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annotate freyd1.agda @ 774:f3a493da92e8

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add simple category version
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Wed, 13 Jun 2018 12:56:38 +0900
parents 917e51be9bbf
children
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 Functor
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
10
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
11 open import Comma1 F G
492
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 491
diff changeset
12 -- open import freyd CommaCategory -- we don't need this yet
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
13
492
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 491
diff changeset
14 open import Category.Cat -- Functor composition
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
15 open NTrans
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
16 open Complete
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
17 open CommaObj
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
18 open CommaHom
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
19 open Limit
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 486
diff changeset
20 open IsLimit
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
21
493
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
22 --
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
23 --
483
265f13adf93b add NIC
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 482
diff changeset
24 -- F : A → C
265f13adf93b add NIC
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 482
diff changeset
25 -- G : A → C
265f13adf93b add NIC
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 482
diff changeset
26 --
493
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
27 -- if A is complete and G preserve limit, Comma Category F↓G is complete
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
28 -- i.e. it has limit for Γ : I → F↓G
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
29 --
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
30 --
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
31 --
483
265f13adf93b add NIC
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 482
diff changeset
32
493
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
33 --- Get a functor Functor I A from a functor I CommaCategory
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
34 ---
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
35 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
36 FIA {I} Γ = record {
482
fd752ad25ac0 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
37 FObj = λ x → obj (FObj Γ x ) ;
fd752ad25ac0 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
38 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
39 isFunctor = record {
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
40 identity = identity
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
41 ; distr = IsFunctor.distr (isFunctor Γ)
fd752ad25ac0 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
42 ; ≈-cong = IsFunctor.≈-cong (isFunctor Γ)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
43 }} where
fd752ad25ac0 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
44 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
45 identity {x} = let open ≈-Reasoning (A) in begin
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
46 arrow (FMap Γ (id1 I x))
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
47 ≈⟨ IsFunctor.identity (isFunctor Γ) ⟩
fd752ad25ac0 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
48 id1 A (obj (FObj Γ x))
fd752ad25ac0 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
49 ∎
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
50
493
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
51 --- Get a nat on A from a nat on CommaCategory
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
52 --
491
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
53 NIA : { I : Category c₁ c₂ ℓ } → ( Γ : Functor I CommaCategory )
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 659
diff changeset
54 (c : Obj CommaCategory ) ( ta : NTrans I CommaCategory ( K I CommaCategory c ) Γ ) → NTrans I A ( K I A (obj c) ) (FIA Γ)
491
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
55 NIA {I} Γ c ta = record {
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
56 TMap = λ x → arrow (TMap ta x )
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
57 ; isNTrans = record { commute = comm1 }
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
58 } where
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
59 comm1 : {a b : Obj I} {f : Hom I a b} →
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 659
diff changeset
60 A [ A [ FMap (FIA Γ) f o arrow (TMap ta a) ] ≈ A [ arrow (TMap ta b) o FMap (K I A (obj c)) f ] ]
491
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
61 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
62
485
da4486523f73 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
63
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 486
diff changeset
64 open LimitPreserve
483
265f13adf93b add NIC
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 482
diff changeset
65
493
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
66 -- Limit on A from Γ : I → CommaCategory
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
67 --
484
fcae3025d900 fix Limit pu a0 and t0 in record definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 483
diff changeset
68 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
69 → ( Γ : Functor I CommaCategory )
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 659
diff changeset
70 → ( glimit : LimitPreserve I A C G )
