Mercurial > hg > Members > kono > Proof > category
annotate equalizer.agda @ 242:5d1ecfec6f41
fix
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
---|---|
date | Sun, 08 Sep 2013 17:47:04 +0900 |
parents | 9e4dc349831e |
children | 40ac16f69708 |
rev | line source |
---|---|
205 | 1 --- |
2 -- | |
3 -- Equalizer | |
4 -- | |
208 | 5 -- e f |
205 | 6 -- c --------> a ----------> b |
208 | 7 -- ^ . ----------> |
205 | 8 -- | . g |
230 | 9 -- |k . |
10 -- | . h | |
11 -- d | |
205 | 12 -- |
13 -- Shinji KONO <kono@ie.u-ryukyu.ac.jp> | |
14 ---- | |
15 | |
230 | 16 open import Category -- https://github.com/konn/category-agda |
205 | 17 open import Level |
18 module equalizer { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } where | |
19 | |
20 open import HomReasoning | |
21 open import cat-utility | |
22 | |
233
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
23 record Equalizer { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {c a b : Obj A} (e : Hom A c a) (f g : Hom A a b) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where |
205 | 24 field |
221 | 25 fe=ge : A [ A [ f o e ] ≈ A [ g o e ] ] |
209 | 26 k : {d : Obj A} (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d c |
215 | 27 ek=h : {d : Obj A} → ∀ {h : Hom A d a} → {eq : A [ A [ f o h ] ≈ A [ g o h ] ] } → A [ A [ e o k {d} h eq ] ≈ h ] |
230 | 28 uniqueness : {d : Obj A} → ∀ {h : Hom A d a} → {eq : A [ A [ f o h ] ≈ A [ g o h ] ] } → {k' : Hom A d c } → |
214 | 29 A [ A [ e o k' ] ≈ h ] → A [ k {d} h eq ≈ k' ] |
209 | 30 equalizer : Hom A c a |
31 equalizer = e | |
206 | 32 |
230 | 33 -- |
225 | 34 -- Flat Equational Definition of Equalizer |
230 | 35 -- |
236
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
36 record Burroni { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {c a b : Obj A} (f g : Hom A a b) (e : Hom A c a) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where |
206 | 37 field |
236
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
38 α : {a b c : Obj A } → (f : Hom A a b) → (g : Hom A a b ) → {e : Hom A c a } → Hom A c a |
214 | 39 γ : {a b c d : Obj A } → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → Hom A d c |
236
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
40 δ : {a b c : Obj A } → {e : Hom A c a } → (f : Hom A a b) → Hom A a c |
242 | 41 cong-α : {a b c : Obj A } → { e : Hom A c a } |
42 → {f g g' : Hom A a b } → A [ g ≈ g' ] → A [ α f g {e} ≈ α f g' {e} ] | |
43 cong-γ : {a _ c d : Obj A } → {f g : Hom A a b} {h h' : Hom A d a } → A [ h ≈ h' ] | |
44 → { γ γ' : {a b c d : Obj A } → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → Hom A d c } | |
45 → A [ γ f g h ≈ γ' f g h' ] | |
240 | 46 cong-δ : {a b c : Obj A } → {f f' : Hom A a b} → A [ f ≈ f' ] → { δ δ' : Hom A a c } → A [ δ ≈ δ' ] |
238
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
47 b1 : A [ A [ f o α {a} {b} {c} f g {e} ] ≈ A [ g o α {a} {b} {c} f g {e} ] ] |
236
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
48 b2 : {d : Obj A } → {h : Hom A d a } → A [ A [ ( α {a} {b} {c} f g {e} ) o (γ {a} {b} {c} f g h) ] ≈ A [ h o α (A [ f o h ]) (A [ g o h ]){id1 A d} ] ] |
