Mercurial > hg > Members > kono > Proof > category
annotate equalizer.agda @ 219:2ae029454fb6
on going ...
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
---|---|
date | Wed, 04 Sep 2013 17:36:32 +0900 |
parents | 749a1ecbc0b5 |
children | 5d96be63053f |
rev | line source |
---|---|
205 | 1 --- |
2 -- | |
3 -- Equalizer | |
4 -- | |
208 | 5 -- e f |
205 | 6 -- c --------> a ----------> b |
208 | 7 -- ^ . ----------> |
205 | 8 -- | . g |
208 | 9 -- |k . |
10 -- | . h | |
205 | 11 -- d |
12 -- | |
13 -- Shinji KONO <kono@ie.u-ryukyu.ac.jp> | |
14 ---- | |
15 | |
16 open import Category -- https://github.com/konn/category-agda | |
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 | |
209 | 23 record Equalizer { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {c a b : Obj A} (f g : Hom A a b) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where |
205 | 24 field |
209 | 25 e : Hom A c a |
215 | 26 ef=eg : A [ A [ f o e ] ≈ A [ g o e ] ] |
209 | 27 k : {d : Obj A} (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d c |
215 | 28 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 ] |
29 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 | 30 A [ A [ e o k' ] ≈ h ] → A [ k {d} h eq ≈ k' ] |
209 | 31 equalizer : Hom A c a |
32 equalizer = e | |
206 | 33 |
209 | 34 record EqEqualizer { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {c a b : Obj A} (f g : Hom A a b) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where |
206 | 35 field |
212 | 36 α : {a b c : Obj A } → (f : Hom A a b) → (g : Hom A a b ) → Hom A c a |
214 | 37 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 |
212 | 38 δ : {a b c : Obj A } → (f : Hom A a b) → Hom A a c |
213 | 39 b1 : A [ A [ f o α {a} {b} {a} f g ] ≈ A [ g o α f g ] ] |
214 | 40 b2 : {d : Obj A } → {h : Hom A d a } → A [ A [ ( α f g) o (γ {a} {b} {c} f g h) ] ≈ A [ h o α (A [ f o h ]) (A [ g o h ]) ] ] |
213 | 41 b3 : A [ A [ α f f o δ {a} {b} {a} f ] ≈ id1 A a ] |
207
22811f7a04e1
Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
206
diff
changeset
|
42 -- b4 : {c d : Obj A } {k : Hom A c a} → A [ β f g ( A [ α f g o k ] ) ≈ k ] |
215 | 43 b4 : {d : Obj A } {k : Hom A d c} → A [ A [ γ {a} {b} {c} {d} f g ( A [ α {a} {b} {c} f g o k ] ) o δ (A [ f o A [ α f g o k ] ] ) ] ≈ k ] |
207
22811f7a04e1
Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
206
diff
changeset
|
44 -- A [ α f g o β f g h ] ≈ h |
214 | 45 β : { d e a b : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → Hom A d c |
46 β {d} {e} {a} {b} f g h = A [ γ {a} {b} {c} f g h o δ (A [ f o h ]) ] | |
207
22811f7a04e1
Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
206
diff
changeset
|
47 |
