Mercurial > hg > Members > kono > Proof > category

annotate freyd.agda @ 441:61550782be4a

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
preinital full subcategory done
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Tue, 30 Aug 2016 15:11:17 +0900
parents ff36c500962e
children 87600d338337
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
304
bd7b3f3d1d4c Freyd Adjoint Functor Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
1 open import Category -- https://github.com/konn/category-agda
bd7b3f3d1d4c Freyd Adjoint Functor Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
2 open import Level
bd7b3f3d1d4c Freyd Adjoint Functor Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
3
bd7b3f3d1d4c Freyd Adjoint Functor Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
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:
diff changeset
5 where
bd7b3f3d1d4c Freyd Adjoint Functor Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
6
307
9872bddec072 small full subcategory done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 306
diff changeset
7 open import cat-utility
312
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
diff changeset
8 open import HomReasoning
304
bd7b3f3d1d4c Freyd Adjoint Functor Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
9 open import Relation.Binary.Core
307
9872bddec072 small full subcategory done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 306
diff changeset
10 open Functor
304
bd7b3f3d1d4c Freyd Adjoint Functor Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
11
311
cf278f4d0b32 on going...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 310
diff changeset
12 -- C is small full subcategory of A ( C is image of F )
304
bd7b3f3d1d4c Freyd Adjoint Functor Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
13
307
9872bddec072 small full subcategory done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 306
diff changeset
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
diff changeset
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
diff changeset
16 : Set (suc ℓ ⊔ (suc c₁ ⊔ suc c₂)) where
306
92475fe5f59e Small Full Subcategory (underconstruction)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 305
diff changeset
17 field
307
9872bddec072 small full subcategory done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 306
diff changeset
18 ≈→≡ : {a b : Obj A } → { x y : Hom A (FObj F a) (FObj F b) } →
436
ef37decef1ca initialFullSubCategory
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 435
diff changeset
19 (x≈y : A [ FMap F x ≈ FMap F y ]) → FMap F x ≡ FMap F y -- codomain of FMap is local small
307
9872bddec072 small full subcategory done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 306
diff changeset
20 full→ : { a b : Obj A } { x : Hom A (FObj F a) (FObj F b) } → A [ FMap F ( FMap← x ) ≈ x ]
305
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 304
diff changeset
21
309
e213595b845e preinitial problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
22 -- pre-initial
e213595b845e preinitial problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
23
311
cf278f4d0b32 on going...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 310
diff changeset
24 record PreInitial {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ)
cf278f4d0b32 on going...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 310
diff changeset
25 (F : Functor A A ) : Set (suc ℓ ⊔ (suc c₁ ⊔ suc c₂)) where
308
7f00cd09274c pre-initial
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 307
diff changeset
26 field
314
d1af69c4aaf8 on going...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 313
diff changeset
27 preinitialObj : ∀{ a : Obj A } → Obj A
d1af69c4aaf8 on going...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 313
diff changeset
28 preinitialArrow : ∀{ a : Obj A } → Hom A ( FObj F (preinitialObj {a} )) a
309
e213595b845e preinitial problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
29
e213595b845e preinitial problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
30 -- initial object
308
7f00cd09274c pre-initial
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 307
diff changeset
31
309
e213595b845e preinitial problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
32 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
diff changeset
33 field
314
d1af69c4aaf8 on going...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 313
diff changeset
34 initial : ∀( a : Obj A ) → Hom A i a
309
e213595b845e preinitial problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
35 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
diff changeset
36
315
0d7fa6fc5979 System T and System F
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 314
diff changeset
37 -- 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
diff changeset
38
312
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
diff changeset
39 open NTrans
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
diff changeset
40 open Limit
313
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 312
diff changeset
41 open SmallFullSubcategory
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 312
diff changeset
42 open PreInitial
440