240 | 49 b3 : {a b d : Obj A} → (f : Hom A a b ) → {h : Hom A d a } → A [ A [ α {a} {b} {d} f f {h} o δ {a} {b} {d} {h} f ] ≈ id1 A a ] |
207
22811f7a04e1
Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
206
diff
changeset
|
50 -- b4 : {c d : Obj A } {k : Hom A c a} → A [ β f g ( A [ α f g o k ] ) ≈ k ] |
236
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
51 b4 : {d : Obj A } {k : Hom A d c} → |
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
52 A [ A [ γ {a} {b} {c} {d} f g ( A [ α {a} {b} {c} f g {e} o k ] ) o ( δ {d} {b} {d} {id1 A d} (A [ f o A [ α {a} {b} {c} f g {e} o k ] ] ) )] ≈ k ] |
207
22811f7a04e1
Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
206
diff
changeset
|
53 -- A [ α f g o β f g h ] ≈ h |
238
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
54 β : { d a b : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → Hom A d c |
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
55 β {d} {a} {b} f g h = A [ γ {a} {b} {c} f g h o δ {d} {b} {d} {id1 A d} (A [ f o h ]) ] |
207
22811f7a04e1
Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
206
diff
changeset
|
56 |
209 | 57 open Equalizer |
225 | 58 open Burroni |
209 | 59 |
225 | 60 -- |
61 -- Some obvious conditions for k (fe = ge) → ( fh = gh ) | |
62 -- | |
219 | 63 |
224 | 64 f1=g1 : { a b c : Obj A } {f g : Hom A a b } → (eq : A [ f ≈ g ] ) → (h : Hom A c a) → A [ A [ f o h ] ≈ A [ g o h ] ] |
65 f1=g1 eq h = let open ≈-Reasoning (A) in (resp refl-hom eq ) | |
66 | |
226 | 67 f1=f1 : { a b : Obj A } (f : Hom A a b ) → A [ A [ f o (id1 A a) ] ≈ A [ f o (id1 A a) ] ] |
230 | 68 f1=f1 f = let open ≈-Reasoning (A) in refl-hom |
226 | 69 |
224 | 70 f1=gh : { a b c d : Obj A } {f g : Hom A a b } → { e : Hom A c a } → { h : Hom A d c } → |
71 (eq : A [ A [ f o e ] ≈ A [ g o e ] ] ) → A [ A [ f o A [ e o h ] ] ≈ A [ g o A [ e o h ] ] ] | |
230 | 72 f1=gh {a} {b} {c} {d} {f} {g} {e} {h} eq = let open ≈-Reasoning (A) in |
224 | 73 begin |
74 f o ( e o h ) | |
75 ≈⟨ assoc ⟩ | |
230 | 76 (f o e ) o h |
224 | 77 ≈⟨ car eq ⟩ |
230 | 78 (g o e ) o h |
224 | 79 ≈↑⟨ assoc ⟩ |
80 g o ( e o h ) | |
81 ∎ | |
219 | 82 |
225 | 83 -- |
84 -- | |
85 -- An isomorphic element c' of c makes another equalizer | |
86 -- | |
230 | 87 -- e eqa f g f |
222 | 88 -- c ----------> a ------->b |
230 | 89 -- |^ |
90 -- || | |
222 | 91 -- h || h-1 |
230 | 92 -- v| |
93 -- c' | |
222 | 94 |
234 | 95 equalizer+iso : {a b c c' : Obj A } {f g : Hom A a b } {e : Hom A c a } |
96 (h-1 : Hom A c' c ) → (h : Hom A c c' ) → | |
97 A [ A [ h o h-1 ] ≈ id1 A c' ] → A [ A [ h-1 o h ] ≈ id1 A c ] → | |
98 ( fe=ge' : A [ A [ f o A [ e o h-1 ] ] ≈ A [ g o A [ e o h-1 ] ] ] ) | |
99 ( eqa : Equalizer A e f g ) | |
100 → Equalizer A (A [ e o h-1 ] ) f g | |
101 equalizer+iso {a} {b} {c} {c'} {f} {g} {e} h-1 h hh-1=1 h-1h=1 fe=ge' eqa = record { | |
222 | 102 fe=ge = fe=ge1 ; |
103 k = λ j eq → A [ h o k eqa j eq ] ; | |
230 | 104 ek=h = ek=h1 ; |
222 | 105 uniqueness = uniqueness1 |
106 } where | |
