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annotate freyd1.agda @ 490:1a42f06e7ae1

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commaNat done
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 13 Mar 2017 09:51:44 +0900
parents 75a60ceb55af
children 04da2c458d44
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
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 ) ;
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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
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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))
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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
485
da4486523f73 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
46 FICG : { I : Category c₁ c₂ ℓ } → ( Γ : Functor I CommaCategory ) → Functor I C
da4486523f73 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
47 FICG {I} Γ = G ○ (FIA Γ)
da4486523f73 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
48
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 486
diff changeset
49 open LimitPreserve
483
265f13adf93b add NIC
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 482
diff changeset
50
484
fcae3025d900 fix Limit pu a0 and t0 in record definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 483
diff changeset
51 LimitC : { I : Category c₁ c₂ ℓ } → ( comp : Complete A I )
485
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
52 → ( Γ : Functor I CommaCategory )
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 486
diff changeset
53 → ( glimit : LimitPreserve A I C G )
485
da4486523f73 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
54 → Limit C I (FICG Γ)
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 486
diff changeset
55 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
56
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
57 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
58 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
59
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
60 tu : { 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
61 → NTrans I C (K C I (FObj F (limit-c comp (FIA Γ)))) (FICG Γ)
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
62 tu {I} comp Γ = record {
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
63 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
64 ; 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
65 } where
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
66 commute : {a b : Obj I} {f : Hom I a b} →
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
67 C [ C [ FMap (FICG Γ) f o C [ hom (FObj Γ a) o FMap F (frev comp Γ a) ] ]
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
68 ≈ 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
69 commute {a} {b} {f} = let open ≈-Reasoning (C) in begin
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
70 FMap (FICG 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
71 ≈⟨⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
72 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
73 ≈⟨ assoc ⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
74 (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
75 ≈⟨ car ( comm (FMap Γ f)) ⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
76 (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
77 ≈↑⟨ assoc ⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
78 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
79 ≈↑⟨ cdr (distr F) ⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
80 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
81 ≈⟨⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
82 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
83 ≈⟨ 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
84 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
85 ≈⟨⟩
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 ( 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
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 (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
89 ≈⟨ cdr ( cdr ( IsFunctor.identity (isFunctor F) ) ) ⟩
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 (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
91 ≈⟨ assoc ⟩
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
92 ( 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
93 ∎
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
94 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
95 → ( 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
96 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
97
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
98 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
99 → ( glimit : LimitPreserve A I C G )
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
100 → Obj CommaCategory
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
101 commaLimit {I} comp Γ glimit = record {
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
102 obj = limit-c comp (FIA Γ)
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
103 ; hom = limitHom comp Γ glimit
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
104 }
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
105
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
106
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
107 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
108 → ( glimit : LimitPreserve A I C G )
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
109 → 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
110 commaNat {I} comp Γ glimit = record {
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
111 TMap = λ x → record {
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
112 arrow = TMap ( limit-u comp (FIA Γ ) ) x
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
113 ; comm = comm1 x
488
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
114 }
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
115 ; isNTrans = record { commute = comm2 }
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
116 } where
488
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
117 comm1 : (x : Obj I ) →
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
118 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
119 ≈ 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
120 comm1 x = let open ≈-Reasoning (C) in begin
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
121 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
122 ≈⟨⟩
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
123 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
124 ≈⟨⟩
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
125 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
126 ≈⟨⟩
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
127 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
128 ≈⟨ t0f=t ( isLimit ( LimitC comp Γ glimit ) ) ⟩
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
129 TMap (tu comp Γ) 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 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
132 ∎
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
133 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
134 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
135 ≈ 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
136 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
137 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
138 ≈⟨ IsNTrans.commute (isNTrans (limit-u comp (FIA Γ))) ⟩
1a42f06e7ae1 commaNat done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 489
diff changeset
139 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
140 ∎
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
141
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
142
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
143 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
144 → ( glimit : LimitPreserve A I C G )
485
da4486523f73 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
145 → ( Γ : Functor I CommaCategory )
da4486523f73 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
146 → Limit CommaCategory I Γ
da4486523f73 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
147 hasLimit {I} comp glimit Γ = record {
488
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
148 a0 = commaLimit {I} comp Γ glimit ;
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
149 t0 = commaNat { I} comp Γ glimit ;
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 486
diff changeset
150 isLimit = record {
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 486
diff changeset
151 limit = λ a t → {!!} ;
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 486
diff changeset
152 t0f=t = λ {a t i } → {!!} ;
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 486
diff changeset
153 limit-uniqueness = λ {a} {t} f t=f → {!!}
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 486
diff changeset
154 }
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
155 }