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annotate freyd1.agda @ 491:04da2c458d44

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comma-a0 commuativity remains
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 13 Mar 2017 10:41:07 +0900
parents 1a42f06e7ae1
children c7b8017bcd4d
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
1 open import Category -- https://github.com/konn/category-agda
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
2 open import Level
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
3
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
4 module freyd1 {c₁ c₂ ℓ c₁' c₂' ℓ' : Level} {A : Category c₁ c₂ ℓ} {C : Category c₁' c₂' ℓ'}
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
5 ( F : Functor A C ) ( G : Functor A C ) where
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
6
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
7 open import cat-utility
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
8 open import HomReasoning
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
9 open import Relation.Binary.Core
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
10 open Functor
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
11
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
12 open import Comma1 F G
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
13 open import freyd CommaCategory
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
14
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
15 open import Category.Cat
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
16
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
17 open NTrans
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
18
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
19
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
20 open Complete
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
21 open CommaObj
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
22 open CommaHom
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
23 open Limit
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 486
diff changeset
24 open IsLimit
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
25
483
265f13adf93b add NIC
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 482
diff changeset
26 -- F : A → C
265f13adf93b add NIC
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 482
diff changeset
27 -- G : A → C
265f13adf93b add NIC
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 482
diff changeset
28 --
265f13adf93b add NIC
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 482
diff changeset
29
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
30 FIA : { I : Category c₁ c₂ ℓ } → ( Γ : Functor I CommaCategory ) → Functor I A
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
31 FIA {I} Γ = record {
482
fd752ad25ac0 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
32 FObj = λ x → obj (FObj Γ x ) ;
fd752ad25ac0 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
33 FMap = λ {a} {b} f → arrow (FMap Γ f ) ;
fd752ad25ac0 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
34 isFunctor = record {
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
35 identity = identity
fd752ad25ac0 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
36 ; distr = IsFunctor.distr (isFunctor Γ)
fd752ad25ac0 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
37 ; ≈-cong = IsFunctor.≈-cong (isFunctor Γ)
fd752ad25ac0 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
38 }} where
fd752ad25ac0 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
39 identity : {x : Obj I } → A [ arrow (FMap Γ (id1 I x)) ≈ id1 A (obj (FObj Γ x)) ]
fd752ad25ac0 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
40 identity {x} = let open ≈-Reasoning (A) in begin
fd752ad25ac0 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
41 arrow (FMap Γ (id1 I x))
fd752ad25ac0 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
42 ≈⟨ IsFunctor.identity (isFunctor Γ) ⟩
fd752ad25ac0 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
43 id1 A (obj (FObj Γ x))
fd752ad25ac0 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
44 ∎
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
45
491
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
46 NIA : { I : Category c₁ c₂ ℓ } → ( Γ : Functor I CommaCategory )
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
47 (c : Obj CommaCategory ) ( ta : NTrans I CommaCategory ( K CommaCategory I c ) Γ ) → NTrans I A ( K A I (obj c) ) (FIA Γ)
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
48 NIA {I} Γ c ta = record {
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
49 TMap = λ x → arrow (TMap ta x )
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
50 ; isNTrans = record { commute = comm1 }
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
51 } where
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
52 comm1 : {a b : Obj I} {f : Hom I a b} →
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
53 A [ A [ FMap (FIA Γ) f o arrow (TMap ta a) ] ≈ A [ arrow (TMap ta b) o FMap (K A I (obj c)) f ] ]
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
54 comm1 {a} {b} {f} = IsNTrans.commute (isNTrans ta )
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
55
485
da4486523f73 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
56
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 486