234 | 107 fe=ge1 : A [ A [ f o A [ e o h-1 ] ] ≈ A [ g o A [ e o h-1 ] ] ] |
108 fe=ge1 = fe=ge' | |
222 | 109 ek=h1 : {d : Obj A} {j : Hom A d a} {eq : A [ A [ f o j ] ≈ A [ g o j ] ]} → |
234 | 110 A [ A [ A [ e o h-1 ] o A [ h o k eqa j eq ] ] ≈ j ] |
222 | 111 ek=h1 {d} {j} {eq} = let open ≈-Reasoning (A) in |
112 begin | |
234 | 113 ( e o h-1 ) o ( h o k eqa j eq ) |
114 ≈↑⟨ assoc ⟩ | |
115 e o ( h-1 o ( h o k eqa j eq ) ) | |
116 ≈⟨ cdr assoc ⟩ | |
117 e o (( h-1 o h) o k eqa j eq ) | |
118 ≈⟨ cdr (car h-1h=1 ) ⟩ | |
119 e o (id1 A c o k eqa j eq ) | |
120 ≈⟨ cdr idL ⟩ | |
121 e o k eqa j eq | |
222 | 122 ≈⟨ ek=h eqa ⟩ |
123 j | |
124 ∎ | |
125 uniqueness1 : {d : Obj A} {h' : Hom A d a} {eq : A [ A [ f o h' ] ≈ A [ g o h' ] ]} {j : Hom A d c'} → | |
234 | 126 A [ A [ A [ e o h-1 ] o j ] ≈ h' ] → |
222 | 127 A [ A [ h o k eqa h' eq ] ≈ j ] |
128 uniqueness1 {d} {h'} {eq} {j} ej=h = let open ≈-Reasoning (A) in | |
129 begin | |
130 h o k eqa h' eq | |
234 | 131 ≈⟨ cdr (uniqueness eqa ( begin |
132 e o ( h-1 o j ) | |
133 ≈⟨ assoc ⟩ | |
134 (e o h-1 ) o j | |
135 ≈⟨ ej=h ⟩ | |
136 h' | |
137 ∎ )) ⟩ | |
138 h o ( h-1 o j ) | |
139 ≈⟨ assoc ⟩ | |
140 (h o h-1 ) o j | |
141 ≈⟨ car hh-1=1 ⟩ | |
142 id1 A c' o j | |
143 ≈⟨ idL ⟩ | |
222 | 144 j |
145 ∎ | |
146 | |
225 | 147 -- |
148 -- If we have two equalizers on c and c', there are isomorphic pair h, h' | |
149 -- | |
150 -- h : c → c' h' : c' → c | |
233
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
151 -- e' = h o e |
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
152 -- e = h' o e' |
225 | 153 |
154 | |
233
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
155 c-iso-l : { c c' a b : Obj A } {f g : Hom A a b } → {e : Hom A c a } { e' : Hom A c' a } |
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
156 ( eqa : Equalizer A e f g) → ( eqa' : Equalizer A e' f g ) |
234 | 157 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) |
241 | 158 → Hom A c c' -- should be e' = c-sio-l o e |
234 | 159 c-iso-l {c} {c'} eqa eqa' keqa = equalizer keqa |
226 | 160 |
233
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
161 c-iso-r : { c c' a b : Obj A } {f g : Hom A a b } {e : Hom A c a } {e' : Hom A c' a} → ( eqa : Equalizer A e f g) → ( eqa' : Equalizer A e' f g ) |
234 | 162 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) |
241 | 163 → Hom A c' c -- e = c-sio-r o e' |
230 | 164 c-iso-r {c} {c'} eqa eqa' keqa = k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) |
223 | 165 |
234 | 166 -- e' f |
230 | 167 -- c'----------> a ------->b f e j = g e j |
168 -- ^ g | |
169 -- |k h | |
229
68b2681ea9df
c in equalizer is equal up to iso done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
228
diff
changeset
|
170 -- | h = e(eqaj) o k jhek = jh (uniqueness) |
230 | 171 -- | |
172 -- c j o (k (eqa ef ef) j ) = id c h = e(eqaj) | |
173 -- | |
174 -- h j e f = h j e g → h = 'j e f | |
229
68b2681ea9df
c in equalizer is equal up to iso done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
228
diff
changeset
|
175 -- h = j e f -> j = j' |
68b2681ea9df
