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annotate limit-to.agda @ 742:20f2700a481c

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nat-ε
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Tue, 05 Dec 2017 01:57:41 +0900
parents 917e51be9bbf
children 06388660995b
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
415
dd086f5fb29f same conflict again ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 414
diff changeset
1 open import Category -- https://github.com/konn/category-agda
350
c483374f542b try equalizer from limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
2 open import Level
c483374f542b try equalizer from limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
3
403
375edfefbf6a maybe CAT
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 402
diff changeset
4 module limit-to where
350
c483374f542b try equalizer from limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
5
c483374f542b try equalizer from limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
6 open import cat-utility
c483374f542b try equalizer from limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
7 open import HomReasoning
c483374f542b try equalizer from limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
8 open import Relation.Binary.Core
c483374f542b try equalizer from limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
9
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
10 open import discrete
427
531b739a1d7c add Nil
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 426
diff changeset
11
365
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
12
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
13 --- Equalizer from Limit ( 2→A IdnexFunctor Γ and IndexNat : K →　Γ)
458
fd79b6d9f350 fix comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 457
diff changeset
14 ---
fd79b6d9f350 fix comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 457
diff changeset
15 ---
387
2f59c2db9d98 what is this ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 386
diff changeset
16 --- f
431
a15f52456c78 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 430
diff changeset
17 --- e -----→
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
18 --- c -----→ a b A
431
a15f52456c78 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 430
diff changeset
19 --- ^ / -----→
387
2f59c2db9d98 what is this ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 386
diff changeset
20 --- |k h g
415
dd086f5fb29f same conflict again ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 414
diff changeset
21 --- | /
426
1290d6876129 Functor MA → A
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 425
diff changeset
22 --- | / ^
1290d6876129 Functor MA → A
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 425
diff changeset
23 --- | / |
1290d6876129 Functor MA → A
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 425
diff changeset
24 --- |/ | Γ
1290d6876129 Functor MA → A
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 425
diff changeset
25 --- d _ |
432
688066ac172e add another method as a comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 431
diff changeset
26 --- |\ |
688066ac172e add another method as a comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 431
diff changeset
27 --- \ K af
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
28 --- \ -----→
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
29 --- \ t0 t1 I
431
a15f52456c78 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 430
diff changeset
30 --- -----→
426
1290d6876129 Functor MA → A
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 425
diff changeset
31 --- ag
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
32 ---
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
33 ---
387
2f59c2db9d98 what is this ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 386
diff changeset
34
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
35 open Complete
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
36 open Limit
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c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
37 open IsLimit
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0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
38 open NTrans
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f589e71875ea bad approach
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 351
diff changeset
39
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99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
40 -- Functor Γ : TwoCat → A
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4329525f5085 fix imit-to-edu for MaybeCat / TwoCat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 423
diff changeset
41
461
8436a018f88a clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
42 IndexFunctor : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) ( a b : Obj A) ( f g : Hom A a b ) → Functor (TwoCat {c₁} {c₂}) A
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
43 IndexFunctor {c₁} {c₂} {ℓ} A a b f g = record {
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
44 FObj = λ a → fobj a
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
45 ; FMap = λ {a} {b} f → fmap {a} {b} f
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
46 ; isFunctor = record {
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
47 identity = λ{x} → identity x
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
48 ; distr = λ {a} {b} {c} {f} {g} → distr1 {a} {b} {c} {f} {g}
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
49 ; ≈-cong = λ {a} {b} {c} {f} → ≈-cong {a} {b} {c} {f}
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
50 }
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
51 } where
461
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
