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annotate freyd.agda @ 481:65e6906782bb

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Completeness of Comma Category begin
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Fri, 10 Mar 2017 23:57:49 +0900
parents 961c236807f1
children fcae3025d900
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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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 : 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
16 field
442
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
17 SFSF : Functor A A
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
18 SFSFMap← : { a b : Obj A } → Hom A (FObj SFSF a) (FObj SFSF b ) → Hom A a b
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
19 full→ : { a b : Obj A } { x : Hom A (FObj SFSF a) (FObj SFSF b) } → A [ FMap SFSF ( SFSFMap← x ) ≈ x ]
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
20
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
21 -- ≈→≡ : {a b : Obj A } → { x y : Hom A (FObj SFSF a) (FObj SFSF b) } →
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
22 -- (x≈y : A [ FMap SFSF x ≈ FMap SFSF y ]) → FMap SFSF x ≡ FMap SFSF y -- codomain of FMap is local small
305
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 304
diff changeset
23
309
e213595b845e preinitial problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
24 -- pre-initial
e213595b845e preinitial problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
25
311
cf278f4d0b32 on going...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 310
diff changeset
26 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
27 (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
28 field
314
d1af69c4aaf8 on going...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 313
diff changeset
29 preinitialObj : ∀{ a : Obj A } → Obj A
d1af69c4aaf8 on going...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 313
diff changeset
30 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
31
e213595b845e preinitial problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
32 -- initial object
308
7f00cd09274c pre-initial
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 307
diff changeset
33
309
e213595b845e preinitial problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
34 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
35 field
314
d1af69c4aaf8 on going...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 313
diff changeset
36 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
37 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
38
315
0d7fa6fc5979 System T and System F
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 314
diff changeset
39 -- 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
40
312
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
diff changeset
41 open NTrans
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
diff changeset
42 open Limit
313
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 312
diff changeset
43 open SmallFullSubcategory
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 312
diff changeset
44 open PreInitial
440
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
45 open Complete
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
46 open Equalizer
443
f526f4b68565 fix IsEqualizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 442
diff changeset
47 open IsEqualizer
312
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
diff changeset
48
309
e213595b845e preinitial problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
49 initialFromPreInitialFullSubcategory : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ)
440
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
50 (comp : Complete A A)
442
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
51 (SFS : SmallFullSubcategory A ) →
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
52 (PI : PreInitial A (SFSF SFS )) → { a0 : Obj A } → HasInitialObject A (limit-c comp (SFSF SFS))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
53 initialFromPreInitialFullSubcategory A comp SFS PI = record {
314
d1af69c4aaf8 on going...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 313
diff changeset
54 initial = initialArrow ;
312
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
diff changeset
55 uniqueness = λ a f → lemma1 a f
702adc45704f is this right direction?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
diff changeset
56 } where
442
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
57 F : Functor A A
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
58 F = SFSF SFS
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
59 FMap← : { a b : Obj A } → Hom A (FObj F a) (FObj F b ) → Hom A a b
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
60 FMap← = SFSFMap← SFS
459
9d24fb809746 freyd trbouled again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
61 a0 : Obj A
9d24fb809746 freyd trbouled again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
62 a0 = limit-c comp F
460
961c236807f1 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 459
diff changeset
63 lim : ( Γ : Functor A A ) → Limit A A Γ (limit-c comp Γ) (limit-u comp Γ)
961c236807f1 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 459
diff changeset
64 lim Γ = isLimit comp Γ
459
9d24fb809746 freyd trbouled again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
65 u : NTrans A A (K A A a0) F
9d24fb809746 freyd trbouled again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
66 u = T0 ( lim F )
443
f526f4b68565 fix IsEqualizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 442
diff changeset
67 equ : {a b : Obj A} → (f g : Hom A a b) → IsEqualizer A (equalizer-e comp f g ) f g
f526f4b68565 fix IsEqualizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 442
diff changeset
68 equ f g = Complete.isEqualizer comp f g
442
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
69 ep : {a b : Obj A} → {f g : Hom A a b} → Obj A
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
70 ep {a} {b} {f} {g} = equalizer-p comp f g
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
71 ee : {a b : Obj A} → {f g : Hom A a b} → Hom A (ep {a} {b} {f} {g} ) a
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
72 ee {a} {b} {f} {g} = equalizer-e comp f g
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
73 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
74 f {a} = TMap u (preinitialObj PI {a} )
314
d1af69c4aaf8 on going...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 313
