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UpreseveLimit detailing
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Wed, 15 Mar 2017 12:10:24 +0900
parents e8b85a05a6b2
children 511fd37d90ec
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497
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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1 open import Category -- https://github.com/konn/category-agda
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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2 open import Level
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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3 open import Category.Sets
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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4
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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5 module freyd2
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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6 where
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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7
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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8 open import HomReasoning
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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9 open import cat-utility
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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10 open import Relation.Binary.Core
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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11 open import Function
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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12
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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13 ----------
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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14 --
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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15 -- a : Obj A
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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16 -- U : A → Sets
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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17 -- U ⋍ Hom (a,-)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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18 --
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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19
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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20 -- A is Locally small
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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21 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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22
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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23 import Relation.Binary.PropositionalEquality
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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24 -- 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>
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25 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>
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26
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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27
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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28 ----
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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29 --
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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30 -- Hom ( a, - ) is Object mapping in co Yoneda Functor
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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31 --
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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32 ----
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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>
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34 open NTrans
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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35 open Functor
498
c981a2f0642f UpreseveLimit detailing
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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36 open Limit
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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37 open IsLimit
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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38 open import Category.Cat
497
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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39
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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40 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>
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41 HomA {c₁} {c₂} {ℓ} A a = record {
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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42 FObj = λ b → Hom A a b
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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43 ; 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>
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44 ; isFunctor = record {
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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45 identity = λ {b} → extensionality A ( λ x → lemma-y-obj1 {b} x ) ;
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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46 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>
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47 ≈-cong = λ eq → extensionality A ( λ x → lemma-y-obj3 x eq )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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48 }
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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49 } where
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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50 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>
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51 lemma-y-obj1 {b} x = let open ≈-Reasoning A in ≈-≡ {_} {_} {_} {A} idL
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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52 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>
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53 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>
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54 lemma-y-obj2 a₁ b c f g x = let open ≈-Reasoning A in ≈-≡ {_} {_} {_} {A} ( begin
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55 A [ A [ g o f ] o x ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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56 ≈↑⟨ assoc ⟩
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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57 A [ g o A [ f o x ] ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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58 ≈⟨⟩
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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59 ( 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>
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60 ∎ )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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61 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>
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62 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>
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63 A [ f o x ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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64 ≈⟨ resp refl-hom eq ⟩
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65 A [ g o x ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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66 ∎ )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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67
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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68
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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69
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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70 record Representable { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) ( U : Functor A (Sets {c₂}) ) (b : Obj A) : Set (suc ℓ ⊔ (suc (suc c₂) ⊔ suc c₁ )) where
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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71 field
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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72 -- FObj U x : A → Set
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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73 -- FMap U f = Set → Set
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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74 -- λ b → Hom (a,b) : A → Set
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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75 -- λ f → Hom (a,-) = λ b → Hom a b
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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76
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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77 repr→ : NTrans A (Sets {c₂}) U (HomA A b )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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78 repr← : NTrans A (Sets {c₂}) (HomA A b) U
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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79 representable→ : {x : Obj A} → Sets [ Sets [ TMap repr→ x o TMap repr← x ] ≈ id1 (Sets {c₂}) (FObj (HomA A b) x )]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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80 representable← : {x : Obj A} → Sets [ Sets [ TMap repr← x o TMap repr→ x ] ≈ id1 (Sets {c₂}) (FObj U x)]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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81
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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82 UpreseveLimit : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → ( U : Functor A (Sets {c₂}) ) (b : Obj A)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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83 { c₁' c₂' ℓ' : Level} ( I : Category c₁' c₂' ℓ' )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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84 ( rep : Representable A U b ) → LimitPreserve A I (Sets {c₂}) U
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85 UpreseveLimit {_} { c₂} A U b I rep = record {
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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86 preserve = λ Γ limita → record {
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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87 limit = λ a t → limitu a Γ t limita
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88 ; t0f=t = λ { a t i } → t0f=tu {a} Γ t limita {i}
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89 ; limit-uniqueness = λ { a t f } t=f → limit-uniquenessu {a} Γ limita t f t=f
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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90 }
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 497
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91 } where
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 497
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92 limitu : ( a : Obj (Sets {c₂}) ) → (Γ : Functor I A ) ( t : NTrans I Sets ( K Sets I a ) (U ○ Γ) ) → ( limita : Limit A I Γ ) → Hom Sets a (FObj U (a0 limita))
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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93 limitu = {!!}
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 497
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94 t0f=tu : { a : Obj (Sets {c₂}) } → (Γ : Functor I A ) ( t : NTrans I Sets ( K Sets I a ) (U ○ Γ) ) → ( limita : Limit A I Γ ) →
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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95 ∀ { i : Obj I } → Sets [ Sets [ TMap (LimitNat A I Sets Γ (a0 limita) (t0 limita) U) i o limitu a Γ t limita ] ≈ TMap t i ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 497
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96 t0f=tu = {!!}
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 497
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97 limit-uniquenessu : { a : Obj (Sets {c₂}) } → (Γ : Functor I A )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 497
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98 → ( limita : Limit A I Γ ) →
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99 ( t : NTrans I Sets ( K Sets I a ) (U ○ Γ) ) → ( f : Hom Sets a (FObj U (a0 limita)) )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 497
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100 → ( ∀ { i : Obj I } → (Sets [ TMap (LimitNat A I Sets Γ (a0 limita) (t0 limita) U) i o f ] ≡ TMap t i ) )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 497
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101 → Sets [ limitu a Γ t limita ≈ f ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 497
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102 limit-uniquenessu = {!!}
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 497
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103