52 T = TwoCat {c₁} {c₂}
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
53 fobj : Obj T → Obj A
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
54 fobj t0 = a
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
55 fobj t1 = b
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
56 fmap : {x y : Obj T } → (Hom T x y ) → Hom A (fobj x) (fobj y)
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
57 fmap {t0} {t0} id-t0 = id1 A a
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
58 fmap {t1} {t1} id-t1 = id1 A b
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
59 fmap {t0} {t1} arrow-f = f
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
60 fmap {t0} {t1} arrow-g = g
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
61 ≈-cong : {a : Obj T} {b : Obj T} {f g : Hom T a b} → T [ f ≈ g ] → A [ fmap f ≈ fmap g ]
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
62 ≈-cong {a} {b} {f} {.f} refl = let open ≈-Reasoning A in refl-hom
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99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
63 identity : (x : Obj T ) → A [ fmap (id1 T x) ≈ id1 A (fobj x) ]
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
64 identity t0 = let open ≈-Reasoning A in refl-hom
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
65 identity t1 = let open ≈-Reasoning A in refl-hom
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
66 distr1 : {a : Obj T} {b : Obj T} {c : Obj T} {f : Hom T a b} {g : Hom T b c} →
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
67 A [ fmap (T [ g o f ]) ≈ A [ fmap g o fmap f ] ]
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
68 distr1 {t0} {t0} {t0} {id-t0 } { id-t0 } = let open ≈-Reasoning A in sym-hom idL
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
69 distr1 {t1} {t1} {t1} { id-t1 } { id-t1 } = let open ≈-Reasoning A in begin
467
ba042c2d3ff5 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 466
diff changeset
70 id b
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
71 ≈↑⟨ idL ⟩
467
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 466
diff changeset
72 id b o id b
457
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
73 ∎
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
74 distr1 {t0} {t0} {t1} { id-t0 } { arrow-f } = let open ≈-Reasoning A in begin
462
e618db534987 clean up fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 461
diff changeset
75 fmap (T [ arrow-f o id-t0 ] )
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
76 ≈⟨⟩
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
77 fmap arrow-f
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
78 ≈↑⟨ idR ⟩
467
ba042c2d3ff5 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 466
diff changeset
79 fmap arrow-f o id a
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
80 ∎
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
81 distr1 {t0} {t0} {t1} { id-t0 } { arrow-g } = let open ≈-Reasoning A in begin
462
e618db534987 clean up fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 461
diff changeset
82 fmap (T [ arrow-g o id-t0 ] )
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
83 ≈⟨⟩
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
84 fmap arrow-g
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
85 ≈↑⟨ idR ⟩
467
ba042c2d3ff5 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 466
diff changeset
86 fmap arrow-g o id a
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
87 ∎
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
88 distr1 {t0} {t1} {t1} { arrow-f } { id-t1 } = let open ≈-Reasoning A in begin
462
e618db534987 clean up fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 461
diff changeset
89 fmap (T [ id-t1 o arrow-f ] )
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
90 ≈⟨⟩
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
91 fmap arrow-f
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
92 ≈↑⟨ idL ⟩
467
ba042c2d3ff5 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 466
diff changeset
93 id b o fmap arrow-f
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
94 ∎
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
95 distr1 {t0} {t1} {t1} { arrow-g } { id-t1 } = let open ≈-Reasoning A in begin
462
e618db534987 clean up fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 461
diff changeset
96 fmap (T [ id-t1 o arrow-g ] )
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
97 ≈⟨⟩
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
98 fmap arrow-g
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
99 ≈↑⟨ idL ⟩
467
ba042c2d3ff5 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 466
diff changeset
100 id b o fmap arrow-g
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
101 ∎
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
102
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
103 --- Nat for Limit
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
104 --
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
105 -- Nat : K → IndexFunctor
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
106 --
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
107
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
108 open Functor
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
109
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
110 IndexNat : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ)
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
111 → {a b : Obj A} (f g : Hom A a b )
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
112 (d : Obj A) → (h : Hom A d a ) → A [ A [ f o h ] ≈ A [ g o h ] ] →
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 670
diff changeset
113 NTrans TwoCat A (K TwoCat A d) (IndexFunctor {c₁} {c₂} {ℓ} A a b f g)
460
