Mercurial > hg > Members > kono > Proof > category
annotate freyd2.agda @ 614:e6be03d94284
Representational Functor preserve limit done
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Tue, 13 Jun 2017 22:53:44 +0900 |
parents | afddfebea797 |
children | a45c32ceca97 |
rev | line source |
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1 open import Category -- https://github.com/konn/category-agda |
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2 open import Level |
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3 open import Category.Sets renaming ( _o_ to _*_ ) |
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4 |
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5 module freyd2 |
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6 where |
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7 |
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8 open import HomReasoning |
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9 open import cat-utility |
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10 open import Relation.Binary.Core |
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11 open import Function |
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12 |
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13 ---------- |
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14 -- |
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15 -- a : Obj A |
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16 -- U : A → Sets |
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17 -- U ⋍ Hom (a,-) |
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18 -- |
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19 |
608 | 20 ---- |
21 -- C is locally small i.e. Hom C i j is a set c₁ | |
22 -- | |
23 record Small { c₁ c₂ ℓ : Level} ( C : Category c₁ c₂ ℓ ) ( I : Set c₁ ) | |
24 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ )) where | |
25 field | |
26 hom→ : {i j : Obj C } → Hom C i j → I | |
27 hom← : {i j : Obj C } → ( f : I ) → Hom C i j | |
28 hom-iso : {i j : Obj C } → { f : Hom C i j } → hom← ( hom→ f ) ≡ f | |
29 | |
30 open Small | |
31 | |
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32 postulate ≈-≡ : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a b : Obj A } { x y : Hom A a b } → (x≈y : A [ x ≈ y ]) → x ≡ y |
497
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33 |
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34 import Relation.Binary.PropositionalEquality |
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35 -- Extensionality a b = {A : Set a} {B : A → Set b} {f g : (x : A) → B x} → (∀ x → f x ≡ g x) → f ≡ g → ( λ x → f x ≡ λ x → g x ) |
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36 postulate extensionality : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → Relation.Binary.PropositionalEquality.Extensionality c₂ c₂ |
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37 |
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38 |
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39 ---- |
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40 -- |
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41 -- Hom ( a, - ) is Object mapping in co Yoneda Functor |
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42 -- |
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43 ---- |
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44 |
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45 open NTrans |
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46 open Functor |
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47 open Limit |
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48 open IsLimit |
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49 open import Category.Cat |
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50 |
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51 HomA : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) (a : Obj A) → Functor A (Sets {c₂}) |
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52 HomA {c₁} {c₂} {ℓ} A a = record { |
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53 FObj = λ b → Hom A a b |
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54 ; FMap = λ {x} {y} (f : Hom A x y ) → λ ( g : Hom A a x ) → A [ f o g ] -- f : Hom A x y → Hom Sets (Hom A a x ) (Hom A a y) |
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55 ; isFunctor = record { |