c in equalizer is equal up to iso done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
228
diff
changeset
|
176 -- |
228 | 177 |
241 | 178 -- e = c-iso-l o e' is assumed by equalizer's degree of freedom |
234 | 179 c-iso→ : { c c' a b : Obj A } {f g : Hom A a b } → {e : Hom A c a } {e' : Hom A c' a} ( eqa : Equalizer A e f g) → ( eqa' : Equalizer A e' f g ) |
180 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) | |
229
68b2681ea9df
c in equalizer is equal up to iso done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
228
diff
changeset
|
181 → A [ A [ c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa ] ≈ id1 A c' ] |
234 | 182 c-iso→ {c} {c'} {a} {b} {f} {g} eqa eqa' keqa = let open ≈-Reasoning (A) in begin |
229
68b2681ea9df
c in equalizer is equal up to iso done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
228
diff
changeset
|
183 c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa |
234 | 184 ≈⟨⟩ |
185 equalizer keqa o k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) | |
186 ≈⟨ ek=h keqa ⟩ | |
229
68b2681ea9df
c in equalizer is equal up to iso done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
228
diff
changeset
|
187 id1 A c' |
68b2681ea9df
c in equalizer is equal up to iso done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
228
diff
changeset
|
188 ∎ |
226 | 189 |
234 | 190 c-iso← : { c c' a b : Obj A } {f g : Hom A a b } → {e : Hom A c a } {e' : Hom A c' a} ( eqa : Equalizer A e f g) → ( eqa' : Equalizer A e' f g ) |
191 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa )) (A [ f o e' ]) (A [ g o e' ]) ) | |
192 → ( keqa' : Equalizer A (k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') )) (A [ f o e ]) (A [ g o e ]) ) | |
241 | 193 -- e' = c-iso-r o e is assumed by equalizer's degree of freedom |
194 → { e'->e : A [ e' ≈ A [ e o equalizer keqa' ] ] } -- implicit assumption , which should be refl | |
234 | 195 → A [ A [ c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa ] ≈ id1 A c ] |
196 c-iso← {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' keqa keqa' {e'->e} = let open ≈-Reasoning (A) in begin | |
197 c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa | |
198 ≈⟨⟩ | |
199 k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) o k eqa' e (fe=ge eqa ) | |
200 ≈⟨⟩ | |
201 equalizer keqa' o k eqa' e (fe=ge eqa ) | |
202 ≈⟨ cdr ( begin | |
203 k eqa' e (fe=ge eqa ) | |
204 ≈⟨ uniqueness eqa' ( begin | |
205 e' o k keqa' (id1 A c) (f1=g1 (fe=ge eqa) (id1 A c)) | |
206 ≈⟨ car e'->e ⟩ | |
207 ( e o equalizer keqa' ) o k keqa' (id1 A c) (f1=g1 (fe=ge eqa) (id1 A c)) | |
208 ≈↑⟨ assoc ⟩ | |
209 e o ( equalizer keqa' o k keqa' (id1 A c) (f1=g1 (fe=ge eqa) (id1 A c))) | |
210 ≈⟨ cdr ( ek=h keqa' ) ⟩ | |
211 e o id1 A c | |
212 ≈⟨ idR ⟩ | |
213 e | |
214 ∎ )⟩ | |
215 k keqa' (id1 A c) ( f1=g1 (fe=ge eqa) (id1 A c) ) | |
216 ∎ )⟩ | |
217 equalizer keqa' o k keqa' (id1 A c) ( f1=g1 (fe=ge eqa) (id1 A c) ) | |
218 ≈⟨ ek=h keqa' ⟩ | |
219 id1 A c | |
220 ∎ | |
221 | |
222 | |
230 | 223 |
225 | 224 ---- |
225 -- | |
226 -- An equalizer satisfies Burroni equations | |