961c236807f1 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 458
diff changeset
114 IndexNat {c₁} {c₂} {ℓ} A {a} {b} f g d h fh=gh = record {
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
115 TMap = λ x → nmap x d h ;
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
116 isNTrans = record {
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
117 commute = λ {x} {y} {f'} → commute1 {x} {y} {f'} d h fh=gh
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
118 }
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
119 } where
461
8436a018f88a clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
120 I = TwoCat {c₁} {c₂}
429
02eefa110eae nat commute in limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 428
diff changeset
121 Γ : Functor I A
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
122 Γ = IndexFunctor {c₁} {c₂} {ℓ} A a b f g
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 670
diff changeset
123 nmap : (x : Obj I ) ( d : Obj (A) ) (h : Hom A d a ) → Hom A (FObj (K I A d) x) (FObj Γ x)
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
124 nmap t0 d h = h
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
125 nmap t1 d h = A [ f o h ]
431
a15f52456c78 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 430
diff changeset
126 commute1 : {x y : Obj I} {f' : Hom I x y} (d : Obj A) (h : Hom A d a ) → A [ A [ f o h ] ≈ A [ g o h ] ]
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 670
diff changeset
127 → A [ A [ FMap Γ f' o nmap x d h ] ≈ A [ nmap y d h o FMap (K I A d) f' ] ]
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
128 commute1 {t0} {t1} {arrow-f} d h fh=gh = let open ≈-Reasoning A in begin
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
129 f o h
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
130 ≈↑⟨ idR ⟩
467
ba042c2d3ff5 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 466
diff changeset
131 (f o h ) o id d
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
132 ∎
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
133 commute1 {t0} {t1} {arrow-g} d h fh=gh = let open ≈-Reasoning A in begin
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
134 g o h
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
135 ≈↑⟨ fh=gh ⟩
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
136 f o h
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
137 ≈↑⟨ idR ⟩
467
ba042c2d3ff5 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 466
diff changeset
138 (f o h ) o id d
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
139 ∎
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
140 commute1 {t0} {t0} {id-t0} d h fh=gh = let open ≈-Reasoning A in begin
467
ba042c2d3ff5 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 466
diff changeset
141 id a o h
429
02eefa110eae nat commute in limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 428
diff changeset
142 ≈⟨ idL ⟩
02eefa110eae nat commute in limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 428
diff changeset
143 h
02eefa110eae nat commute in limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 428
diff changeset
144 ≈↑⟨ idR ⟩
467
ba042c2d3ff5 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 466
diff changeset
145 h o id d
429
02eefa110eae nat commute in limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 428
diff changeset
146 ∎
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
147 commute1 {t1} {t1} {id-t1} d h fh=gh = let open ≈-Reasoning A in begin
467
ba042c2d3ff5 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 466
diff changeset
148 id b o ( f o h )
429
02eefa110eae nat commute in limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 428
diff changeset
149 ≈⟨ idL ⟩
428
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 427
diff changeset
150 f o h
429
02eefa110eae nat commute in limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 428
diff changeset
151 ≈↑⟨ idR ⟩
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
152 ( f o h ) o id d
428
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 427
diff changeset
153 ∎
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
154
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
155
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 670
diff changeset
156 equlimit : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) {a b : Obj A} → (f g : Hom A a b) (lim : Limit TwoCat A (IndexFunctor A a b f g) ) →
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
157 Hom A (a0 lim) a
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
158 equlimit A {a} {b} f g lim = TMap (Limit.t0 lim) discrete.t0
460
961c236807f1 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 458
diff changeset
159
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
160 lim-to-equ : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ)
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
161 → {a b : Obj A} (f g : Hom A a b )
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 670
diff changeset
162 (lim : Limit TwoCat A (IndexFunctor A a b f g) )
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
163 → IsEqualizer A (equlimit A f g lim) f g
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
164 lim-to-equ {c₁} {c₂} {ℓ} A {a} {b} f g lim = record {
601
2e7b5a777984 prove fe=ge in limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 508
diff changeset
165 fe=ge = fe=ge0
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
166 ; k = λ {d} h fh=gh → k {d} h fh=gh
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
167 ; ek=h = λ {d} {h} {fh=gh} → ek=h d h fh=gh
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
168 ; uniqueness = λ {d} {h} {fh=gh} {k'} → uniquness d h fh=gh k'
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
169 } where
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