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 659
diff changeset
71 → Limit I C (G ○ (FIA Γ))
492
c7b8017bcd4d on going..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 491
diff changeset
72 LimitC {I} comp Γ glimit = plimit glimit (climit comp (FIA Γ))
486
56cf6581c5f6 add some lemma but no use
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 485
diff changeset
73
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
74 tu : { I : Category c₁ c₂ ℓ } → ( comp : Complete A I) → ( Γ : Functor I CommaCategory )
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 659
diff changeset
75 → NTrans I C (K I C (FObj F (limit-c comp (FIA Γ)))) (G ○ (FIA Γ))
489
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
76 tu {I} comp Γ = record {
493
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
77 TMap = λ i → C [ hom ( FObj Γ i ) o FMap F ( TMap (t0 ( climit comp (FIA Γ))) i) ]
489
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
78 ; 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
79 } where
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
80 commute : {a b : Obj I} {f : Hom I a b} →
496
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 495
diff changeset
81 C [ C [ FMap (G ○ (FIA Γ)) f o C [ hom (FObj Γ a) o FMap F (TMap (t0 ( climit comp (FIA Γ))) a) ] ]
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 659
diff changeset
82 ≈ C [ C [ hom (FObj Γ b) o FMap F (TMap (t0 ( climit comp (FIA Γ))) b) ] o FMap (K I C (FObj F (limit-c comp (FIA Γ)))) f ] ]
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
83 commute {a} {b} {f} = let open ≈-Reasoning (C) in begin
496
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 495
diff changeset
84 FMap (G ○ (FIA Γ)) f o ( hom (FObj Γ a) o FMap F (TMap (t0 ( climit comp (FIA Γ))) a ))
488
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 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
87 ≈⟨ assoc ⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
88 (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
89 ≈⟨ car ( comm (FMap Γ 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 (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
91 ≈↑⟨ assoc ⟩
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 (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
93 ≈↑⟨ cdr (distr F) ⟩
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 [ 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
95 ≈⟨⟩
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 (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
97 ≈⟨ cdr ( fcong F ( IsNTrans.commute (isNTrans (t0 ( climit comp (FIA Γ))) ))) ⟩
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 659
diff changeset
98 hom (FObj Γ b) o ( FMap F ( A [ (TMap (t0 ( climit comp (FIA Γ))) b) o FMap (K I A (a0 (climit comp (FIA Γ)))) f ] ))
488
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
99 ≈⟨⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
100 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
101 ≈⟨ cdr ( distr F ) ⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
102 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
103 ≈⟨ cdr ( cdr ( IsFunctor.identity (isFunctor F) ) ) ⟩
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
104 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
105 ≈⟨ assoc ⟩
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 659
diff changeset
106 ( hom (FObj Γ b) o FMap F (TMap (t0 ( climit comp (FIA Γ))) b)) o FMap (K I C (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
107 ∎
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
108 limitHom : { I : Category c₁ c₂ ℓ } → (comp : Complete A I) → ( Γ : Functor I CommaCategory )
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 659
diff changeset
109 → ( glimit : LimitPreserve I A C G ) → Hom C (FObj F (limit-c comp (FIA Γ ) )) (FObj G (limit-c comp (FIA Γ) ))
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
110 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
111
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
112 commaLimit : { I : Category c₁ c₂ ℓ } → ( Complete A I) → ( Γ : Functor I CommaCategory )
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 659
diff changeset
113 → ( glimit : LimitPreserve I A C G )
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
114 → Obj CommaCategory
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
115 commaLimit {I} comp Γ glimit = record {
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
116 obj = limit-c comp (FIA Γ)
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
117 ; hom = limitHom comp Γ glimit
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
118 }
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
119
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
120
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
121 commaNat : { I : Category c₁ c₂ ℓ } → ( comp : Complete A I) → ( Γ : Functor I CommaCategory )
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 659
diff changeset
122 → ( glimit : LimitPreserve I A C G )
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 659
diff changeset
123 → NTrans I CommaCategory (K I CommaCategory (commaLimit {I} comp Γ glimit)) Γ
488
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
124 commaNat {I} comp Γ glimit = record {
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
125 TMap = λ x → record {
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
126 arrow = TMap ( limit-u comp (FIA Γ ) ) x
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
127 ; comm = comm1 x
488
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
128 }
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
129 ; isNTrans = record { commute = comm2 }
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