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
43 open Complete
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
44 open Equalizer
312
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
diff changeset
45
309
e213595b845e preinitial problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
46 initialFromPreInitialFullSubcategory : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ)
e213595b845e preinitial problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
47 (F : Functor A A ) ( FMap← : { a b : Obj A } → Hom A (FObj F a) (FObj F b ) → Hom A a b )
440
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
48 (comp : Complete A A)
309
e213595b845e preinitial problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
49 (SFS : SmallFullSubcategory A F FMap← ) →
440
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
50 (PI : PreInitial A F ) → { a0 : Obj A } → HasInitialObject A (limit-c comp F)
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
51 initialFromPreInitialFullSubcategory A F FMap← comp SFS PI = record {
314
d1af69c4aaf8 on going...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 313
diff changeset
52 initial = initialArrow ;
312
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
diff changeset
53 uniqueness = λ a f → lemma1 a f
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
diff changeset
54 } where
440
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
55 lim : ( Γ : Functor A A ) → Limit A A Γ (limit-c comp Γ) (limit-u comp Γ)
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
56 lim Γ = isLimit comp Γ
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
57 equ : {a b : Obj A} → (f g : Hom A a b) → Equalizer A (equalizer-e comp f g ) f g
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
58 equ f g = isEqualizer comp f g
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
59 a0 = limit-c comp F
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
60 u = limit-u comp F
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
61 ee : {a b : Obj A} → {f g : Hom A a b} → Obj A
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
62 ee {a} {b} {f} {g} = equalizer-p comp f g
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
63 ep : {a b : Obj A} → {f g : Hom A a b} → Hom A (ee {a} {b} {f} {g} ) a
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
64 ep {a} {b} {f} {g} = equalizer-e comp f g
437
9be298a02c35 add rest of equation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 436
diff changeset
65 f : {a : Obj A} -> Hom A a0 (FObj F (preinitialObj PI {a} ) )
440
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
66 f {a} = TMap u (preinitialObj PI {a} )
314
d1af69c4aaf8 on going...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 313
diff changeset
67 initialArrow : ∀( a : Obj A ) → Hom A a0 a
437
9be298a02c35 add rest of equation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 436
diff changeset
68 initialArrow a = A [ preinitialArrow PI {a} o f ]
440
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
69 equ-fi : { a : Obj A} -> {f' : Hom A a0 a} ->
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
70 Equalizer A ep ( A [ preinitialArrow PI {a} o f ] ) f'
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
71 equ-fi {a} {f'} = equ ( A [ preinitialArrow PI {a} o f ] ) f'
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
72 e=id : {e : Hom A a0 a0} -> ( {c : Obj A} -> A [ A [ TMap u c o e ] ≈ TMap u c ] ) -> A [ e ≈ id1 A a0 ]
438
ce9edc8088e8 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 437
diff changeset
73 e=id {e} uce=uc = let open ≈-Reasoning (A) in
437
9be298a02c35 add rest of equation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 436
diff changeset
74 begin
9be298a02c35 add rest of equation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 436
diff changeset
75 e
440
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
76 ≈↑⟨ limit-uniqueness (lim F) e ( \{i} -> uce=uc ) ⟩
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
77 limit (lim F) a0 u
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
78 ≈⟨ limit-uniqueness (lim F) (id1 A a0) ( \{i} -> idR ) ⟩
437
9be298a02c35 add rest of equation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 436
diff changeset
79 id1 A a0
9be298a02c35 add rest of equation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 436
diff changeset
80 ∎
440
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
81 kfuc=uc : {c k1 : Obj A} -> {p : Hom A k1 a0} -> A [ A [ TMap u c o
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
82 A [ p o A [ preinitialArrow PI {k1} o TMap u (preinitialObj PI) ] ] ]
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
83 ≈ TMap u c ]
441
61550782be4a preinital full subcategory done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
84 kfuc=uc {c} {k1} {p} = let open ≈-Reasoning (A) in
61550782be4a preinital full subcategory done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
85 begin
61550782be4a preinital full subcategory done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
86 TMap u c o ( p o ( preinitialArrow PI {k1} o TMap u (preinitialObj PI) ))
61550782be4a preinital full subcategory done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