diff changeset
75 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
76 initialArrow a = A [ preinitialArrow PI {a} o f ]
442
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
77 equ-fi : { a : Obj A} → {f' : Hom A a0 a} →
443
f526f4b68565 fix IsEqualizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 442
diff changeset
78 IsEqualizer A ee ( A [ preinitialArrow PI {a} o f ] ) f'
440
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
79 equ-fi {a} {f'} = equ ( A [ preinitialArrow PI {a} o f ] ) f'
442
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
80 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
81 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
82 begin
9be298a02c35 add rest of equation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 436
diff changeset
83 e
442
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
84 ≈↑⟨ limit-uniqueness (lim F) e ( λ {i} → uce=uc ) ⟩
440
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
85 limit (lim F) a0 u
442
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
86 ≈⟨ limit-uniqueness (lim F) (id1 A a0) ( λ {i} → idR ) ⟩
437
9be298a02c35 add rest of equation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 436
diff changeset
87 id1 A a0
9be298a02c35 add rest of equation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 436
diff changeset
88 ∎
442
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
89 kfuc=uc : {c k1 : Obj A} → {p : Hom A k1 a0} → A [ A [ TMap u c o
440
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
90 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
91 ≈ TMap u c ]
441
61550782be4a preinital full subcategory done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
92 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
93 begin
61550782be4a preinital full subcategory done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
94 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
95 ≈⟨ cdr assoc ⟩
61550782be4a preinital full subcategory done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
96 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
97 ≈⟨ assoc ⟩
61550782be4a preinital full subcategory done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
98 (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
99 ≈↑⟨ car ( full→ SFS ) ⟩
61550782be4a preinital full subcategory done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
100 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
101 ≈⟨ nat u ⟩
442
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
102 TMap u c o FMap (K A A a0) (FMap← (TMap u c o p o preinitialArrow PI))
441
61550782be4a preinital full subcategory done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
103 ≈⟨⟩
442
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
104 TMap u c o id1 A a0
441
61550782be4a preinital full subcategory done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
105 ≈⟨ idR ⟩
61550782be4a preinital full subcategory done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
106 TMap u c
61550782be4a preinital full subcategory done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
107 ∎
442
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
108 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
109 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
110 -- if equalizer has right inverse, f = g
438
ce9edc8088e8 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 437
diff changeset
111 lemma2 : (a b : Obj A) {c : Obj A} ( f g : Hom A a b )
443
f526f4b68565 fix IsEqualizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 442
diff changeset
112 {e : Hom A c a } {e' : Hom A a c } ( equ : IsEqualizer A e f g ) (inv-e : A [ A [ e o e' ] ≈ id1 A a ] )
442
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
113 → A [ f ≈ g ]
438
ce9edc8088e8 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 437
diff changeset
114 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
115 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
116 begin
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
117 f
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
118 ≈↑⟨ idR ⟩
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
119 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
120 ≈↑⟨ cdr inv-e ⟩
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
121 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
122 ≈⟨ assoc ⟩
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
123 ( f o e ) o e'
441
61550782be4a preinital full subcategory done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 440
diff changeset
124 ≈⟨ 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
125 ≈↑⟨ assoc ⟩
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
126 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
127 ≈⟨ cdr inv-e ⟩
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
128 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
129 ≈⟨ idR ⟩
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
130 g
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
131 ∎
439
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 438
diff changeset
132 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
133 lemma1 a f' = let open ≈-Reasoning (A) in
436
ef37decef1ca initialFullSubCategory
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 435
diff changeset
134 sym (
ef37decef1ca initialFullSubCategory
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 435
diff changeset
135 begin
ef37decef1ca initialFullSubCategory
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 435
diff changeset
136 initialArrow a
ef37decef1ca initialFullSubCategory
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 435
diff changeset
137 ≈⟨⟩
440
ff36c500962e fix limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 439
diff changeset
138 preinitialArrow PI {a} o f
442
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
139 ≈⟨ lemma2 a0 a (A [ preinitialArrow PI {a} o f ]) f' {ee {a0} {a} {A [ preinitialArrow PI {a} o f ]} {f'} } (equ-fi )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 441
diff changeset
140 (kfuc=1 {ep} {ee} ) ⟩
438
ce9edc8088e8 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 437
diff changeset
141 f'
436
ef37decef1ca initialFullSubCategory
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 435
diff changeset
142 ∎ )
435
9f014f34b988 f=g if equalizer k has right inverse
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 350
diff changeset
143
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
144
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
145