209 | 48 open Equalizer |
49 open EqEqualizer | |
50 | |
217 | 51 -- Equalizer is unique up to iso |
52 | |
53 equalizer-iso : { c c' a b : Obj A } {f g : Hom A a b } → ( eqa : Equalizer A {c} f g) → ( eqa' : Equalizer A {c'} f g ) | |
54 → Hom A c c' --- != id1 A c | |
55 equalizer-iso {c} eqa eqa' = k eqa' (e eqa) (ef=eg eqa) | |
56 | |
219 | 57 equalizer-iso-eq : { c c' a b : Obj A } {f g : Hom A a b } → ( eqa : Equalizer A {c} f g) → ( eqa' : Equalizer A {c'} f g ) → {eff : Equalizer A {c} f f} |
58 → A [ A [ equalizer-iso eqa eqa' o equalizer-iso eqa' eqa ] ≈ id1 A c' ] | |
59 equalizer-iso-eq {c} {c'} {f} {g} eqa eqa' {eff} = let open ≈-Reasoning (A) in | |
60 begin | |
61 equalizer-iso eqa eqa' o equalizer-iso eqa' eqa | |
62 ≈⟨⟩ | |
63 k eqa' (e eqa) (ef=eg eqa) o k eqa (e eqa') (ef=eg eqa') | |
64 ≈⟨ car (uniqueness eqa' {!!}) ⟩ | |
65 {!!} o k eqa (e eqa') (ef=eg eqa') | |
66 ≈⟨ {!!} ⟩ | |
67 id1 A c' | |
68 ∎ | |
69 | |
70 -- ke = e' k'e' = e → k k' = 1 , k' k = 1 | |
71 -- ke = e' | |
72 -- k'ke = k'e' = e kk' = 1 | |
73 | |
74 -- x e = e -> x = id? | |
75 | |
76 f1=g1 : { a b : Obj A } {f g : Hom A a b } → (eq : A [ f ≈ g ] ) → A [ A [ f o id1 A a ] ≈ A [ g o id1 A a ] ] | |
77 f1=g1 eq = let open ≈-Reasoning (A) in (resp refl-hom eq ) | |
78 | |
79 | |
80 equalizer-eq-k : { a b : Obj A } {f g : Hom A a b } → (eq : A [ f ≈ g ] ) → ( eqa : Equalizer A {a} f g) → | |
81 A [ e eqa ≈ id1 A a ] → | |
82 A [ k eqa (id1 A a) (f1=g1 eq) ≈ id1 A a ] | |
83 equalizer-eq-k {a} {b} {f} {g} eq eqa e=1 = let open ≈-Reasoning (A) in | |
84 begin | |
85 k eqa (id1 A a) (f1=g1 eq) | |
86 ≈⟨ uniqueness eqa ( begin | |
87 e eqa o id1 A a | |
88 ≈⟨ idR ⟩ | |
89 e eqa | |
90 ≈⟨ e=1 ⟩ | |
91 id1 A a | |
92 ∎ )⟩ | |
93 id1 A a | |
94 ∎ | |
95 | |
217 | 96 -- e eqa f g f |
97 -- c ----------> a ------->b | |
218 | 98 -- ^ ---> d ---> |
99 -- | i h | |
100 -- j| k' (d' → d) | |
101 -- | k (d' → a) | |
102 -- d' | |
217 | 103 |
218 | 104 equalizer+h : {a b c d : Obj A } {f g : Hom A a b } ( eqa : Equalizer A {c} f g) (i : Hom A c d ) → (h : Hom A d a ) → (h-1 : Hom A a d ) |
105 → A [ A [ h o i ] ≈ e eqa ] → A [ A [ h-1 o h ] ≈ id1 A d ] | |
217 | 106 → Equalizer A {c} (A [ f o h ]) (A [ g o h ] ) |
218 | 107 equalizer+h {a} {b} {c} {d} {f} {g} eqa i h h-1 eq h-1-id = record { |
108 e = i ; -- A [ h-1 o e eqa ] ; -- Hom A a d | |
217 | 109 ef=eg = ef=eg1 ; |
110 k = λ j eq' → k eqa (A [ h o j ]) (fhj=ghj j eq' ) ; | |
111 ek=h = ek=h1 ; | |
112 uniqueness = uniqueness1 | |
113 } where | |
114 fhj=ghj : {d' : Obj A } → (j : Hom A d' d ) → | |
115 A [ A [ A [ f o h ] o j ] ≈ A [ A [ g o h ] o j ] ] → | |
116 A [ A [ f o A [ h o j ] ] ≈ A [ g o A [ h o j ] ] ] | |