diff changeset
57 open LimitPreserve
483
265f13adf93b add NIC
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 482
diff changeset
58
484
fcae3025d900 fix Limit pu a0 and t0 in record definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 483
diff changeset
59 LimitC : { I : Category c₁ c₂ ℓ } → ( comp : Complete A I )
485
da4486523f73 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
60 → ( Γ : Functor I CommaCategory )
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 486
diff changeset
61 → ( glimit : LimitPreserve A I C G )
491
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
62 → Limit C I (G ○ (FIA Γ))
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 486
diff changeset
63 LimitC {I} comp Γ glimit = plimit glimit (FIA Γ) (climit comp (FIA Γ))
486
56cf6581c5f6 add some lemma but no use
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 485
diff changeset
64
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
65 frev : { I : Category c₁ c₂ ℓ } → (comp : Complete A I) → ( Γ : Functor I CommaCategory ) (i : Obj I ) → Hom A (limit-c comp (FIA Γ)) (obj (FObj Γ i))
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
66 frev comp Γ i = TMap (t0 ( climit comp (FIA Γ))) i
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
67
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
68 tu : { I : Category c₁ c₂ ℓ } → ( comp : Complete A I) → ( Γ : Functor I CommaCategory )
491
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
69 → NTrans I C (K C I (FObj F (limit-c comp (FIA Γ)))) (G ○ (FIA Γ))
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
70 tu {I} comp Γ = record {
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
71 TMap = λ i → C [ hom ( FObj Γ i ) o FMap F (frev comp Γ i) ]
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
72 ; isNTrans = record { commute = λ {a} {b} {f} → commute {a} {b} {f} }
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
73 } where
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
74 commute : {a b : Obj I} {f : Hom I a b} →
491
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
75 C [ C [ FMap (G ○ (FIA Γ)) f o C [ hom (FObj Γ a) o FMap F (frev comp Γ a) ] ]
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
76 ≈ C [ C [ hom (FObj Γ b) o FMap F (frev comp Γ b) ] o FMap (K C I (FObj F (limit-c comp (FIA Γ)))) f ] ]
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
77 commute {a} {b} {f} = let open ≈-Reasoning (C) in begin
491
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
78 FMap (G ○ (FIA Γ)) f o ( hom (FObj Γ a) o FMap F (frev comp Γ a) )
488
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
79 ≈⟨⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
80 FMap G (arrow (FMap Γ f ) ) o ( hom (FObj Γ a) o FMap F ( TMap (t0 ( climit comp (FIA Γ))) a ))
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
81 ≈⟨ assoc ⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
82 (FMap G (arrow (FMap Γ f ) ) o hom (FObj Γ a)) o FMap F ( TMap (t0 ( climit comp (FIA Γ))) a )
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
83 ≈⟨ car ( comm (FMap Γ f)) ⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
84 (hom (FObj Γ b) o FMap F (arrow (FMap Γ f)) ) o FMap F ( TMap (t0 ( climit comp (FIA Γ))) a )
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
85 ≈↑⟨ assoc ⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
86 hom (FObj Γ b) o ( FMap F (arrow (FMap Γ f)) o FMap F ( TMap (t0 ( climit comp (FIA Γ))) a ) )
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
87 ≈↑⟨ cdr (distr F) ⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
88 hom (FObj Γ b) o ( FMap F (A [ arrow (FMap Γ f) o TMap (t0 ( climit comp (FIA Γ))) a ] ) )
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
89 ≈⟨⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
90 hom (FObj Γ b) o ( FMap F (A [ FMap (FIA Γ) f o TMap (t0 ( climit comp (FIA Γ))) a ] ) )
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
91 ≈⟨ cdr ( fcong F ( IsNTrans.commute (isNTrans (t0 ( climit comp (FIA Γ))) ))) ⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
92 hom (FObj Γ b) o ( FMap F ( A [ (TMap (t0 ( climit comp (FIA Γ))) b) o FMap (K A I (a0 (climit comp (FIA Γ)))) f ] ))
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
93 ≈⟨⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
94 hom (FObj Γ b) o ( FMap F ( A [ (TMap (t0 ( climit comp (FIA Γ))) b) o id1 A (limit-c comp (FIA Γ)) ] ))
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
95 ≈⟨ cdr ( distr F ) ⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
96 hom (FObj Γ b) o ( FMap F (TMap (t0 ( climit comp (FIA Γ))) b) o FMap F (id1 A (limit-c comp (FIA Γ))))
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
97 ≈⟨ cdr ( cdr ( IsFunctor.identity (isFunctor F) ) ) ⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
98 hom (FObj Γ b) o ( FMap F (TMap (t0 ( climit comp (FIA Γ))) b) o id1 C (FObj F (limit-c comp (FIA Γ))))
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
99 ≈⟨ assoc ⟩
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
100 ( hom (FObj Γ b) o FMap F (frev comp Γ b)) o FMap (K C I (FObj F (limit-c comp (FIA Γ)))) f
488
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
101 ∎
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
102 limitHom : { I : Category c₁ c₂ ℓ } → (comp : Complete A I) → ( Γ : Functor I CommaCategory )
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