170 I : Category c₁ c₂ c₂
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
171 I = TwoCat
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
172 Γ : Functor I A
461
8436a018f88a clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 460
diff changeset
173 Γ = IndexFunctor A a b f g
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
174 e : Hom A (a0 lim) a
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
175 e = equlimit A f g lim
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
176 c : Obj A
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
177 c = a0 lim
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 670
diff changeset
178 inat : (d : Obj A) (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → NTrans I A (K I A d) Γ
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
179 inat = IndexNat A f g
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
180 fe=ge0 : A [ A [ f o (equlimit A f g lim ) ] ≈ A [ g o (equlimit A f g lim ) ] ]
601
2e7b5a777984 prove fe=ge in limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 508
diff changeset
181 fe=ge0 = let open ≈-Reasoning A in begin
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
182 f o (equlimit A f g lim )
601
2e7b5a777984 prove fe=ge in limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 508
diff changeset
183 ≈⟨⟩
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
184 FMap Γ arrow-f o TMap (Limit.t0 lim) discrete.t0
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
185 ≈⟨ IsNTrans.commute ( isNTrans (Limit.t0 lim)) {discrete.t0} {discrete.t1} {arrow-f} ⟩
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 670
diff changeset
186 TMap (Limit.t0 lim) discrete.t1 o FMap (K (TwoCat {c₁} {c₂} ) A (a0 lim)) id-t0
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
187 ≈↑⟨ IsNTrans.commute ( isNTrans (Limit.t0 lim)) {discrete.t0} {discrete.t1} {arrow-g} ⟩
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
188 FMap Γ arrow-g o TMap (Limit.t0 lim) discrete.t0
601
2e7b5a777984 prove fe=ge in limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 508
diff changeset
189 ≈⟨⟩
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
190 g o (equlimit A f g lim )
601
2e7b5a777984 prove fe=ge in limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 508
diff changeset
191 ∎
350
c483374f542b try equalizer from limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
192 k : {d : Obj A} (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d c
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
193 k {d} h fh=gh = limit (isLimit lim) d (inat d h fh=gh )
431
a15f52456c78 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 430
diff changeset
194 ek=h : (d : Obj A ) (h : Hom A d a ) → ( fh=gh : A [ A [ f o h ] ≈ A [ g o h ] ] ) → A [ A [ e o k h fh=gh ] ≈ h ]
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
195 ek=h d h fh=gh = let open ≈-Reasoning A in begin
430
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
196 e o k h fh=gh
460
961c236807f1 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 458
diff changeset
197 ≈⟨⟩
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
198 TMap (Limit.t0 lim) discrete.t0 o k h fh=gh
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
199 ≈⟨ t0f=t (isLimit lim) {d} {inat d h fh=gh } {discrete.t0} ⟩
484
fcae3025d900 fix Limit pu a0 and t0 in record definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 474
diff changeset
200 TMap (inat d h fh=gh) discrete.t0
460
961c236807f1 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 458
diff changeset
201 ≈⟨⟩
430
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
202 h
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
203 ∎
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
204 uniq-nat : {i : Obj I} → (d : Obj A ) (h : Hom A d a ) ( k' : Hom A d c )
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
205 ( fh=gh : A [ A [ f o h ] ≈ A [ g o h ] ]) → A [ A [ e o k' ] ≈ h ] →
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
206 A [ A [ TMap (Limit.t0 lim) i o k' ] ≈ TMap (inat d h fh=gh) i ]
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
207 uniq-nat {t0} d h k' fh=gh ek'=h = let open ≈-Reasoning A in begin
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
208 TMap (Limit.t0 lim) discrete.t0 o k'
430
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
209 ≈⟨⟩
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
210 e o k'
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
211 ≈⟨ ek'=h ⟩
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
212 h
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
213 ≈⟨⟩
484
fcae3025d900 fix Limit pu a0 and t0 in record definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 474
diff changeset
214 TMap (inat d h fh=gh) discrete.t0
430
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
215 ∎
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
216 uniq-nat {t1} d h k' fh=gh ek'=h = let open ≈-Reasoning A in begin
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
217 TMap (Limit.t0 lim) t1 o k'
460
961c236807f1 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 458
diff changeset
218 ≈↑⟨ car (idR) ⟩
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
219 ( TMap (Limit.t0 lim) t1 o id c ) o k'
460
961c236807f1 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 458
diff changeset
220 ≈⟨⟩
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 670
diff changeset
221 ( TMap (Limit.t0 lim) t1 o FMap (K I A c) arrow-f ) o k'
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
222 ≈↑⟨ car ( nat1 (Limit.t0 lim) arrow-f ) ⟩
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
223 ( FMap Γ arrow-f o TMap (Limit.t0 lim) discrete.t0 ) o k'
430
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
224 ≈⟨⟩
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