227 -- | |
241 | 228 -- congs are not yet done |
225 | 229 ---- |
230 | |
236
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
231 lemma-equ1 : {a b c : Obj A} (f g : Hom A a b) → (e : Hom A c a ) → |
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
232 ( eqa : {a b c : Obj A} → (f g : Hom A a b) → {e : Hom A c a } → Equalizer A e f g ) |
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
233 → Burroni A {c} {a} {b} f g e |
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
234 lemma-equ1 {a} {b} {c} f g e eqa = record { |
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
235 α = λ {a} {b} {c} f g {e} → equalizer (eqa {a} {b} {c} f g {e} ) ; -- Hom A c a |
242 | 236 γ = λ {a} {b} {c} {d} f g h → k (eqa f g ) {d} ( A [ h o (equalizer ( eqa (A [ f o h ] ) (A [ g o h ] ))) ] ) |
237 (lemma-equ4 {a} {b} {c} {d} f g h ) ; -- Hom A c d | |
236
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
238 δ = λ {a} {b} {c} {e} f → k (eqa {a} {b} {c} f f {e} ) (id1 A a) (lemma-equ2 f); -- Hom A a c |
240 | 239 cong-α = cong-α1 ; |
240 cong-γ = cong-γ1 ; | |
241 cong-δ = cong-δ1 ; | |
238
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
242 b1 = fe=ge (eqa {a} {b} {c} f g {e}) ; |
226 | 243 b2 = lemma-b2 ; |
244 b3 = lemma-b3 ; | |
230 | 245 b4 = lemma-b4 |
211 | 246 } where |
216 | 247 -- |
248 -- e eqa f g f | |
249 -- c ----------> a ------->b | |
230 | 250 -- ^ g |
251 -- | | |
216 | 252 -- |k₁ = e eqa (f o (e (eqa f g))) (g o (e (eqa f g)))) |
230 | 253 -- | |
216 | 254 -- d |
230 | 255 -- |
256 -- | |
216 | 257 -- e o id1 ≈ e → k e ≈ id |
258 | |
211 | 259 lemma-equ2 : {a b : Obj A} (f : Hom A a b) → A [ A [ f o id1 A a ] ≈ A [ f o id1 A a ] ] |
260 lemma-equ2 f = let open ≈-Reasoning (A) in refl-hom | |
240 | 261 lemma-b3 : {a b d : Obj A} (f : Hom A a b ) { h : Hom A d a } → A [ A [ equalizer (eqa f f ) o k (eqa f f) (id1 A a) (lemma-equ2 f) ] ≈ id1 A a ] |
262 lemma-b3 {a} {b} {d} f {h} = let open ≈-Reasoning (A) in | |
230 | 263 begin |
233
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
264 equalizer (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) |
215 | 265 ≈⟨ ek=h (eqa f f ) ⟩ |
211 | 266 id1 A a |
267 ∎ | |
230 | 268 lemma-equ4 : {a b c d : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → |
233
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
269 A [ A [ f o A [ h o equalizer (eqa (A [ f o h ]) (A [ g o h ])) ] ] ≈ A [ g o A [ h o equalizer (eqa (A [ f o h ]) (A [ g o h ])) ] ] ] |
214 | 270 lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in |
212 | 271 begin |
233
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
272 f o ( h o equalizer (eqa (f o h) ( g o h ))) |
212 | 273 ≈⟨ assoc ⟩ |
233
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
274 (f o h) o equalizer (eqa (f o h) ( g o h )) |
221 | 275 ≈⟨ fe=ge (eqa (A [ f o h ]) (A [ g o h ])) ⟩ |
233
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
276 (g o h) o equalizer (eqa (f o h) ( g o h )) |
212 | 277 ≈↑⟨ assoc ⟩ |
233
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