103 → ( glimit : LimitPreserve A I C G ) → Hom C (FObj F (limit-c comp (FIA Γ ) )) (FObj G (limit-c comp (FIA Γ) ))
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
104 limitHom comp Γ glimit = limit (isLimit (LimitC comp Γ glimit )) (FObj F ( limit-c comp (FIA Γ))) (tu comp Γ )
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
105
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
106 commaLimit : { I : Category c₁ c₂ ℓ } → ( Complete A I) → ( Γ : Functor I CommaCategory )
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
107 → ( glimit : LimitPreserve A I C G )
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
108 → Obj CommaCategory
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
109 commaLimit {I} comp Γ glimit = record {
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
110 obj = limit-c comp (FIA Γ)
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
111 ; hom = limitHom comp Γ glimit
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
112 }
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
113
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
114
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
115 commaNat : { I : Category c₁ c₂ ℓ } → ( comp : Complete A I) → ( Γ : Functor I CommaCategory )
488
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
116 → ( glimit : LimitPreserve A I C G )
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
117 → NTrans I CommaCategory (K CommaCategory I (commaLimit {I} comp Γ glimit)) Γ
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
118 commaNat {I} comp Γ glimit = record {
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
119 TMap = λ x → record {
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
120 arrow = TMap ( limit-u comp (FIA Γ ) ) x
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
121 ; comm = comm1 x
488
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
122 }
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
123 ; isNTrans = record { commute = comm2 }
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
124 } where
488
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
125 comm1 : (x : Obj I ) →
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
126 C [ C [ FMap G (TMap (limit-u comp (FIA Γ)) x) o hom (FObj (K CommaCategory I (commaLimit comp Γ glimit)) x) ]
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
127 ≈ C [ hom (FObj Γ x) o FMap F (TMap (limit-u comp (FIA Γ)) x) ] ]
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
128 comm1 x = let open ≈-Reasoning (C) in begin
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
129 FMap G (TMap (limit-u comp (FIA Γ)) x) o hom (FObj (K CommaCategory I (commaLimit comp Γ glimit)) x)
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
130 ≈⟨⟩
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
131 FMap G (TMap (limit-u comp (FIA Γ)) x) o hom (commaLimit comp Γ glimit)
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
132 ≈⟨⟩
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
133 FMap G (TMap (limit-u comp (FIA Γ)) x) o limit (isLimit (LimitC comp Γ glimit )) (FObj F ( limit-c comp (FIA Γ))) (tu comp Γ )
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
134 ≈⟨⟩
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
135 TMap (t0 ( LimitC comp Γ glimit )) x o limit (isLimit (LimitC comp Γ glimit )) (FObj F ( limit-c comp (FIA Γ))) (tu comp Γ )
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
136 ≈⟨ t0f=t ( isLimit ( LimitC comp Γ glimit ) ) ⟩
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
137 TMap (tu comp Γ) x
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
138 ≈⟨⟩
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
139 hom (FObj Γ x) o FMap F (TMap (limit-u comp (FIA Γ)) x)
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
140 ∎
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
141 comm2 : {a b : Obj I} {f : Hom I a b} →
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
142 CommaCategory [ CommaCategory [ FMap Γ f o record { arrow = TMap (limit-u comp (FIA Γ)) a ; comm = comm1 a } ]
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
143 ≈ CommaCategory [ record { arrow = TMap (limit-u comp (FIA Γ)) b ; comm = comm1 b } o FMap (K CommaCategory I (commaLimit comp Γ glimit)) f ] ]
490
1a42f06e7ae1 commaNat done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 489
diff changeset
144 comm2 {a} {b} {f} = let open ≈-Reasoning (A) in begin
1a42f06e7ae1 commaNat done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 489
diff changeset
145 FMap (FIA Γ) f o TMap (limit-u comp (FIA Γ)) a
1a42f06e7ae1 commaNat done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 489
diff changeset
146 ≈⟨ IsNTrans.commute (isNTrans (limit-u comp (FIA Γ))) ⟩
1a42f06e7ae1 commaNat done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 489
diff changeset
147 TMap (limit-u comp (FIA Γ)) b o FMap (K A I (limit-c comp (FIA Γ))) f
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
148 ∎
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
149
491
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
150 comma-a0 : { I : Category c₁ c₂ ℓ } → ( comp : Complete A I) → ( Γ : Functor I CommaCategory )
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
151 → ( glimit : LimitPreserve A I C G ) (a : CommaObj) → ( t : NTrans I CommaCategory (K CommaCategory I a) Γ ) → Hom CommaCategory a (commaLimit comp Γ glimit)
