |
| 515 id1 A (FObj (U ○ Γ) i) o TMap t i | |
| 516 ≈⟨ idL ⟩ | |
| 517 TMap t i | |
| 518 ∎ | |
| 296 | 519 -- ta = TMap (Functor*Nat I A U tb) , FMap U ( TMap tb i ) o f ≈ TMap t i |
| 293 | 520 limit-uniqueness1 : {a : Obj A} {t : NTrans I A (K A I a) (U ○ Γ)} {f : Hom A a (FObj U lim)} |
| 291 | 521 → ({i : Obj I} → A [ A [ TMap ta i o f ] ≈ TMap t i ]) → |
| 522 A [ limit1 a t ≈ f ] | |
| 295 | 523 limit-uniqueness1 {a} {t} {f} lim=t = let open ≈-Reasoning (A) in begin |
| 524 limit1 a t | |
| 525 ≈⟨⟩ | |
| 526 FMap U (limit limitb (FObj F a) (tF a t )) o TMap η a | |
| 296 | 527 ≈⟨ car ( fcong U (limit-uniqueness limitb ( λ {i} → lemma1 i) )) ⟩ |
| 298 | 528 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
|
529 ≈⟨ car (distr U ) ⟩ |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
530 ( (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
|
531 ≈⟨ sym assoc ⟩ |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
532 (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
|
533 ≈⟨ cdr (nat η) ⟩ |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
534 (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
|
535 ≈⟨ assoc ⟩ |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
536 ((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
|
537 ≈⟨ car ( IsAdjunction.adjoint1 ( Adjunction.isAdjunction adj)) ⟩ |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
538 id (FObj U lim) o f |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
539 ≈⟨ idL ⟩ |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
540 f |
| 296 | 541 ∎ where |
| 542 lemma1 : (i : Obj I) → B [ B [ TMap tb i o B [ TMap ε lim o FMap F f ] ] ≈ TMap (tF a t) i ] | |
| 543 lemma1 i = let open ≈-Reasoning (B) in begin | |
| 544 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
|
545 ≈⟨ assoc ⟩ |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
546 ( 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
|
547 ≈⟨ car ( nat ε ) ⟩ |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
548 ( 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
|
549 ≈↑⟨ assoc ⟩ |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
550 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
|
551 ≈↑⟨ cdr ( distr F ) ⟩ |
|
537570f6a44f
limit preservation proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
296
diff
changeset
|
552 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
|
553 ≈⟨ cdr ( fcong F (lim=t {i}) ) ⟩ |
| 296 | 554 TMap ε (FObj Γ i) o FMap F (TMap t i) |
| 555 ≈⟨⟩ | |
| 556 TMap (tF a t) i | |
| 557 ∎ | |
| 295 | 558 |
| 296 | 559 |
| 560 |