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add if U is iso to representable functor then preserve limit
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Wed, 15 Mar 2017 11:19:54 +0900
parents
children c981a2f0642f
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497
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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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
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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36
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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37 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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38 HomA {c₁} {c₂} {ℓ} A a = record {
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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39 FObj = λ b → Hom A a b
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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40 ; 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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41 ; isFunctor = record {
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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42 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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43 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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44 ≈-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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45 }
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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46 } where
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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47 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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48 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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49 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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50 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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51 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>
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52 A [ A [ g o f ] o x ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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53 ≈↑⟨ assoc ⟩
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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54 A [ g o A [ f o x ] ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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55 ≈⟨⟩
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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56 ( 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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57 ∎ )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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58 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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59 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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60 A [ f o x ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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61 ≈⟨ resp refl-hom eq ⟩
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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62 A [ g o x ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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63 ∎ )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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64
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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65
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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 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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68 field
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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69 -- FObj U x : A → Set
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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70 -- FMap U f = Set → Set
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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71 -- λ b → Hom (a,b) : A → Set
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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72 -- 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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73
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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74 repr→ : NTrans A (Sets {c₂}) U (HomA A b )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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75 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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76 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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77 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>
parents:
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78
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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79 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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80 { 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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81 ( rep : Representable A U b ) → LimitPreserve A I (Sets {c₂}) U
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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82 UpreseveLimit = {!!}