117 fhj=ghj j eq' = let open ≈-Reasoning (A) in | |
118 begin | |
119 f o ( h o j ) | |
120 ≈⟨ assoc ⟩ | |
121 (f o h ) o j | |
122 ≈⟨ eq' ⟩ | |
123 (g o h ) o j | |
124 ≈↑⟨ assoc ⟩ | |
125 g o ( h o j ) | |
126 ∎ | |
127 ef=eg1 : A [ A [ A [ f o h ] o i ] ≈ A [ A [ g o h ] o i ] ] | |
128 ef=eg1 = let open ≈-Reasoning (A) in | |
129 begin | |
130 ( f o h ) o i | |
131 ≈↑⟨ assoc ⟩ | |
132 f o (h o i ) | |
133 ≈⟨ cdr eq ⟩ | |
134 f o (e eqa) | |
135 ≈⟨ ef=eg eqa ⟩ | |
136 g o (e eqa) | |
137 ≈↑⟨ cdr eq ⟩ | |
138 g o (h o i ) | |
139 ≈⟨ assoc ⟩ | |
140 ( g o h ) o i | |
141 ∎ | |
218 | 142 ek=h1 : {d' : Obj A} {k' : Hom A d' d} {eq' : A [ A [ A [ f o h ] o k' ] ≈ A [ A [ g o h ] o k' ] ]} → |
143 A [ A [ i o k eqa (A [ h o k' ]) (fhj=ghj k' eq') ] ≈ k' ] | |
144 ek=h1 {d'} {k'} {eq'} = let open ≈-Reasoning (A) in | |
217 | 145 begin |
218 | 146 i o k eqa (h o k' ) (fhj=ghj k' eq') -- h-1 (h o i ) o k eqa (h o k' ) = h-1 (h o k') |
147 ≈↑⟨ idL ⟩ | |
148 (id1 A d ) o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) | |
149 ≈↑⟨ car h-1-id ⟩ | |
150 ( h-1 o h ) o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) | |
151 ≈↑⟨ assoc ⟩ | |
152 h-1 o ( h o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) ) | |
153 ≈⟨ cdr assoc ⟩ | |
154 h-1 o ( (h o i ) o k eqa (h o k' ) (fhj=ghj k' eq')) | |
155 ≈⟨ cdr (car eq ) ⟩ | |
156 h-1 o ( (e eqa) o k eqa (h o k' ) (fhj=ghj k' eq')) | |
157 ≈⟨ cdr (ek=h eqa) ⟩ | |
158 h-1 o ( h o k' ) | |
159 ≈⟨ assoc ⟩ | |
160 ( h-1 o h ) o k' | |
161 ≈⟨ car h-1-id ⟩ | |
162 id1 A d o k' | |
163 ≈⟨ idL ⟩ | |
164 k' | |
217 | 165 ∎ |
166 uniqueness1 : {d' : Obj A} {h' : Hom A d' d} {eq' : A [ A [ A [ f o h ] o h' ] ≈ A [ A [ g o h ] o h' ] ]} {k' : Hom A d' c} → | |
167 A [ A [ i o k' ] ≈ h' ] → A [ k eqa (A [ h o h' ]) (fhj=ghj h' eq') ≈ k' ] | |
168 uniqueness1 {d'} {h'} {eq'} {k'} ik=h = let open ≈-Reasoning (A) in | |
169 begin | |
170 k eqa (A [ h o h' ]) (fhj=ghj h' eq') | |
171 ≈⟨ uniqueness eqa ( begin | |
172 e eqa o k' | |
173 ≈↑⟨ car eq ⟩ | |
174 (h o i ) o k' | |
175 ≈↑⟨ assoc ⟩ | |
176 h o (i o k') | |
177 ≈⟨ cdr ik=h ⟩ | |
178 h o h' | |
179 ∎ ) ⟩ | |
180 k' | |
181 ∎ | |
215 | 182 |
218 | 183 h+equalizer : {a b c d : Obj A } {f g : Hom A a b } ( eqa : Equalizer A {c} f g) (h : Hom A b d ) |
184 → (h-1 : Hom A d b ) → A [ A [ h-1 o h ] ≈ id1 A b ] | |
185 → Equalizer A {c} (A [ h o f ]) (A [ h o g ] ) | |
186 h+equalizer {a} {b} {c} {d} {f} {g} eqa h h-1 h-1-id = record { | |
187 e = e eqa ; | |
188 ef=eg = ef=eg1 ; | |
189 k = λ j eq' → k eqa j (fj=gj j eq') ; | |
190 ek=h = ek=h1 ; | |
191 uniqueness = uniqueness1 | |
192 } where | |