130 } where
488
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
131 comm1 : (x : Obj I ) →
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 659
diff changeset
132 C [ C [ FMap G (TMap (limit-u comp (FIA Γ)) x) o hom (FObj (K I CommaCategory (commaLimit comp Γ glimit)) x) ]
488
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
133 ≈ 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
134 comm1 x = let open ≈-Reasoning (C) in begin
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 659
diff changeset
135 FMap G (TMap (limit-u comp (FIA Γ)) x) o hom (FObj (K I CommaCategory (commaLimit comp Γ glimit)) x)
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
136 ≈⟨⟩
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
137 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
138 ≈⟨⟩
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
139 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
140 ≈⟨⟩
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
141 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
142 ≈⟨ t0f=t ( isLimit ( LimitC comp Γ glimit ) ) ⟩
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
143 TMap (tu comp Γ) x
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
144 ≈⟨⟩
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
145 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
146 ∎
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
147 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
148 CommaCategory [ CommaCategory [ FMap Γ f o record { arrow = TMap (limit-u comp (FIA Γ)) a ; comm = comm1 a } ]
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 659
diff changeset
149 ≈ CommaCategory [ record { arrow = TMap (limit-u comp (FIA Γ)) b ; comm = comm1 b } o FMap (K I CommaCategory (commaLimit comp Γ glimit)) f ] ]
490
1a42f06e7ae1 commaNat done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 489
diff changeset
150 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
151 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
152 ≈⟨ IsNTrans.commute (isNTrans (limit-u comp (FIA Γ))) ⟩
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 659
diff changeset
153 TMap (limit-u comp (FIA Γ)) b o FMap (K I A (limit-c comp (FIA Γ))) f
489
75a60ceb55af on going ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 488
diff changeset
154 ∎
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
155
495
633df882db86 fryed1 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 494
diff changeset
156 gnat : { I : Category c₁ c₂ ℓ } → ( Γ : Functor I CommaCategory )
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 659
diff changeset
157 → (a : CommaObj) → ( t : NTrans I CommaCategory (K I CommaCategory a) Γ ) →
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 659
diff changeset
158 NTrans I C (K I C (FObj F (obj a))) (G ○ FIA Γ)
495
633df882db86 fryed1 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 494
diff changeset
159 gnat {I} Γ a t = record {
494
ba6a67523523 unique direction 2 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 493
diff changeset
160 TMap = λ i → C [ hom ( FObj Γ i ) o FMap F ( TMap (NIA Γ a t) i ) ]
ba6a67523523 unique direction 2 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 493
diff changeset
161 ; isNTrans = record { commute = λ {x y f} → comm1 x y f }
ba6a67523523 unique direction 2 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 493
diff changeset
162 } where
ba6a67523523 unique direction 2 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 493
diff changeset
163 comm1 : (x y : Obj I) (f : Hom I x y ) →
ba6a67523523 unique direction 2 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 493
diff changeset
164 C [ C [ FMap (G ○ FIA Γ) f o C [ hom (FObj Γ x) o FMap F (TMap (NIA Γ a t) x) ] ]
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 659
diff changeset
165 ≈ C [ C [ hom (FObj Γ y) o FMap F (TMap (NIA Γ a t) y) ] o FMap (K I C (FObj F (obj a))) f ] ]
494
ba6a67523523 unique direction 2 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 493
diff changeset
166 comm1 x y f = let open ≈-Reasoning (C) in begin
ba6a67523523 unique direction 2 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 493
diff changeset
167 FMap (G ○ FIA Γ) f o ( hom (FObj Γ x) o FMap F (TMap (NIA Γ a t) x ))
ba6a67523523 unique direction 2 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 493
diff changeset
168 ≈⟨⟩
ba6a67523523 unique direction 2 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 493
diff changeset
169 FMap G (FMap (FIA Γ) f) o ( hom (FObj Γ x) o FMap F (TMap (NIA Γ a t) x ))
ba6a67523523 unique direction 2 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 493
diff changeset
170 ≈⟨ assoc ⟩
ba6a67523523 unique direction 2 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 493
diff changeset
171 (FMap G (FMap (FIA Γ) f) o ( hom (FObj Γ x))) o FMap F (TMap (NIA Γ a t) x )
ba6a67523523 unique direction 2 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 493
diff changeset
172 ≈⟨ car ( comm (FMap Γ f) ) ⟩
ba6a67523523 unique direction 2 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 493
diff changeset
173 ( hom (FObj Γ y) o FMap F (FMap (FIA Γ) f )) o FMap F (TMap (NIA Γ a t) x )
ba6a67523523 unique direction 2 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 493
diff changeset
174 ≈↑⟨ assoc ⟩
ba6a67523523 unique direction 2 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 493
diff changeset
175 hom (FObj Γ y) o ( FMap F (FMap (FIA Γ) f ) o FMap F (TMap (NIA Γ a t) x ))
ba6a67523523 unique direction 2 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 493
diff changeset
176 ≈↑⟨ cdr (distr F) ⟩
ba6a67523523 unique direction 2 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 493
diff changeset