278 g o ( h o equalizer (eqa (f o h) ( g o h ))) |
212 | 279 ∎ |
230 | 280 lemma-b2 : {d : Obj A} {h : Hom A d a} → A [ |
233
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
281 A [ equalizer (eqa f g) o k (eqa f g) (A [ h o equalizer (eqa (A [ f o h ]) (A [ g o h ])) ]) (lemma-equ4 {a} {b} {c} f g h) ] |
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
282 ≈ A [ h o equalizer (eqa (A [ f o h ]) (A [ g o h ])) ] ] |
226 | 283 lemma-b2 {d} {h} = let open ≈-Reasoning (A) in |
212 | 284 begin |
233
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
285 equalizer (eqa f g) o k (eqa f g) (h o equalizer (eqa (f o h) (g o h))) (lemma-equ4 {a} {b} {c} f g h) |
215 | 286 ≈⟨ ek=h (eqa f g) ⟩ |
233
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
287 h o equalizer (eqa (f o h ) ( g o h )) |
212 | 288 ∎ |
230 | 289 |
290 lemma-b4 : {d : Obj A} {j : Hom A d c} → A [ | |
236
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
291 A [ k (eqa f g) (A [ A [ equalizer (eqa f g) o j ] o |
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
292 equalizer (eqa (A [ f o A [ equalizer (eqa f g {e}) o j ] ]) (A [ g o A [ equalizer (eqa f g {e} ) o j ] ])) ]) |
233
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
293 (lemma-equ4 {a} {b} {c} f g (A [ equalizer (eqa f g) o j ])) o |
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
294 k (eqa (A [ f o A [ equalizer (eqa f g) o j ] ]) (A [ f o A [ equalizer (eqa f g) o j ] ])) (id1 A d) (lemma-equ2 (A [ f o A [ equalizer (eqa f g) o j ] ])) ] |
222 | 295 ≈ j ] |
230 | 296 lemma-b4 {d} {j} = let open ≈-Reasoning (A) in |
215 | 297 begin |
236
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
298 ( k (eqa f g) (( ( equalizer (eqa f g) o j ) o equalizer (eqa (( f o ( equalizer (eqa f g {e}) o j ) )) (( g o ( equalizer (eqa f g {e}) o j ) ))) )) |
233
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
299 (lemma-equ4 {a} {b} {c} f g (( equalizer (eqa f g) o j ))) o |
236
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
300 k (eqa (( f o ( equalizer (eqa f g) o j ) )) (( f o ( equalizer (eqa f g) o j ) ))) (id1 A d) (lemma-equ2 (( f o ( equalizer (eqa f g) o j ) ))) ) |
235
8835015a3e1a
passed let;s remove yellow
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
234
diff
changeset
|
301 ≈⟨ car ((uniqueness (eqa f g) ( begin |
8835015a3e1a
passed let;s remove yellow
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
234
diff
changeset
|
302 equalizer (eqa f g) o j |
8835015a3e1a
passed let;s remove yellow
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
234
diff
changeset
|
303 ≈↑⟨ idR ⟩ |
8835015a3e1a
passed let;s remove yellow
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
234
diff
changeset
|
304 (equalizer (eqa f g) o j ) o id1 A d |
241 | 305 ≈⟨⟩ -- here we decide e (fej) (gej)' is id1 A d |
236
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
306 ((equalizer (eqa f g) o j) o equalizer (eqa (f o equalizer (eqa f g {e}) o j) (g o equalizer (eqa f g {e}) o j))) |
235
8835015a3e1a
passed let;s remove yellow
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
