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annotate freyd.agda @ 435:9f014f34b988

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f=g if equalizer k has right inverse
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 28 Aug 2016 18:59:40 +0900
parents c483374f542b
children ef37decef1ca
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304
bd7b3f3d1d4c Freyd Adjoint Functor Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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1 open import Category -- https://github.com/konn/category-agda
bd7b3f3d1d4c Freyd Adjoint Functor Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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2 open import Level
bd7b3f3d1d4c Freyd Adjoint Functor Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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3
bd7b3f3d1d4c Freyd Adjoint Functor Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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4 module freyd {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ)
bd7b3f3d1d4c Freyd Adjoint Functor Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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5 where
bd7b3f3d1d4c Freyd Adjoint Functor Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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6
307
9872bddec072 small full subcategory done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 306
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7 open import cat-utility
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702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
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8 open import HomReasoning
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bd7b3f3d1d4c Freyd Adjoint Functor Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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9 open import Relation.Binary.Core
307
9872bddec072 small full subcategory done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 306
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10 open Functor
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bd7b3f3d1d4c Freyd Adjoint Functor Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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11
311
cf278f4d0b32 on going...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 310
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12 -- C is small full subcategory of A ( C is image of F )
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bd7b3f3d1d4c Freyd Adjoint Functor Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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13
307
9872bddec072 small full subcategory done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 306
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14 record SmallFullSubcategory {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ)
9872bddec072 small full subcategory done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 306
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15 (F : Functor A A ) ( FMap← : { a b : Obj A } → Hom A (FObj F a) (FObj F b ) → Hom A a b )
9872bddec072 small full subcategory done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 306
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16 : Set (suc ℓ ⊔ (suc c₁ ⊔ suc c₂)) where
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92475fe5f59e Small Full Subcategory (underconstruction)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 305
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17 field
307
9872bddec072 small full subcategory done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 306
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18 ≈→≡ : {a b : Obj A } → { x y : Hom A (FObj F a) (FObj F b) } →
9872bddec072 small full subcategory done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 306
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19 (x≈y : A [ FMap F x ≈ FMap F y ]) → FMap F x ≡ FMap F y -- co-comain of FMap is local small
9872bddec072 small full subcategory done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 306
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20 full→ : { a b : Obj A } { x : Hom A (FObj F a) (FObj F b) } → A [ FMap F ( FMap← x ) ≈ x ]
9872bddec072 small full subcategory done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 306
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21 full← : { a b : Obj A } { x : Hom A a b } → A [ FMap← ( FMap F x ) ≈ x ]
305
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 304
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22
309
e213595b845e preinitial problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
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23 -- pre-initial
e213595b845e preinitial problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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24
311
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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25 record PreInitial {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 310
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26 (F : Functor A A ) : Set (suc ℓ ⊔ (suc c₁ ⊔ suc c₂)) where
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7f00cd09274c pre-initial
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 307
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27 field
314
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 313
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28 preinitialObj : ∀{ a : Obj A } → Obj A
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 313
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29 preinitialArrow : ∀{ a : Obj A } → Hom A ( FObj F (preinitialObj {a} )) a
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e213595b845e preinitial problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
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30
e213595b845e preinitial problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
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31 -- initial object
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7f00cd09274c pre-initial
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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32
309
e213595b845e preinitial problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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33 record HasInitialObject {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (i : Obj A) : Set (suc ℓ ⊔ (suc c₁ ⊔ suc c₂)) where
e213595b845e preinitial problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
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34 field
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 313
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35 initial : ∀( a : Obj A ) → Hom A i a
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e213595b845e preinitial problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
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36 uniqueness : ( a : Obj A ) → ( f : Hom A i a ) → A [ f ≈ initial a ]
e213595b845e preinitial problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
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37
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0d7fa6fc5979 System T and System F
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 314
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38 -- A complete catagory has initial object if it has PreInitial-SmallFullSubcategory
309
e213595b845e preinitial problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
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39
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702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
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40 open NTrans
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
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41 open Limit
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 312
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42 open SmallFullSubcategory
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 312
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43 open PreInitial
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702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
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44
309
e213595b845e preinitial problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
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45 initialFromPreInitialFullSubcategory : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ)
e213595b845e preinitial problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
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46 (F : Functor A A ) ( FMap← : { a b : Obj A } → Hom A (FObj F a) (FObj F b ) → Hom A a b )
312
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
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47 (lim : ( Γ : Functor A A ) → { a0 : Obj A } { u : NTrans A A ( K A A a0 ) Γ } → Limit A A Γ a0 u ) -- completeness
435
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
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48 ( equ : {a b c : Obj A} → (f g : Hom A a b) → {e : Hom A c a } → Equalizer A e f g )
309
e213595b845e preinitial problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
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49 (SFS : SmallFullSubcategory A F FMap← ) →
313
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 312
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50 (PI : PreInitial A F ) → { a0 : Obj A } → { u : (a : Obj A) → NTrans A A (K A A a) F }
312
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
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51 → HasInitialObject A a0
435
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
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52 initialFromPreInitialFullSubcategory A F FMap← lim equ SFS PI {a0} {u} = record {
314
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 313
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53 initial = initialArrow ;
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702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
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54 uniqueness = λ a f → lemma1 a f
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
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55 } where
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 313
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56 initialArrow : ∀( a : Obj A ) → Hom A a0 a
d1af69c4aaf8 on going...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 313
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57 initialArrow a = A [ preinitialArrow PI {a} o limit (lim F {FObj F (preinitialObj PI)} {u (FObj F (preinitialObj PI))} ) a0 (u a0 ) ]
d1af69c4aaf8 on going...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 313
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58 lemma1 : (a : Obj A) (f : Hom A a0 a) → A [ f ≈ initialArrow a ]
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702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
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59 lemma1 a f = let open ≈-Reasoning (A) in
350
c483374f542b try equalizer from limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 315
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60 sym (
312
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
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61 begin
350
c483374f542b try equalizer from limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 315
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62 initialArrow a
c483374f542b try equalizer from limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 315
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63 ≈⟨⟩
c483374f542b try equalizer from limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 315
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64 preinitialArrow PI {a} o limit (lim F) a0 (u a0)
313
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 312
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65 ≈⟨ {!!} ⟩
350
c483374f542b try equalizer from limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 315
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66 f
c483374f542b try equalizer from limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 315
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67 ∎ )
435
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
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68 -- if equalizer has right inverse, f = g
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
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69 lemma2 : (a b c : Obj A) ( f g : Hom A a b ) (e : Hom A c a ) (e' : Hom A a c ) ( equ : Equalizer A e f g ) (inv-e : A [ A [ e o e' ] ≈ id1 A a ] )
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
70 -> A [ f ≈ g ]
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
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71 lemma2 a b c f g e e' equ inv-e = let open ≈-Reasoning (A) in
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
72 let open Equalizer in
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
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73 begin
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
74 f
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
75 ≈↑⟨ idR ⟩
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
76 f o id1 A a
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
77 ≈↑⟨ cdr inv-e ⟩
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
78 f o ( e o e' )
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
79 ≈⟨ assoc ⟩
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
80 ( f o e ) o e'
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
81 ≈⟨ car ( fe=ge equ ) ⟩
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
82 ( g o e ) o e'
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
83 ≈↑⟨ assoc ⟩
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
84 g o ( e o e' )
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
85 ≈⟨ cdr inv-e ⟩
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
86 g o id1 A a
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
87 ≈⟨ idR ⟩
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
88 g
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
89 ∎
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
90