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annotate freyd2.agda @ 609:d686d7ae38e0

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on goging
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 12 Jun 2017 16:35:34 +0900
parents 7194ba55df56
children 3fb4d834c349
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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497
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
1 open import Category -- https://github.com/konn/category-agda
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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2 open import Level
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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3 open import Category.Sets
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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4
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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5 module freyd2
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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6 where
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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7
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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8 open import HomReasoning
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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9 open import cat-utility
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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10 open import Relation.Binary.Core
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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11 open import Function
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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12
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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13 ----------
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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14 --
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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15 -- a : Obj A
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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16 -- U : A → Sets
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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17 -- U ⋍ Hom (a,-)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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18 --
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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19
608
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
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20 ----
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
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21 -- C is locally small i.e. Hom C i j is a set c₁
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
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22 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
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23 record Small { c₁ c₂ ℓ : Level} ( C : Category c₁ c₂ ℓ ) ( I : Set c₁ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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24 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ )) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
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25 field
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
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26 hom→ : {i j : Obj C } → Hom C i j → I
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
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27 hom← : {i j : Obj C } → ( f : I ) → Hom C i j
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
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28 hom-iso : {i j : Obj C } → { f : Hom C i j } → hom← ( hom→ f ) ≡ f
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
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29
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
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30 open Small
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
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31
497
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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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
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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33
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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34 import Relation.Binary.PropositionalEquality
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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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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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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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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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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37
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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38
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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39 ----
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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40 --
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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41 -- Hom ( a, - ) is Object mapping in co Yoneda Functor
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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42 --
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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43 ----
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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44
e8b85a05a6b2 add if U is iso to representable functor then preserve limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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45 open NTrans
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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46 open Functor
498
c981a2f0642f UpreseveLimit detailing
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 497
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47 open Limit
c981a2f0642f UpreseveLimit detailing
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 497
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48 open IsLimit
c981a2f0642f UpreseveLimit detailing
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 497
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49 open import Category.Cat
497
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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50
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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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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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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52 HomA {c₁} {c₂} {ℓ} A a = record {
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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53 FObj = λ b → Hom A a b
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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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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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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55 ; isFunctor = record {
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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56 identity = λ {b} → extensionality A ( λ x → lemma-y-obj1 {b} x ) ;
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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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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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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58 ≈-cong = λ eq → extensionality A ( λ x → lemma-y-obj3 x eq )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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59 }
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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60 } where
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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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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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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62 lemma-y-obj1 {b} x = let open ≈-Reasoning A in ≈-≡ {_} {_} {_} {A} idL
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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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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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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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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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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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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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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66 A [ A [ g o f ] o x ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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67 ≈↑⟨ assoc ⟩
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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68 A [ g o A [ f o x ] ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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69 ≈⟨⟩
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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70 ( λ x → A [ g o x ] ) ( ( λ x → A [ f o x ] ) x )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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71 ∎ )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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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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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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73 lemma-y-obj3 {_} {_} {f} {g} x eq = let open ≈-Reasoning A in ≈-≡ {_} {_} {_} {A} ( begin