177 hom (FObj Γ y) o ( FMap F ( A [ FMap (FIA Γ) f o TMap (NIA Γ a t) x ]) )
ba6a67523523 unique direction 2 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 493
diff changeset
178 ≈⟨ cdr ( fcong F ( IsNTrans.commute ( isNTrans ( NIA Γ a t )))) ⟩
ba6a67523523 unique direction 2 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 493
diff changeset
179 hom (FObj Γ y) o ( FMap F ( A [ TMap (NIA Γ a t) y o id1 A (obj a) ]) )
ba6a67523523 unique direction 2 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 493
diff changeset
180 ≈⟨ cdr ( fcong F ( IsCategory.identityR (Category.isCategory A))) ⟩
ba6a67523523 unique direction 2 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 493
diff changeset
181 hom (FObj Γ y) o FMap F (TMap (NIA Γ a t) y)
ba6a67523523 unique direction 2 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 493
diff changeset
182 ≈↑⟨ idR ⟩
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 659
diff changeset
183 ( hom (FObj Γ y) o FMap F (TMap (NIA Γ a t) y) ) o FMap (K I C (FObj F (obj a))) f
494
ba6a67523523 unique direction 2 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 493
diff changeset
184 ∎
ba6a67523523 unique direction 2 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 493
diff changeset
185
493
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
186
491
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
187 comma-a0 : { I : Category c₁ c₂ ℓ } → ( comp : Complete A I) → ( Γ : Functor I CommaCategory )
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 659
diff changeset
188 → ( glimit : LimitPreserve I A C G ) (a : CommaObj) → ( t : NTrans I CommaCategory (K I CommaCategory a) Γ ) → Hom CommaCategory a (commaLimit comp Γ glimit)
491
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
189 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
190 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
191 ; comm = comm1
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
192 } where
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
193 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
194 ≈ 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
195 comm1 = let open ≈-Reasoning (C) in begin
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
196 FMap G (limit (isLimit (climit comp (FIA Γ))) (obj a) (NIA Γ a t)) o hom a
495
633df882db86 fryed1 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 494
diff changeset
197 ≈↑⟨ limit-uniqueness (isLimit (LimitC comp Γ glimit )) ( λ {i} → begin
633df882db86 fryed1 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 494
diff changeset
198 TMap (t0 (LimitC comp Γ glimit)) i o ( FMap G (limit (isLimit (climit comp (FIA Γ))) (obj a) (NIA Γ a t)) o hom a )
633df882db86 fryed1 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 494
diff changeset
199 ≈⟨ assoc ⟩
633df882db86 fryed1 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 494
diff changeset
200 ( TMap (t0 (LimitC comp Γ glimit)) i o FMap G (limit (isLimit (climit comp (FIA Γ))) (obj a) (NIA Γ a t))) o hom a
633df882db86 fryed1 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 494
diff changeset
201 ≈⟨⟩
633df882db86 fryed1 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 494
diff changeset
202 ( FMap G ( TMap (limit-u comp (FIA Γ )) i ) o FMap G (limit (isLimit (climit comp (FIA Γ))) (obj a) (NIA Γ a t))) o hom a
633df882db86 fryed1 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 494
diff changeset
203 ≈↑⟨ car ( distr G ) ⟩
633df882db86 fryed1 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 494
diff changeset
204 FMap G ( A [ TMap (limit-u comp (FIA Γ )) i o limit (isLimit (climit comp (FIA Γ))) (obj a) (NIA Γ a t) ] ) o hom a
633df882db86 fryed1 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 494
diff changeset
205 ≈⟨ car ( fcong G ( t0f=t (isLimit (climit comp (FIA Γ ))))) ⟩
633df882db86 fryed1 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 494
diff changeset
206 FMap G (arrow (TMap t i)) o hom a
633df882db86 fryed1 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 494
diff changeset
207 ≈⟨ comm ( TMap t i) ⟩
633df882db86 fryed1 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 494
diff changeset
208 hom ( FObj Γ i ) o FMap F ( TMap (NIA Γ a t) i )
633df882db86 fryed1 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 494
diff changeset
209 ≈⟨⟩
633df882db86 fryed1 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 494
diff changeset
210 TMap (gnat Γ a t) i
633df882db86 fryed1 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 494
diff changeset
211 ∎
633df882db86 fryed1 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 494
diff changeset
212 ) ⟩
633df882db86 fryed1 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 494
diff changeset
213 limit (isLimit (LimitC comp Γ glimit )) (FObj F (obj a)) (gnat Γ a t )
633df882db86 fryed1 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 494
diff changeset
214 ≈⟨ limit-uniqueness (isLimit (LimitC comp Γ glimit )) ( λ {i} → begin
493
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
215 TMap (t0 (LimitC comp Γ glimit )) i o
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
216 ( limit (isLimit (LimitC comp Γ glimit )) (FObj F ( limit-c comp (FIA Γ))) (tu comp Γ )
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
217 o FMap F (limit (isLimit (climit comp (FIA Γ))) (obj a) (NIA Γ a t)) )
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
218 ≈⟨ assoc ⟩
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
219 ( TMap (t0 (LimitC comp Γ glimit )) i o
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
220 ( limit (isLimit (LimitC comp Γ glimit )) (FObj F ( limit-c comp (FIA Γ))) (tu comp Γ ) ))
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