225 (f o e ) o k'
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
226 ≈↑⟨ assoc ⟩
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
227 f o ( e o k' )
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
228 ≈⟨ cdr ek'=h ⟩
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
229 f o h
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
230 ≈⟨⟩
466
44bd77c80555 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 462
diff changeset
231 TMap (inat d h fh=gh) t1
430
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
232 ∎
431
a15f52456c78 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 430
diff changeset
233 uniquness : (d : Obj A ) (h : Hom A d a ) → ( fh=gh : A [ A [ f o h ] ≈ A [ g o h ] ] ) →
372
b4855a3ebd34 add more lemma in limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 371
diff changeset
234 ( k' : Hom A d c )
457
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
235 → A [ A [ e o k' ] ≈ h ] → A [ k h fh=gh ≈ k' ]
0ba86e29f492 limit-to and discrete clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 443
diff changeset
236 uniquness d h fh=gh k' ek'=h = let open ≈-Reasoning A in begin
430
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
237 k h fh=gh
495
633df882db86 fryed1 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
238 ≈⟨ limit-uniqueness (isLimit lim) ( λ{i} → uniq-nat {i} d h k' fh=gh ek'=h ) ⟩
430
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
239 k'
46c057cb3d82 limit-to done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 429
diff changeset
240 ∎
368
b18585089d2e add more parameter to nat in lim-to-equ
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 367
diff changeset
241
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
242
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
243 --- Product from Limit ( given Discrete→A functor Γ and pnat : K →　Γ)
469
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
244
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
245 open DiscreteHom
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
246
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
247 plimit : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (S : Set c₁)
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 670
diff changeset
248 → ( Γ : Functor (DiscreteCat S ) A ) → (lim : Limit ( DiscreteCat S ) A Γ ) → Obj A
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
249 plimit A S Γ lim = a0 lim
469
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
250
472
f3d6d0275a0a discrete equality as a dom equality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 469
diff changeset
251 discrete-identity : { c₁ : Level} { S : Set c₁} { a : DiscreteObj {c₁} S } → (f : DiscreteHom a a ) → (DiscreteCat S) [ f ≈ id1 (DiscreteCat S) a ]
f3d6d0275a0a discrete equality as a dom equality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 469
diff changeset
252 discrete-identity f = refl
f3d6d0275a0a discrete equality as a dom equality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 469
diff changeset
253
474
2d32ded94aaf clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 472
diff changeset
254 pnat : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (S : Set c₁)
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 670
diff changeset
255 → (Γ : Functor (DiscreteCat S) A )
474
2d32ded94aaf clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 472
diff changeset
256 → {q : Obj A } ( qi : (i : Obj ( DiscreteCat S)) → Hom A q (FObj Γ i) )
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 670
diff changeset
257 → NTrans (DiscreteCat S )A (K (DiscreteCat S) A q) Γ
474
2d32ded94aaf clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 472
diff changeset
258 pnat A S Γ {q} qi = record {
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
259 TMap = qi ;
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
260 isNTrans = record { commute = λ {a} {b} {f} → commute {a} {b} {f} }
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
261 } where
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 670
diff changeset
262 commute : {a b : Obj (DiscreteCat S) } {f : Hom (DiscreteCat S) a b} →
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 670
diff changeset
263 A [ A [ FMap Γ f o qi a ] ≈ A [ qi b o FMap (K (DiscreteCat S) A q) f ] ]
469
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
264 commute {a} {b} {f} with discrete f
472
f3d6d0275a0a discrete equality as a dom equality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 469
diff changeset
265 commute {a} {.a} {f} | refl = let open ≈-Reasoning A in begin
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
266 FMap Γ f o qi a
472
f3d6d0275a0a discrete equality as a dom equality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 469
diff changeset
267 ≈⟨ car ( fcong Γ (discrete-identity f )) ⟩
f3d6d0275a0a discrete equality as a dom equality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 469
diff changeset
268 FMap Γ (id1 (DiscreteCat S) a ) o qi a
f3d6d0275a0a discrete equality as a dom equality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 469
diff changeset
269 ≈⟨ car ( IsFunctor.identity (isFunctor Γ) ) ⟩
f3d6d0275a0a discrete equality as a dom equality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 469
diff changeset
270 id1 A (FObj Γ a) o qi a
f3d6d0275a0a discrete equality as a dom equality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 469
diff changeset
271 ≈⟨ idL ⟩
f3d6d0275a0a discrete equality as a dom equality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 469
diff changeset
272 qi a
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
273 ≈↑⟨ idR ⟩
472
f3d6d0275a0a discrete equality as a dom equality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 469