87 ≈⟨ cdr assoc ⟩
61550782be4a preinital full subcategory done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
88 TMap u c o ((p o preinitialArrow PI) o TMap u (preinitialObj PI))
61550782be4a preinital full subcategory done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
89 ≈⟨ assoc ⟩
61550782be4a preinital full subcategory done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
90 (TMap u c o ( p o ( preinitialArrow PI {k1} ))) o TMap u (preinitialObj PI)
61550782be4a preinital full subcategory done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
91 ≈↑⟨ car ( full→ SFS ) ⟩
61550782be4a preinital full subcategory done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
92 FMap F (FMap← (TMap u c o p o preinitialArrow PI)) o TMap u (preinitialObj PI)
61550782be4a preinital full subcategory done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
93 ≈⟨ nat u ⟩
61550782be4a preinital full subcategory done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
94 TMap u c o FMap (K A A (limit-c comp F)) (FMap← (TMap u c o p o preinitialArrow PI))
61550782be4a preinital full subcategory done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
95 ≈⟨⟩
61550782be4a preinital full subcategory done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
96 TMap u c o id1 A (limit-c comp F)
61550782be4a preinital full subcategory done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
97 ≈⟨ idR ⟩
61550782be4a preinital full subcategory done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
98 TMap u c
61550782be4a preinital full subcategory done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
99 ∎
440
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
100 kfuc=1 : {k1 : Obj A} -> {p : Hom A k1 a0} -> A [ A [ p o A [ preinitialArrow PI {k1} o TMap u (preinitialObj PI) ] ] ≈ id1 A a0 ]
439
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 438
diff changeset
101 kfuc=1 {k1} {p} = e=id ( kfuc=uc )
435
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
102 -- if equalizer has right inverse, f = g
438
ce9edc8088e8 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 437
diff changeset
103 lemma2 : (a b : Obj A) {c : Obj A} ( f g : Hom A a b )
ce9edc8088e8 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 437
diff changeset
104 {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 ] )
435
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
105 -> A [ f ≈ g ]
438
ce9edc8088e8 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 437
diff changeset
106 lemma2 a b {c} f g {e} {e'} equ inv-e = let open ≈-Reasoning (A) in
435
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
107 let open Equalizer in
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
108 begin
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
109 f
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
110 ≈↑⟨ idR ⟩
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
111 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
112 ≈↑⟨ cdr inv-e ⟩
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
113 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
114 ≈⟨ assoc ⟩
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
115 ( f o e ) o e'
441
61550782be4a preinital full subcategory done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
116 ≈⟨ car ( fe=ge equ ) ⟩ ( g o e ) o e'
435
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
117 ≈↑⟨ assoc ⟩
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
118 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
119 ≈⟨ cdr inv-e ⟩
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
120 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
121 ≈⟨ idR ⟩
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
122 g
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
123 ∎
439
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 438
diff changeset
124 lemma1 : (a : Obj A) (f' : Hom A a0 a) → A [ f' ≈ initialArrow a ]
438
ce9edc8088e8 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 437
diff changeset
125 lemma1 a f' = let open ≈-Reasoning (A) in
436
ef37decef1ca initialFullSubCategory
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 435
diff changeset
126 sym (
ef37decef1ca initialFullSubCategory
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 435
diff changeset
127 begin
ef37decef1ca initialFullSubCategory
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 435
diff changeset
128 initialArrow a
ef37decef1ca initialFullSubCategory
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 435
diff changeset
129 ≈⟨⟩
440
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
130 preinitialArrow PI {a} o f
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
131 ≈⟨ lemma2 a0 a (A [ preinitialArrow PI {a} o f ]) f' {ep {a0} {a} {A [ preinitialArrow PI {a} o f ]} {f'} } (equ-fi )
439
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 438
diff changeset
132 (kfuc=1 {ee} {ep} ) ⟩
438
ce9edc8088e8 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 437
diff changeset
133 f'
436
ef37decef1ca initialFullSubCategory
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 435
diff changeset
134 ∎ )
435
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
135