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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74 A [ f o x ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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75 ≈⟨ resp refl-hom eq ⟩
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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76 A [ g o x ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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77 ∎ )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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78
609
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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79 -- Representable U ≈　Hom(A,-)
502
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 499
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80
609
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
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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
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 499
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82 field
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 499
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83 -- FObj U x : A → Set
609
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
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84 -- FMap U f = Set → Set (locally small)
502
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 499
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85 -- λ b → Hom (a,b) : A → Set
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 499
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86 -- λ f → Hom (a,-) = λ b → Hom a b
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 499
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87
609
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
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88 repr→ : NTrans A (Sets {c₂}) U (HomA A a )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
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89 repr← : NTrans A (Sets {c₂}) (HomA A a) U
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
90 reprId→ : {x : Obj A} → Sets [ Sets [ TMap repr→ x o TMap repr← x ] ≈ id1 (Sets {c₂}) (FObj (HomA A a) x )]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
91 reprId← : {x : Obj A} → Sets [ Sets [ TMap repr← x o TMap repr→ x ] ≈ id1 (Sets {c₂}) (FObj U x)]
608
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
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92
609
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
93 open Representable
608
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
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94 open import freyd
502
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 499
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95
608
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
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96 _↓_ : { c₁ c₂ ℓ : Level} { c₁' c₂' ℓ' : Level} { A : Category c₁ c₂ ℓ } { B : Category c₁' c₂' ℓ' }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
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97 → ( F G : Functor A B )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
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98 → Category (c₂' ⊔ c₁) (ℓ' ⊔ c₂) ℓ
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
diff changeset
99 _↓_ { c₁} {c₂} {ℓ} {c₁'} {c₂'} {ℓ'} { A } { B } F G = CommaCategory
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
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100 where open import Comma1 F G
498
c981a2f0642f UpreseveLimit detailing
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 497
diff changeset
101
608
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
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102 open import freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
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103 open SmallFullSubcategory
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
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104 open Complete
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
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105 open PreInitial
609
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
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106 open HasInitialObject
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
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107 open import Comma1
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
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108 open CommaObj
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
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109 open LimitPreserve
608
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
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110
609
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
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111 -- Representable Functor U preserve limit , so K{*}↓U is complete
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
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112
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
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113 UpreserveLimit : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (I : Category c₁ c₂ ℓ)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
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114 (comp : Complete A A)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
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115 (U : Functor A (Sets) )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
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116 (a : Obj A )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
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117 (R : Representable A U a ) → LimitPreserve A I Sets U
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
118 UpreserveLimit A I comp U a R = record {
d686d7ae38e0 on goging
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
119 preserve = λ Γ lim → record {
d686d7ae38e0 on goging
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
120 limit = λ a t → {!!}
d686d7ae38e0 on goging
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
121 ; t0f=t = λ {a t i} → ?
d686d7ae38e0 on goging
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
122 ; limit-uniqueness = λ {a} {t} {f} t0f=t → {!!}
d686d7ae38e0 on goging
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
123 }
d686d7ae38e0 on goging
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
124 }
d686d7ae38e0 on goging
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
125
d686d7ae38e0 on goging
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
126 -- K{*}↓U has preinitial full subcategory then U is representable
d686d7ae38e0 on goging
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
127 -- K{*}↓U is complete, so it has initial object
608
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
diff changeset
128
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
diff changeset
129 UisRepresentable : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
diff changeset
130 (comp : Complete A A)
609
d686d7ae38e0 on goging
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
131 (U : Functor A (Sets {c₂}) )
d686d7ae38e0 on goging
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
132 (a : Obj Sets )
d686d7ae38e0 on goging
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
133 (x : Obj ( K (Sets) A a ↓ U) )
d686d7ae38e0 on goging
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
134 ( init : HasInitialObject {c₁} {c₂} {ℓ} ( K (Sets) A a ↓ U ) x )
d686d7ae38e0 on goging
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
135 → Representable A U (obj x)
d686d7ae38e0 on goging
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
136 UisRepresentable A comp U a x init = record {
d686d7ae38e0 on goging
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
137 repr→ = record { TMap = {!!} ; isNTrans = record { commute = {!!} } }
d686d7ae38e0 on goging
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
138 ; repr← = record { TMap = {!!} ; isNTrans = record { commute = {!!} } }
d686d7ae38e0 on goging
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
139 ; reprId→ = λ {y} →　?
d686d7ae38e0 on goging
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
140 ; reprId← = λ {y} →　?
d686d7ae38e0 on goging
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
141 }
608
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
diff changeset
142
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
diff changeset
143 -- K{*}↓U has preinitial full subcategory if U is representable
609
d686d7ae38e0 on goging
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
144 -- if U is representable, K{*}↓U has initial Object ( so it has preinitial full subcategory )
608
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
diff changeset
145
609
d686d7ae38e0 on goging
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
146 KUhasInitialObj : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ)
608
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
diff changeset
147 (comp : Complete A A)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
diff changeset
148 (U : Functor A (Sets) )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
diff changeset
149 (a : Obj A )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
diff changeset
150 (R : Representable A U a ) →
609
d686d7ae38e0 on goging
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
151 HasInitialObject {c₁} {c₂} {ℓ} ( K (Sets) A (FObj U a) ↓ U ) ( record { obj = a ; hom = id1 Sets (FObj U a) } )
d686d7ae38e0 on goging
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
152 KUhasInitialObj A comp U a R = record {
d686d7ae38e0 on goging
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
153 initial = λ b → {!!}
d686d7ae38e0 on goging
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
154 ; uniqueness = λ b f → {!!}
d686d7ae38e0 on goging
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
155 }
608
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 502
diff changeset
156