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56 identity = λ {b} → extensionality A ( λ x → lemma-y-obj1 {b} x ) ; |
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57 distr = λ {a} {b} {c} {f} {g} → extensionality A ( λ x → lemma-y-obj2 a b c f g x ) ; |
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58 ≈-cong = λ eq → extensionality A ( λ x → lemma-y-obj3 x eq ) |
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59 } |
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60 } where |
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61 lemma-y-obj1 : {b : Obj A } → (x : Hom A a b) → A [ id1 A b o x ] ≡ x |
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62 lemma-y-obj1 {b} x = let open ≈-Reasoning A in ≈-≡ A idL |
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63 lemma-y-obj2 : (a₁ b c : Obj A) (f : Hom A a₁ b) (g : Hom A b c ) → (x : Hom A a a₁ )→ |
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64 A [ A [ g o f ] o x ] ≡ (Sets [ _[_o_] A g o _[_o_] A f ]) x |
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65 lemma-y-obj2 a₁ b c f g x = let open ≈-Reasoning A in ≈-≡ A ( begin |
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66 A [ A [ g o f ] o x ] |
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67 ≈↑⟨ assoc ⟩ |
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68 A [ g o A [ f o x ] ] |
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69 ≈⟨⟩ |
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70 ( λ x → A [ g o x ] ) ( ( λ x → A [ f o x ] ) x ) |
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71 ∎ ) |
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72 lemma-y-obj3 : {b c : Obj A} {f g : Hom A b c } → (x : Hom A a b ) → A [ f ≈ g ] → A [ f o x ] ≡ A [ g o x ] |
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73 lemma-y-obj3 {_} {_} {f} {g} x eq = let open ≈-Reasoning A in ≈-≡ A ( begin |
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74 A [ f o x ] |
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75 ≈⟨ resp refl-hom eq ⟩ |
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76 A [ g o x ] |
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77 ∎ ) |
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78 |
609 | 79 -- Representable U ≈ Hom(A,-) |
502 | 80 |
609 | 81 record Representable { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) ( U : Functor A (Sets {c₂}) ) (a : Obj A) : Set (suc ℓ ⊔ (suc (suc c₂) ⊔ suc c₁ )) where |
502 | 82 field |
83 -- FObj U x : A → Set | |
609 | 84 -- FMap U f = Set → Set (locally small) |
502 | 85 -- λ b → Hom (a,b) : A → Set |
86 -- λ f → Hom (a,-) = λ b → Hom a b | |
87 | |
609 | 88 repr→ : NTrans A (Sets {c₂}) U (HomA A a ) |
89 repr← : NTrans A (Sets {c₂}) (HomA A a) U | |
90 reprId→ : {x : Obj A} → Sets [ Sets [ TMap repr→ x o TMap repr← x ] ≈ id1 (Sets {c₂}) (FObj (HomA A a) x )] | |
91 reprId← : {x : Obj A} → Sets [ Sets [ TMap repr← x o TMap repr→ x ] ≈ id1 (Sets {c₂}) (FObj U x)] | |
608 | 92 |
609 | 93 open Representable |
608 | 94 open import freyd |
502 | 95 |
608 | 96 _↓_ : { c₁ c₂ ℓ : Level} { c₁' c₂' ℓ' : Level} { A : Category c₁ c₂ ℓ } { B : Category c₁' c₂' ℓ' } |
97 → ( F G : Functor A B ) | |
98 → Category (c₂' ⊔ c₁) (ℓ' ⊔ c₂) ℓ | |
99 _↓_ { c₁} {c₂} {ℓ} {c₁'} {c₂'} {ℓ'} { A } { B } F G = CommaCategory | |
100 where open import Comma1 F G | |
498
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101 |
608 | 102 open import freyd |
103 open SmallFullSubcategory | |
104 open Complete | |
105 open PreInitial | |
609 | 106 open HasInitialObject |
107 open import Comma1 | |
108 open CommaObj | |
109 open LimitPreserve | |
608 | 110 |
609 | 111 -- Representable Functor U preserve limit , so K{*}↓U is complete |
610 | 112 -- |
113 -- HomA A b = λ a → Hom A a b : Functor A Sets | |
114 -- : Functor Sets A | |
115 | |
116 UpreserveLimit0 : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (I : Category c₁ c₂ ℓ) | |
612 | 117 (b : Obj A ) |
610 | 118 (Γ : Functor I A) (limita : Limit A I Γ) → |
119 IsLimit Sets I (HomA A b ○ Γ) (FObj (HomA A b) (a0 limita)) (LimitNat A I Sets Γ (a0 limita) (t0 limita) (HomA A b)) | |
612 | 120 UpreserveLimit0 {c₁} {c₂} {ℓ} A I b Γ lim = record { |
611
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121 limit = λ a t → ψ a t |
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122 ; t0f=t = λ {a t i} → t0f=t0 a t i |
614
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123 ; limit-uniqueness = λ {b} {t} {f} t0f=t → limit-uniqueness0 {b} {t} {f} t0f=t |