234
diff
changeset
|
307 ∎ ))) ⟩ |
8835015a3e1a
passed let;s remove yellow
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
234
diff
changeset
|
308 j o k (eqa (( f o ( equalizer (eqa f g) o j ) )) (( f o ( equalizer (eqa f g) o j ) ))) (id1 A d) (lemma-equ2 (( f o ( equalizer (eqa f g) o j ) ))) |
8835015a3e1a
passed let;s remove yellow
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
234
diff
changeset
|
309 ≈⟨ cdr ((uniqueness (eqa (( f o ( equalizer (eqa f g) o j ) )) (( f o ( equalizer (eqa f g) o j ) ))) ( begin |
236
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
310 equalizer (eqa (f o equalizer (eqa f g {e} ) o j) (f o equalizer (eqa f g {e}) o j)) o id1 A d |
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
311 ≈⟨ idR ⟩ |
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
312 equalizer (eqa (f o equalizer (eqa f g {e}) o j) (f o equalizer (eqa f g {e}) o j)) |
241 | 313 ≈⟨⟩ -- here we decide e (fej) (fej)' is id1 A d |
235
8835015a3e1a
passed let;s remove yellow
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
234
diff
changeset
|
314 id1 A d |
8835015a3e1a
passed let;s remove yellow
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
234
diff
changeset
|
315 ∎ ))) ⟩ |
8835015a3e1a
passed let;s remove yellow
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
234
diff
changeset
|
316 j o id1 A d |
8835015a3e1a
passed let;s remove yellow
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
234
diff
changeset
|
317 ≈⟨ idR ⟩ |
222 | 318 j |
235
8835015a3e1a
passed let;s remove yellow
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
234
diff
changeset
|
319 ∎ |
242 | 320 cong-α1 : {a b c : Obj A } → { e : Hom A c a } |
321 → {f g g' : Hom A a b } → A [ g ≈ g' ] → A [ equalizer (eqa {a} {b} {c} f g {e} )≈ equalizer (eqa {a} {b} {c} f g' {e} ) ] | |
322 cong-α1 {a} {b} {c} {e} {f} {g} {g'} eq = let open ≈-Reasoning (A) in refl-hom | |
323 cong-γ1 : {a _ c d : Obj A } → {f g : Hom A a b} {h h' : Hom A d a } → A [ h ≈ h' ] → | |
324 A [ k (eqa f g ) {d} ( A [ h o (equalizer ( eqa (A [ f o h ] ) (A [ g o h ] ))) ] ) (lemma-equ4 {a} {b} {c} {d} f g h ) | |
325 ≈ k (eqa f g ) {d} ( A [ h' o (equalizer ( eqa (A [ f o h' ] ) (A [ g o h' ] ))) ] ) (lemma-equ4 {a} {b} {c} {d} f g h' ) ] | |
326 cong-γ1 = {!!} | |
327 cong-δ1 : {a b c : Obj A } → {f f' : Hom A a b} → A [ f ≈ f' ] → { δ δ' : Hom A a c } → A [ δ ≈ δ' ] | |
328 cong-δ1 = {!!} | |
211 | 329 |
330 | |
238
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
331 lemma-equ2 : {a b c : Obj A} (f g : Hom A a b) (e : Hom A c a ) |
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
332 → ( bur : Burroni A {c} {a} {b} f g e ) → Equalizer A {c} {a} {b} (α bur f g) f g |
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
333 lemma-equ2 {a} {b} {c} f g e bur = record { |
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
334 fe=ge = fe=ge1 ; |
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
335 k = k1 ; |
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
336 ek=h = λ {d} {h} {eq} → ek=h1 {d} {h} {eq} ; |
