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annotate freyd1.agda @ 489:75a60ceb55af

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 12 Mar 2017 22:10:54 +0900
parents 016087cfa75a
children 1a42f06e7ae1
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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
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
32 FObj = λ x → obj (FObj Γ x ) ;
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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)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
37 ; ≈-cong = IsFunctor.≈-cong (isFunctor Γ)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
38 }} where
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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)) ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
40 identity {x} = let open ≈-Reasoning (A) in begin
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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) ⟩
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
43 id1 A (obj (FObj Γ x))
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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
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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
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
47 FICG {I} Γ = G ○ (FIA Γ)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
48
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
49 FICF : { 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
50 FICF {I} Γ = F ○ (FIA Γ)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
51
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 486
diff changeset
52 open LimitPreserve
483
265f13adf93b add NIC
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 482
diff changeset
53
484
fcae3025d900 fix Limit pu a0 and t0 in record definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 483
diff changeset
54 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
55 → ( Γ : Functor I CommaCategory )
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 486
diff changeset
56 → ( glimit : LimitPreserve A I C G )
485
da4486523f73 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
57 → Limit C I (FICG Γ)
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 486
diff changeset
58 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
59
489
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
60 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
61 frev comp Γ i = TMap (t0 ( climit comp (FIA Γ))) i
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
62
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
63 tu : { I : Category c₁ c₂ ℓ } → ( comp : Complete A I) → ( Γ : Functor I CommaCategory )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
64 → NTrans I C (K C I (FObj F (limit-c comp (FIA Γ)))) (FICG Γ)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
65 tu {I} comp Γ = record {
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
66 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
67 ; isNTrans = record { commute = λ {a} {b} {f} → commute {a} {b} {f} }
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
68 } where
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
69 commute : {a b : Obj I} {f : Hom I a b} →
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
70 C [ C [ FMap (FICG Γ) f o C [ hom (FObj Γ a) o FMap F (frev comp Γ a) ] ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
71 ≈ 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 ] ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
72 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
73 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
74 ≈⟨⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
75 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
76 ≈⟨ assoc ⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
77 (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
78 ≈⟨ car ( comm (FMap Γ f)) ⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
79 (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
80 ≈↑⟨ assoc ⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
81 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
82 ≈↑⟨ cdr (distr F) ⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
83 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
84 ≈⟨⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
85 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
86 ≈⟨ 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
87 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
88 ≈⟨⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
89 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
90 ≈⟨ cdr ( distr F ) ⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
91 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
92 ≈⟨ cdr ( cdr ( IsFunctor.identity (isFunctor F) ) ) ⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
93 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
94 ≈⟨ assoc ⟩
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
95 ( 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
96 ∎
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
97 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
98 → ( glimit : LimitPreserve A I C G ) → Hom C (FObj F (limit-c comp (FIA Γ ) )) (FObj G (limit-c comp (FIA Γ) ))
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
99 limitHom comp Γ glimit = limit (isLimit (LimitC comp Γ glimit )) (FObj F ( limit-c comp (FIA Γ))) (tu comp Γ )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
100
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
101 commaLimit : { I : Category c₁ c₂ ℓ } → ( Complete A I) → ( Γ : Functor I CommaCategory )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
102 → ( glimit : LimitPreserve A I C G )
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
103 → Obj CommaCategory
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
104 commaLimit {I} comp Γ glimit = record {
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
105 obj = limit-c comp (FIA Γ)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
106 ; hom = limitHom comp Γ glimit
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
107 }
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
108
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
109
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
110 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
111 → ( glimit : LimitPreserve A I C G )
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
112 → 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
113 commaNat {I} comp Γ glimit = record {
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
114 TMap = λ x → record {
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
115 arrow = TMap ( limit-u comp (FIA Γ ) ) x
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
116 ; comm = comm1 x
488
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
117 }
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
118 ; isNTrans = record { commute = comm2 }
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
119 } where
488
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
120 comm1 : (x : Obj I ) →
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
121 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
122 ≈ 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
123 comm1 x = let open ≈-Reasoning (C) in begin
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
124 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
125 ≈⟨⟩
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
126 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
127 ≈⟨⟩
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
128 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
129 ≈⟨⟩
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
130 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
131 ≈⟨ t0f=t ( isLimit ( LimitC comp Γ glimit ) ) ⟩
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
132 TMap (tu comp Γ) x
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
133 ≈⟨⟩
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
134 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
135 ∎
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
136 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
137 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
138 ≈ CommaCategory [ record { arrow = TMap (limit-u comp (FIA Γ)) b ; comm = comm1 b } o FMap (K CommaCategory I (commaLimit comp Γ glimit)) f ] ]
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
139 comm2 {a} {b} {f} = let open ≈-Reasoning (CommaCategory) in begin
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
140 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
141 ≈⟨ {!!} ⟩
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
142 record { arrow = TMap (limit-u comp (FIA Γ)) b ; comm = comm1 b } o FMap (K CommaCategory I (commaLimit comp Γ glimit)) f
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
143 ∎
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
144
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
145
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
146 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
147 → ( glimit : LimitPreserve A I C G )
485
da4486523f73 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
148 → ( Γ : Functor I CommaCategory )
da4486523f73 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
149 → Limit CommaCategory I Γ
da4486523f73 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
150 hasLimit {I} comp glimit Γ = record {
488
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
151 a0 = commaLimit {I} comp Γ glimit ;
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
152 t0 = commaNat { I} comp Γ glimit ;
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 486
diff changeset
153 isLimit = record {
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 486
diff changeset
154 limit = λ a t → {!!} ;
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 486
diff changeset
155 t0f=t = λ {a t i } → {!!} ;
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 486
diff changeset
156 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
157 }
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
158 }