610 | 124 } where |
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125 hat0 : NTrans I Sets (K Sets I (FObj (HomA A b) (a0 lim))) (HomA A b ○ Γ) |
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126 hat0 = LimitNat A I Sets Γ (a0 lim) (t0 lim) (HomA A b) |
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127 haa0 : Obj Sets |
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128 haa0 = FObj (HomA A b) (a0 lim) |
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129 ta : (a : Obj Sets) ( x : a ) ( t : NTrans I Sets (K Sets I a) (HomA A b ○ Γ)) → NTrans I A (K A I b ) Γ |
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130 ta a x t = record { TMap = λ i → (TMap t i ) x ; isNTrans = record { commute = commute1 } } where |
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131 commute1 : {a₁ b₁ : Obj I} {f : Hom I a₁ b₁} → |
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132 A [ A [ FMap Γ f o TMap t a₁ x ] ≈ A [ TMap t b₁ x o FMap (K A I b) f ] ] |
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133 commute1 {a₁} {b₁} {f} = let open ≈-Reasoning A in begin |
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134 FMap Γ f o TMap t a₁ x |
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135 ≈⟨⟩ |
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136 ( ( FMap (HomA A b ○ Γ ) f ) * TMap t a₁ ) x |
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|
137 ≈⟨ ≡-≈ ( cong (λ k → k x ) (IsNTrans.commute (isNTrans t)) ) ⟩ |
b1b5c6b4c570
natural transformation in representable functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
610
diff
changeset
|
138 ( TMap t b₁ * ( FMap (K Sets I a) f ) ) x |
b1b5c6b4c570
natural transformation in representable functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
610
diff
changeset
|
139 ≈⟨⟩ |
b1b5c6b4c570
natural transformation in representable functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
610
diff
changeset
|
140 ( TMap t b₁ * id1 Sets a ) x |
b1b5c6b4c570
natural transformation in representable functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
610
diff
changeset
|
141 ≈⟨⟩ |
b1b5c6b4c570
natural transformation in representable functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
610
diff
changeset
|
142 TMap t b₁ x |
b1b5c6b4c570
natural transformation in representable functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
610
diff
changeset
|
143 ≈↑⟨ idR ⟩ |
b1b5c6b4c570
natural transformation in representable functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
610
diff
changeset
|
144 TMap t b₁ x o id1 A b |
b1b5c6b4c570
natural transformation in representable functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
610
diff
changeset
|
145 ≈⟨⟩ |
b1b5c6b4c570
natural transformation in representable functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
610
diff
changeset
|
146 TMap t b₁ x o FMap (K A I b) f |
b1b5c6b4c570
natural transformation in representable functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
610
diff
changeset
|
147 ∎ |
b1b5c6b4c570
natural transformation in representable functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
610
diff
changeset
|
148 ψ : (X : Obj Sets) ( t : NTrans I Sets (K Sets I X) (HomA A b ○ Γ)) |
b1b5c6b4c570
natural transformation in representable functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
610
diff
changeset
|
149 → Hom Sets X (FObj (HomA A b) (a0 lim)) |
b1b5c6b4c570
natural transformation in representable functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
610
diff
changeset
|
150 ψ X t x = FMap (HomA A b) (limit (isLimit lim) b (ta X x t )) (id1 A b ) |
b1b5c6b4c570
natural transformation in representable functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
610
diff
changeset
|
151 t0f=t0 : (a : Obj Sets ) ( t : NTrans I Sets (K Sets I a) (HomA A b ○ Γ)) (i : Obj I) |
b1b5c6b4c570
natural transformation in representable functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
610
diff
changeset
|
152 → Sets [ Sets [ TMap (LimitNat A I Sets Γ (a0 lim) (t0 lim) (HomA A b)) i o ψ a t ] ≈ TMap t i ] |
612 | 153 t0f=t0 a t i = let open ≈-Reasoning A in extensionality A ( λ x → ≈-≡ A ( begin |
154 ( Sets [ TMap (LimitNat A I Sets Γ (a0 lim) (t0 lim) (HomA A b)) i o ψ a t ] ) x | |
155 ≈⟨⟩ | |
613 | 156 FMap (HomA A b) ( TMap (t0 lim) i) (FMap (HomA A b) (limit (isLimit lim) b (ta a x t )) (id1 A b )) |
157 ≈⟨⟩ | |
158 TMap (t0 lim) i o (limit (isLimit lim) b (ta a x t ) o id1 A b ) | |
159 ≈⟨ cdr idR ⟩ | |