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
337 uniqueness = λ {d} {h} {eq} {k'} ek=h → uniqueness1 {d} {h} {eq} {k'} ek=h |
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
338 } where |
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
339 k1 : {d : Obj A} (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d c |
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
340 k1 {d} h fh=gh = β bur {d} {a} {b} f g h |
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
341 fe=ge1 : A [ A [ f o (α bur f g) ] ≈ A [ g o (α bur f g) ] ] |
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
342 fe=ge1 = b1 bur |
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
343 ek=h1 : {d : Obj A} → ∀ {h : Hom A d a} → {eq : A [ A [ f o h ] ≈ A [ g o h ] ] } → A [ A [ (α bur f g) o k1 {d} h eq ] ≈ h ] |
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
344 ek=h1 {d} {h} {eq} = let open ≈-Reasoning (A) in |
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
345 begin |
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
346 α bur f g o k1 h eq |
239 | 347 ≈⟨⟩ |
348 α bur f g o ( γ bur {a} {b} {c} f g h o δ bur {d} {b} {d} {id1 A d} (f o h) ) | |
349 ≈⟨ assoc ⟩ | |
350 ( α bur f g o γ bur {a} {b} {c} f g h ) o δ bur {d} {b} {d} {id1 A d} (f o h) | |
351 ≈⟨ car (b2 bur) ⟩ | |
352 ( h o ( α bur ( f o h ) ( g o h ))) o δ bur {d} {b} {d} {id1 A d} (f o h) | |
353 ≈↑⟨ assoc ⟩ | |
354 h o ((( α bur ( f o h ) ( g o h ))) o δ bur {d} {b} {d} {id1 A d} (f o h) ) | |
240 | 355 ≈↑⟨ cdr ( car ( cong-α bur eq)) ⟩ |
356 h o ((( α bur ( f o h ) ( f o h ))) o δ bur {d} {b} {d} {id1 A d} (f o h) ) | |
357 ≈⟨ cdr (b3 bur {d} {b} {d} (f o h) {id1 A d} ) ⟩ | |
239 | 358 h o id1 A d |
240 | 359 ≈⟨ idR ⟩ |
238
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
360 h |
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
361 ∎ |
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
362 uniqueness1 : {d : Obj A} → ∀ {h : Hom A d a} → {eq : A [ A [ f o h ] ≈ A [ g o h ] ] } → {k' : Hom A d c } → |
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
363 A [ A [ (α bur f g) o k' ] ≈ h ] → A [ k1 {d} h eq ≈ k' ] |
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
364 uniqueness1 {d} {h} {eq} {k'} ek=h = let open ≈-Reasoning (A) in |
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
365 begin |
240 | 366 k1 {d} h eq |
239 | 367 ≈⟨⟩ |
240 | 368 γ bur {a} {b} {c} f g h o δ bur {d} {b} {d} {id1 A d} (f o h) |
369 ≈↑⟨ car (cong-γ bur {a} {b} {c} {d} ek=h ) ⟩ | |
370 γ bur f g (A [ α bur f g o k' ]) o δ bur {d} {b} {d} {id1 A d} (f o h) | |
371 ≈↑⟨ cdr (cong-δ bur {d} {a} {d} (resp {d} {d} {a} {id1 A d} refl-hom ek=h )) ⟩ | |
372 γ bur f g (A [ α bur f g o k' ]) o δ bur (A [ f o A [ α bur f g o k' ] ]) | |
373 ≈⟨ b4 bur ⟩ | |
238
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
374 k' |
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
375 ∎ |
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
376 |
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
377 |
225 | 378 -- end |
212 | 379 |
380 | |
381 | |
215 | 382 |
383 |