221 o FMap F (limit (isLimit (climit comp (FIA Γ))) (obj a) (NIA Γ a t))
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
222 ≈⟨ car ( t0f=t ( isLimit (LimitC comp Γ glimit )) ) ⟩
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
223 TMap (tu comp Γ ) i o FMap F (limit (isLimit (climit comp (FIA Γ))) (obj a) (NIA Γ a t))
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
224 ≈⟨⟩
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
225 ( hom ( FObj Γ i ) o FMap F ( TMap (t0 ( climit comp (FIA Γ))) i) )
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
226 o FMap F (limit (isLimit (climit comp (FIA Γ))) (obj a) (NIA Γ a t))
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
227 ≈↑⟨ assoc ⟩
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
228 hom ( FObj Γ i ) o
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
229 ((FMap F ( TMap (t0 ( climit comp (FIA Γ))) i) ) o FMap F (limit (isLimit (climit comp (FIA Γ))) (obj a) (NIA Γ a t)) )
659
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 496
diff changeset
230 ≈↑⟨ cdr ( distr F) ⟩
493
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
231 hom ( FObj Γ i ) o
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
232 FMap F ( A [ TMap (t0 ( climit comp (FIA Γ))) i o limit (isLimit (climit comp (FIA Γ))) (obj a) (NIA Γ a t) ] )
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
233 ≈⟨ cdr ( fcong F ( t0f=t (isLimit (climit comp (FIA Γ))))) ⟩
494
ba6a67523523 unique direction 2 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 493
diff changeset
234 hom ( FObj Γ i ) o FMap F ( TMap (NIA Γ a t) i )
495
633df882db86 fryed1 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 494
diff changeset
235 ≈⟨⟩
633df882db86 fryed1 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 494
diff changeset
236 TMap (gnat Γ a t ) i
493
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
237 ∎
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
238 ) ⟩
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
239 limit (isLimit (LimitC comp Γ glimit )) (FObj F ( limit-c comp (FIA Γ))) (tu comp Γ )
de9ce7e0d97c on going using limit-uniquness directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 492
diff changeset
240 o FMap F (limit (isLimit (climit comp (FIA Γ))) (obj a) (NIA Γ a t))
491
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
241 ≈⟨⟩
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
242 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
243 ∎
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
244
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
245 comma-t0f=t : { I : Category c₁ c₂ ℓ } → ( comp : Complete A I) → ( Γ : Functor I CommaCategory )
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 659
diff changeset
246 → ( glimit : LimitPreserve I A C G ) (a : CommaObj) → ( t : NTrans I CommaCategory (K I CommaCategory a) Γ ) (i : Obj I )
491
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
247 → 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
248 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
249 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
250 ≈⟨ t0f=t (isLimit ( climit comp (FIA Γ) ) ) ⟩
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
251 TMap (NIA {I} Γ a t ) i
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
252 ∎
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
253
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
254 comma-uniqueness : { I : Category c₁ c₂ ℓ } → ( comp : Complete A I) → ( Γ : Functor I CommaCategory )
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 659
diff changeset
255 → ( glimit : LimitPreserve I A C G ) (a : CommaObj) → ( t : NTrans I CommaCategory (K I CommaCategory a) Γ )
491
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
256 → ( 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
257 → ( ∀ { 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
258 → 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
259 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
260 limit (isLimit ( climit comp (FIA Γ) ) ) (obj a ) (NIA {I} Γ a t )
495
633df882db86 fryed1 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 494
diff changeset
261 ≈⟨ limit-uniqueness (isLimit ( climit comp (FIA Γ) ) ) t=f ⟩
491
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
262 arrow f
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
263 ∎
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
264
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
265 hasLimit : { I : Category c₁ c₂ ℓ } → ( comp : Complete A I )
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 659
diff changeset
266 → ( glimit : LimitPreserve I A C G )
485
da4486523f73 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
267 → ( Γ : Functor I CommaCategory )
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 659
diff changeset
268 → Limit I CommaCategory Γ
485
da4486523f73 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
269 hasLimit {I} comp glimit Γ = record {
488
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
270 a0 = commaLimit {I} comp Γ glimit ;
016087cfa75a commaLimit done, commaNat trying..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
271 t0 = commaNat { I} comp Γ glimit ;
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 486
diff changeset
272 isLimit = record {
491
04da2c458d44 comma-a0 commuativity remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 490
diff changeset
273 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
274 t0f=t = λ {a t i } → comma-t0f=t comp Γ glimit a t i ;
495
633df882db86 fryed1 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 494
diff changeset
275 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
276 }
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
277 }