160 TMap (t0 lim) i o limit (isLimit lim) b (ta a x t ) | |
161 ≈⟨ t0f=t (isLimit lim) ⟩ | |
162 TMap (ta a x t) i | |
163 ≈⟨⟩ | |
612 | 164 TMap t i x |
165 ∎ )) | |
614
e6be03d94284
Representational Functor preserve limit done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
613
diff
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|
166 limit-uniqueness0 : {a : Obj Sets} {t : NTrans I Sets (K Sets I a) (HomA A b ○ Γ)} {f : Hom Sets a (FObj (HomA A b) (a0 lim))} → |
e6be03d94284
Representational Functor preserve limit done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
613
diff
changeset
|
167 ({i : Obj I} → Sets [ Sets [ TMap (LimitNat A I Sets Γ (a0 lim) (t0 lim) (HomA A b)) i o f ] ≈ TMap t i ]) → |
e6be03d94284
Representational Functor preserve limit done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
613
diff
changeset
|
168 Sets [ ψ a t ≈ f ] |
e6be03d94284
Representational Functor preserve limit done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
613
diff
changeset
|
169 limit-uniqueness0 {a} {t} {f} t0f=t = let open ≈-Reasoning A in extensionality A ( λ x → ≈-≡ A ( begin |
e6be03d94284
Representational Functor preserve limit done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
613
diff
changeset
|
170 ψ a t x |
e6be03d94284
Representational Functor preserve limit done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
613
diff
changeset
|
171 ≈⟨⟩ |
e6be03d94284
Representational Functor preserve limit done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
613
diff
changeset
|
172 FMap (HomA A b) (limit (isLimit lim) b (ta a x t )) (id1 A b ) |
e6be03d94284
Representational Functor preserve limit done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
613
diff
changeset
|
173 ≈⟨⟩ |
e6be03d94284
Representational Functor preserve limit done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
613
diff
changeset
|
174 limit (isLimit lim) b (ta a x t ) o id1 A b |
e6be03d94284
Representational Functor preserve limit done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
613
diff
changeset
|
175 ≈⟨ idR ⟩ |
e6be03d94284
Representational Functor preserve limit done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
613
diff
changeset
|
176 limit (isLimit lim) b (ta a x t ) |
e6be03d94284
Representational Functor preserve limit done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
613
diff
changeset
|
177 ≈⟨ limit-uniqueness (isLimit lim) ( λ {i} → ≡-≈ ( cong ( λ g → g x )( t0f=t {i} ))) ⟩ |
e6be03d94284
Representational Functor preserve limit done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
613
diff
changeset
|
178 f x |
e6be03d94284
Representational Functor preserve limit done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
613
diff
changeset
|
179 ∎ )) |
610 | 180 |
609 | 181 |
182 UpreserveLimit : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (I : Category c₁ c₂ ℓ) | |
612 | 183 (b : Obj A ) → LimitPreserve A I Sets (HomA A b) |
184 UpreserveLimit A I b = record { | |
185 preserve = λ Γ lim → UpreserveLimit0 A I b Γ lim | |
610 | 186 } |
609 | 187 |
188 -- K{*}↓U has preinitial full subcategory then U is representable | |
189 -- K{*}↓U is complete, so it has initial object | |
608 | 190 |
191 UisRepresentable : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) | |
192 (comp : Complete A A) | |
609 | 193 (U : Functor A (Sets {c₂}) ) |
194 (a : Obj Sets ) | |
195 (x : Obj ( K (Sets) A a ↓ U) ) | |
196 ( init : HasInitialObject {c₁} {c₂} {ℓ} ( K (Sets) A a ↓ U ) x ) | |
197 → Representable A U (obj x) | |
198 UisRepresentable A comp U a x init = record { | |
199 repr→ = record { TMap = {!!} ; isNTrans = record { commute = {!!} } } | |
200 ; repr← = record { TMap = {!!} ; isNTrans = record { commute = {!!} } } | |
201 ; reprId→ = λ {y} → ? | |
202 ; reprId← = λ {y} → ? | |
203 } | |
608 | 204 |
205 -- K{*}↓U has preinitial full subcategory if U is representable | |
609 | 206 -- if U is representable, K{*}↓U has initial Object ( so it has preinitial full subcategory ) |
608 | 207 |
609 | 208 KUhasInitialObj : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) |
608 | 209 (comp : Complete A A) |
210 (U : Functor A (Sets) ) | |
211 (a : Obj A ) | |
212 (R : Representable A U a ) → | |
609 | 213 HasInitialObject {c₁} {c₂} {ℓ} ( K (Sets) A (FObj U a) ↓ U ) ( record { obj = a ; hom = id1 Sets (FObj U a) } ) |
214 KUhasInitialObj A comp U a R = record { | |
215 initial = λ b → {!!} | |
216 ; uniqueness = λ b f → {!!} | |
217 } | |
608 | 218 |