diff changeset
274 qi a o id q
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
275 ≈⟨⟩
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 670
diff changeset
276 qi a o FMap (K (DiscreteCat S) A q) f
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
277 ∎
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
278
469
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
279 lim-to-product : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) ( S : Set c₁ )
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 670
diff changeset
280 → ( Γ : Functor (DiscreteCat S) A )
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 670
diff changeset
281 → (lim : Limit (DiscreteCat S) A Γ )
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 670
diff changeset
282 → IProduct (Obj (DiscreteCat S)) A (FObj Γ)
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
283 lim-to-product A S Γ lim = record {
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 670
diff changeset
284 iprod = plimit A S Γ lim
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
285 ; pi = λ i → TMap (Limit.t0 lim) i
508
3ce21b2a671a IProduct is written in Sets
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 495
diff changeset
286 ; isIProduct = record {
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
287 iproduct = λ {q} qi → iproduct {q} qi ;
691
917e51be9bbf change argument of Limit and K
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 670
diff changeset
288 pif=q = λ {q} {qi} {i} → pif=q {q} qi {i} ;
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
289 ip-uniqueness = λ {q } { h } → ip-uniqueness {q} {h} ;
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
290 ip-cong = λ {q } { qi } { qi' } qi=qi' → ip-cong {q} {qi} {qi'} qi=qi'
508
3ce21b2a671a IProduct is written in Sets
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 495
diff changeset
291 }
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
292 } where
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
293 D = DiscreteCat S
469
65ab0da524b8 discrete f ≡ refl should be passed, but it doesn't
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 468
diff changeset
294 I = Obj ( DiscreteCat S )
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
295 ai = λ i → FObj Γ i
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
296 p = a0 lim
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
297 pi = λ i → TMap (Limit.t0 lim) i
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
298 iproduct : {q : Obj A} → ( qi : (i : I) → Hom A q (ai i) ) → Hom A q p
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
299 iproduct {q} qi = limit (isLimit lim) q (pnat A S Γ qi )
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
300 pif=q : {q : Obj A} → ( qi : (i : I) → Hom A q (ai i) ) → ∀ { i : I } → A [ A [ ( pi i ) o ( iproduct qi ) ] ≈ (qi i) ]
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
301 pif=q {q} qi {i} = t0f=t (isLimit lim) {q} {pnat A S Γ qi } {i}
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
302 ipu : {i : Obj D} → (q : Obj A) (h : Hom A q p ) → A [ A [ TMap (Limit.t0 lim) i o h ] ≈ A [ pi i o h ] ]
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
303 ipu {i} q h = let open ≈-Reasoning A in refl-hom
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
304 ip-uniqueness : {q : Obj A} { h : Hom A q p } → A [ iproduct ( λ (i : I) → A [ (pi i) o h ] ) ≈ h ]
495
633df882db86 fryed1 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
305 ip-uniqueness {q} {h} = limit-uniqueness (isLimit lim) {q} {pnat A S Γ (λ i → A [ pi i o h ] )} (ipu q h)
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
306 ipc : {q : Obj A} → { qi : (i : I) → Hom A q (ai i) } → { qi' : (i : I) → Hom A q (ai i) }
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
307 → (i : I ) → A [ qi i ≈ qi' i ] →
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
308 A [ A [ TMap (Limit.t0 lim) i o iproduct qi' ] ≈ TMap (pnat A S Γ qi) i ]
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
309 ipc {q} {qi} {qi'} i qi=qi' = let open ≈-Reasoning A in begin
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
310 TMap (Limit.t0 lim) i o iproduct qi'
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
311 ≈⟨⟩
670
99065a1e56ea remove comp from limit-to
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 662
diff changeset
312 TMap (Limit.t0 lim) i o limit (isLimit lim) q (pnat A S Γ qi' )
487
c257347a27f3 found limit in freyd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 484
diff changeset
313 ≈⟨ t0f=t (isLimit lim) {q} {pnat A S Γ qi'} {i} ⟩
474
2d32ded94aaf clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 472
diff changeset
314 TMap (pnat A S Γ qi') i
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
315 ≈⟨⟩
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
316 qi' i
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
317 ≈↑⟨ qi=qi' ⟩
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
318 qi i
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
319 ≈⟨⟩
474
2d32ded94aaf clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 472
diff changeset
320 TMap (pnat A S Γ qi) i
468
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
321 ∎
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
322 ip-cong : {q : Obj A} → { qi : (i : I) → Hom A q (ai i) } → { qi' : (i : I) → Hom A q (ai i) }
c375d8f93a2c discrete category and product from a limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 467
diff changeset
323 → ( ∀ (i : I ) → A [ qi i ≈ qi' i ] ) → A [ iproduct qi ≈ iproduct qi' ]
495
633df882db86 fryed1 done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 487
diff changeset
324 ip-cong {q} {qi} {qi'} qi=qi' = limit-uniqueness (isLimit lim) {q} {pnat A S Γ qi} (λ {i} → ipc {q} {qi} {qi'} i (qi=qi' i))