193 fj=gj : {e : Obj A} → (j : Hom A e a ) → A [ A [ A [ h o f ] o j ] ≈ A [ A [ h o g ] o j ] ] → A [ A [ f o j ] ≈ A [ g o j ] ] | |
194 fj=gj j eq = let open ≈-Reasoning (A) in | |
195 begin | |
196 f o j | |
197 ≈↑⟨ idL ⟩ | |
198 id1 A b o ( f o j ) | |
199 ≈↑⟨ car h-1-id ⟩ | |
200 (h-1 o h ) o ( f o j ) | |
201 ≈↑⟨ assoc ⟩ | |
202 h-1 o (h o ( f o j )) | |
203 ≈⟨ cdr assoc ⟩ | |
204 h-1 o ((h o f) o j ) | |
205 ≈⟨ cdr eq ⟩ | |
206 h-1 o ((h o g) o j ) | |
207 ≈↑⟨ cdr assoc ⟩ | |
208 h-1 o (h o ( g o j )) | |
209 ≈⟨ assoc ⟩ | |
210 (h-1 o h) o ( g o j ) | |
211 ≈⟨ car h-1-id ⟩ | |
212 id1 A b o ( g o j ) | |
213 ≈⟨ idL ⟩ | |
214 g o j | |
215 ∎ | |
216 ef=eg1 : A [ A [ A [ h o f ] o e eqa ] ≈ A [ A [ h o g ] o e eqa ] ] | |
217 ef=eg1 = let open ≈-Reasoning (A) in | |
218 begin | |
219 ( h o f ) o e eqa | |
220 ≈↑⟨ assoc ⟩ | |
221 h o (f o e eqa ) | |
222 ≈⟨ cdr (ef=eg eqa) ⟩ | |
223 h o (g o e eqa ) | |
224 ≈⟨ assoc ⟩ | |
225 ( h o g ) o e eqa | |
226 ∎ | |
227 ek=h1 : {d₁ : Obj A} {j : Hom A d₁ a} {eq : A [ A [ A [ h o f ] o j ] ≈ A [ A [ h o g ] o j ] ]} → | |
228 A [ A [ e eqa o k eqa j (fj=gj j eq) ] ≈ j ] | |
229 ek=h1 {d₁} {j} {eq} = ek=h eqa | |
230 uniqueness1 : {d₁ : Obj A} {j : Hom A d₁ a} {eq : A [ A [ A [ h o f ] o j ] ≈ A [ A [ h o g ] o j ] ]} {k' : Hom A d₁ c} → | |
231 A [ A [ e eqa o k' ] ≈ j ] → A [ k eqa j (fj=gj j eq) ≈ k' ] | |
232 uniqueness1 = uniqueness eqa | |
233 | |
234 | |
211 | 235 lemma-equ1 : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → {a b c : Obj A} (f g : Hom A a b) → |
236 ( {a b c : Obj A} → (f g : Hom A a b) → Equalizer A {c} f g ) → EqEqualizer A {c} f g | |
209 | 237 lemma-equ1 A {a} {b} {c} f g eqa = record { |
216 | 238 α = λ f g → e (eqa f g ) ; -- Hom A c a |
214 | 239 γ = λ {a} {b} {c} {d} f g h → k (eqa f g ) {d} ( A [ h o (e ( eqa (A [ f o h ] ) (A [ g o h ] ))) ] ) (lemma-equ4 {a} {b} {c} {d} f g h ) ; -- Hom A c d |
213 | 240 δ = λ {a} f → k (eqa f f) (id1 A a) (lemma-equ2 f); -- Hom A a c |
211 | 241 b1 = ef=eg (eqa f g) ; |
212 | 242 b2 = lemma-equ5 ; |
243 b3 = lemma-equ3 ; | |
215 | 244 b4 = lemma-equ6 |
211 | 245 } where |
216 | 246 -- |
247 -- e eqa f g f | |
248 -- c ----------> a ------->b | |
249 -- ^ g | |
250 -- | | |
251 -- |k₁ = e eqa (f o (e (eqa f g))) (g o (e (eqa f g)))) | |
252 -- | | |
253 -- d | |
254 -- | |
255 -- | |
256 -- e o id1 ≈ e → k e ≈ id | |
257 | |
211 | 258 lemma-equ2 : {a b : Obj A} (f : Hom A a b) → A [ A [ f o id1 A a ] ≈ A [ f o id1 A a ] ] |
259 lemma-equ2 f = let open ≈-Reasoning (A) in refl-hom | |
213 | 260 lemma-equ3 : A [ A [ e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) ] ≈ id1 A a ] |
261 lemma-equ3 = let open ≈-Reasoning (A) in | |
211 | 262 begin |
263 e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) | |
