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annotate freyd1.agda @ 482:fd752ad25ac0

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sat, 11 Mar 2017 01:35:06 +0900
parents 65e6906782bb
children 265f13adf93b
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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
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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 tb : { c₁' c₂' ℓ' : Level} (B : Category c₁' c₂' ℓ') { c₁ c₂ ℓ : Level} ( I : Category c₁ c₂ ℓ ) ( Γ : Functor I B )
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
21 ( lim : Obj B ) ( tb : NTrans I B ( K B I lim ) Γ ) →
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
22 ( U : Functor B C) → NTrans I C ( K C I (FObj U lim) ) (U ○ Γ)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
23 tb B I Γ lim tb U = record {
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
24 TMap = TMap (Functor*Nat I C U tb) ;
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
25 isNTrans = record { commute = λ {a} {b} {f} → let open ≈-Reasoning (C) in begin
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
26 FMap (U ○ Γ) f o TMap (Functor*Nat I C U tb) a
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
27 ≈⟨ nat ( Functor*Nat I C U tb ) ⟩
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
28 TMap (Functor*Nat I C U tb) b o FMap (U ○ K B I lim) f
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
29 ≈⟨ cdr (IsFunctor.identity (isFunctor U) ) ⟩
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
30 TMap (Functor*Nat I C U tb) b o FMap (K C I (FObj U lim)) f
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
31 ∎
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
32 } }
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
33
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
34 open Complete
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
35
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
36 open CommaObj
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
37 open CommaHom
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
38 open Limit
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
39
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
40 FIA : { I : Category c₁ c₂ ℓ } → ( Γ : Functor I CommaCategory ) → Functor I A
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
41 FIA {I} Γ = record {
482
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
42 FObj = λ x → obj (FObj Γ x ) ;
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
43 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
44 isFunctor = record {
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
45 identity = identity
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
46 ; distr = IsFunctor.distr (isFunctor Γ)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
47 ; ≈-cong = IsFunctor.≈-cong (isFunctor Γ)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
48 }} where
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
49 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
50 identity {x} = let open ≈-Reasoning (A) in begin
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
51 arrow (FMap Γ (id1 I x))
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
52 ≈⟨ IsFunctor.identity (isFunctor Γ) ⟩
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
53 id1 A (obj (FObj Γ x))
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
54 ∎
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
55
482
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
56
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
57 NIA : { I : Category c₁ c₂ ℓ } → ( Γ : Functor I CommaCategory )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
58 (c : Obj CommaCategory ) ( ta : NTrans I CommaCategory ( K CommaCategory I c ) Γ ) → NTrans I A ( K A I (obj c) ) (FIA Γ)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
59 NIA {I} Γ c ta = record {
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
60 TMap = λ x → arrow (TMap ta x )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
61 ; isNTrans = record { commute = comm1 }
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
62 } where
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
63 comm1 : {a b : Obj I} {f : Hom I a b} →
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
64 A [ A [ FMap (FIA Γ) f o arrow (TMap ta a) ] ≈ A [ arrow (TMap ta b) o FMap (K A I (obj c)) f ] ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
65 comm1 {a} {b} {f} = IsNTrans.commute (isNTrans ta )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
66
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
67 commaLimit : { I : Category c₁ c₂ ℓ } → ( Complete A I) → ( Γ : Functor I CommaCategory ) → Obj CommaCategory
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
68 commaLimit {I} comp Γ = record {
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
69 obj = limit-c comp (FIA Γ)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
70 ; hom = limitHom
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
71 } where
482
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
72 ll = ( limit (isLimit comp (FIA Γ)) (limit-c comp (FIA Γ)) {!!})
481
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
73 limitHom : Hom C (FObj F (limit-c comp (FIA Γ ) )) (FObj G (limit-c comp (FIA Γ) ))
482
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
74 limitHom = C [ FMap G ll o C [ {!!} o FMap F ll ] ]
481
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
75
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
76
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
77 commaNat : { I : Category c₁ c₂ ℓ } → ( comp : Complete A I) → ( Γ : Functor I CommaCategory )
482
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
78 → (c : Obj CommaCategory )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
79 → ( ta : NTrans I CommaCategory ( K CommaCategory I c ) Γ )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
80 → NTrans I CommaCategory (K CommaCategory I c) Γ
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
81 commaNat {I} comp Γ c ta = record {
481
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
82 TMap = λ x → tmap x
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
83 ; isNTrans = record { commute = {!!} }
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
84 } where
482
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
85 tmap : (i : Obj I) → Hom CommaCategory (FObj (K CommaCategory I c) i) (FObj Γ i)
481
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
86 tmap i = record {
482
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
87 arrow = A [ arrow ( TMap ta i) o A [ {!!} o limit ( isLimit comp (FIA Γ ) ) (obj c) ( NIA Γ c ta ) ] ]
481
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
88 ; comm = {!!}
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
89 }
65e6906782bb Completeness of Comma Category begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
90 commute : {a b : Obj I} {f : Hom I a b} →
482
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
91 CommaCategory [ CommaCategory [ FMap Γ f o tmap a ] ≈ CommaCategory [ tmap b o FMap (K CommaCategory I c) f ] ]
481
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
92 commute {a} {b} {f} = {!!}
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
93
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
94
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
95 hasLimit : { I : Category c₁ c₂ ℓ } → ( comp : Complete A I )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
96 → ( G-preserve-limit : ( Γ : Functor I A )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
97 ( lim : Obj A ) ( ta : NTrans I A ( K A I lim ) Γ ) → ( limita : Limit A I Γ lim ta ) → Limit C I (G ○ Γ) (FObj G lim) (tb A I Γ lim ta G ) )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
98 → ( Γ : Functor I CommaCategory ) ( ta : NTrans I CommaCategory ( K CommaCategory I (commaLimit comp Γ) ) Γ )
482
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 481
diff changeset
99 → Limit CommaCategory I Γ (commaLimit comp Γ ) ( commaNat comp Γ (commaLimit comp Γ) ta )
481
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
100 hasLimit {I} comp gpresrve Γ ta = record {
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
101 limit = λ a t → {!!} ;
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
102 t0f=t = λ {a t i } → {!!} ;
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
103 limit-uniqueness = λ {a} {t} f t=f → {!!}
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
104 }