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
152 comma-a0 {I} comp Γ glimit a t = record {
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
153 arrow = limit (isLimit ( climit comp (FIA Γ) ) ) (obj a ) (NIA {I} Γ a t )
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
154 ; comm = comm1
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
155 } where
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
156 comm1 : C [ C [ FMap G (limit (isLimit (climit comp (FIA Γ))) (obj a) (NIA Γ a t)) o hom a ]
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
157 ≈ C [ hom (commaLimit comp Γ glimit) o FMap F (limit (isLimit (climit comp (FIA Γ))) (obj a) (NIA Γ a t)) ] ]
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
158 comm1 = let open ≈-Reasoning (C) in begin
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
159 FMap G (limit (isLimit (climit comp (FIA Γ))) (obj a) (NIA Γ a t)) o hom a
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
160 ≈⟨ {!!} ⟩
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
161 limit (isLimit (LimitC comp Γ glimit )) (FObj F ( limit-c comp (FIA Γ))) (tu comp Γ ) o FMap F (limit (isLimit (climit comp (FIA Γ))) (obj a) (NIA Γ a t))
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
162 ≈⟨⟩
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
163 hom (commaLimit comp Γ glimit) o FMap F (limit (isLimit (climit comp (FIA Γ))) (obj a) (NIA Γ a t))
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
164 ∎
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
165
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
166 comma-t0f=t : { I : Category c₁ c₂ ℓ } → ( comp : Complete A I) → ( Γ : Functor I CommaCategory )
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
167 → ( glimit : LimitPreserve A I C G ) (a : CommaObj) → ( t : NTrans I CommaCategory (K CommaCategory I a) Γ ) (i : Obj I )
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
168 → CommaCategory [ CommaCategory [ TMap (commaNat comp Γ glimit) i o comma-a0 comp Γ glimit a t ] ≈ TMap t i ]
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
169 comma-t0f=t {I} comp Γ glimit a t i = let open ≈-Reasoning (A) in begin
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
170 TMap ( limit-u comp (FIA Γ ) ) i o limit (isLimit ( climit comp (FIA Γ) ) ) (obj a ) (NIA {I} Γ a t )
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
171 ≈⟨ t0f=t (isLimit ( climit comp (FIA Γ) ) ) ⟩
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
172 TMap (NIA {I} Γ a t ) i
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
173 ∎
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
174
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
175
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
176 comma-uniqueness : { I : Category c₁ c₂ ℓ } → ( comp : Complete A I) → ( Γ : Functor I CommaCategory )
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
177 → ( glimit : LimitPreserve A I C G ) (a : CommaObj) → ( t : NTrans I CommaCategory (K CommaCategory I a) Γ )
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
178 → ( f : Hom CommaCategory a (commaLimit comp Γ glimit))
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
179 → ( ∀ { i : Obj I } → CommaCategory [ CommaCategory [ TMap ( commaNat { I} comp Γ glimit ) i o f ] ≈ TMap t i ] )
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
180 → CommaCategory [ comma-a0 comp Γ glimit a t ≈ f ]
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
181 comma-uniqueness {I} comp Γ glimit a t f t=f = let open ≈-Reasoning (A) in begin
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
182 limit (isLimit ( climit comp (FIA Γ) ) ) (obj a ) (NIA {I} Γ a t )
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
183 ≈⟨ limit-uniqueness (isLimit ( climit comp (FIA Γ) ) ) (arrow f) t=f ⟩
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
184 arrow f
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
185 ∎
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
186
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
187
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
188 hasLimit : { I : Category c₁ c₂ ℓ } → ( comp : Complete A I )
488
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
189 → ( glimit : LimitPreserve A I C G )
485
da4486523f73 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
190 → ( Γ : Functor I CommaCategory )
da4486523f73 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
191 → Limit CommaCategory I Γ
da4486523f73 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
192 hasLimit {I} comp glimit Γ = record {
488
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
193 a0 = commaLimit {I} comp Γ glimit ;
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
194 t0 = commaNat { I} comp Γ glimit ;
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 486
diff changeset
195 isLimit = record {
491
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
196 limit = λ a t → comma-a0 comp Γ glimit a t ;
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
197 t0f=t = λ {a t i } → comma-t0f=t comp Γ glimit a t i ;
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
198 limit-uniqueness = λ {a} {t} f t=f → comma-uniqueness {I} comp Γ glimit a t f t=f
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 486
diff changeset
199 }
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
200 }