215 | 264 ≈⟨ ek=h (eqa f f ) ⟩ |
211 | 265 id1 A a |
266 ∎ | |
214 | 267 lemma-equ4 : {a b c d : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → |
212 | 268 A [ A [ f o A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ] ] ≈ A [ g o A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ] ] ] |
214 | 269 lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in |
212 | 270 begin |
271 f o ( h o e (eqa (f o h) ( g o h ))) | |
272 ≈⟨ assoc ⟩ | |
273 (f o h) o e (eqa (f o h) ( g o h )) | |
274 ≈⟨ ef=eg (eqa (A [ f o h ]) (A [ g o h ])) ⟩ | |
275 (g o h) o e (eqa (f o h) ( g o h )) | |
276 ≈↑⟨ assoc ⟩ | |
277 g o ( h o e (eqa (f o h) ( g o h ))) | |
278 ∎ | |
279 lemma-equ5 : {d : Obj A} {h : Hom A d a} → A [ | |
214 | 280 A [ e (eqa f g) o k (eqa f g) (A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ]) (lemma-equ4 {a} {b} {c} f g h) ] |
212 | 281 ≈ A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ] ] |
282 lemma-equ5 {d} {h} = let open ≈-Reasoning (A) in | |
283 begin | |
215 | 284 e (eqa f g) o k (eqa f g) (h o e (eqa (f o h) (g o h))) (lemma-equ4 {a} {b} {c} f g h) |
285 ≈⟨ ek=h (eqa f g) ⟩ | |
212 | 286 h o e (eqa (f o h ) ( g o h )) |
287 ∎ | |
215 | 288 lemma-equ6 : {d : Obj A} {k₁ : Hom A d c} → A [ |
289 A [ k (eqa f g) (A [ A [ e (eqa f g) o k₁ ] o e (eqa (A [ f o A [ e (eqa f g) o k₁ ] ]) (A [ g o A [ e (eqa f g) o k₁ ] ])) ]) | |
290 (lemma-equ4 {a} {b} {c} f g (A [ e (eqa f g) o k₁ ])) o | |
291 k (eqa (A [ f o A [ e (eqa f g) o k₁ ] ]) (A [ f o A [ e (eqa f g) o k₁ ] ])) (id1 A d) (lemma-equ2 (A [ f o A [ e (eqa f g) o k₁ ] ])) ] | |
292 ≈ k₁ ] | |
293 lemma-equ6 {d} {k₁} = let open ≈-Reasoning (A) in | |
294 begin | |
295 ( k (eqa f g) (( ( e (eqa f g) o k₁ ) o e (eqa (( f o ( e (eqa f g) o k₁ ) )) (( g o ( e (eqa f g) o k₁ ) ))) )) | |
296 (lemma-equ4 {a} {b} {c} f g (( e (eqa f g) o k₁ ))) o | |
297 k (eqa (( f o ( e (eqa f g) o k₁ ) )) (( f o ( e (eqa f g) o k₁ ) ))) (id1 A d) (lemma-equ2 (( f o ( e (eqa f g) o k₁ ) ))) ) | |
298 ≈⟨ car ( uniqueness (eqa f g) ( begin | |
299 e (eqa f g) o k₁ | |
300 ≈⟨ {!!} ⟩ | |
301 (e (eqa f g) o k₁) o e (eqa (f o e (eqa f g) o k₁) (g o e (eqa f g) o k₁)) | |
302 ∎ )) ⟩ | |
303 k₁ o k (eqa (( f o ( e (eqa f g) o k₁ ) )) (( f o ( e (eqa f g) o k₁ ) ))) (id1 A d) (lemma-equ2 (( f o ( e (eqa f g) o k₁ ) ))) | |
304 ≈⟨ cdr ( uniqueness (eqa (( f o ( e (eqa f g) o k₁ ) )) (( f o ( e (eqa f g) o k₁ ) ))) ( begin | |
305 e (eqa (f o e (eqa f g) o k₁) (f o e (eqa f g) o k₁)) o id1 A d | |
306 ≈⟨ {!!} ⟩ | |
307 id1 A d | |
308 ∎ )) ⟩ | |
309 k₁ o id1 A d | |
310 ≈⟨ idR ⟩ | |
311 k₁ | |
312 ∎ | |
211 | 313 |
314 | |
212 | 315 |
316 | |
317 | |
215 | 318 |
319 |