Mercurial > hg > Members > kono > Proof > category
annotate equalizer.agda @ 226:27f2c77c963f
fix
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Thu, 05 Sep 2013 20:10:51 +0900 |
parents | 1a9f20917fbd |
children | 591efd381c82 |
rev | line source |
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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 |
221 | 26 fe=ge : 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 |
225 | 34 -- |
35 -- Flat Equational Definition of Equalizer | |
36 -- | |
37 record Burroni { 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 | 38 field |
212 | 39 α : {a b c : Obj A } → (f : Hom A a b) → (g : Hom A a b ) → Hom A c a |
214 | 40 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 | 41 δ : {a b c : Obj A } → (f : Hom A a b) → Hom A a c |
213 | 42 b1 : A [ A [ f o α {a} {b} {a} f g ] ≈ A [ g o α f g ] ] |
214 | 43 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 | 44 b3 : A [ A [ α f f o δ {a} {b} {a} f ] ≈ id1 A a ] |
207
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45 -- b4 : {c d : Obj A } {k : Hom A c a} → A [ β f g ( A [ α f g o k ] ) ≈ k ] |
215 | 46 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
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47 -- A [ α f g o β f g h ] ≈ h |
214 | 48 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 |
49 β {d} {e} {a} {b} f g h = A [ γ {a} {b} {c} f g h o δ (A [ f o h ]) ] | |
207
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50 |
209 | 51 open Equalizer |
225 | 52 open Burroni |
209 | 53 |
225 | 54 -- |
55 -- Some obvious conditions for k (fe = ge) → ( fh = gh ) | |
56 -- | |
219 | 57 |
224 | 58 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 ] ] |
59 f1=g1 eq h = let open ≈-Reasoning (A) in (resp refl-hom eq ) | |
60 | |
226 | 61 f1=f1 : { a b : Obj A } (f : Hom A a b ) → A [ A [ f o (id1 A a) ] ≈ A [ f o (id1 A a) ] ] |
62 f1=f1 f = let open ≈-Reasoning (A) in refl-hom | |
63 | |
224 | 64 f1=gh : { a b c d : Obj A } {f g : Hom A a b } → { e : Hom A c a } → { h : Hom A d c } → |
65 (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 ] ] ] | |
66 f1=gh {a} {b} {c} {d} {f} {g} {e} {h} eq = let open ≈-Reasoning (A) in | |
67 begin | |
68 f o ( e o h ) | |
69 ≈⟨ assoc ⟩ | |
70 (f o e ) o h | |
71 ≈⟨ car eq ⟩ | |
72 (g o e ) o h | |
73 ≈↑⟨ assoc ⟩ | |
74 g o ( e o h ) | |
75 ∎ | |
219 | 76 |
225 | 77 -- |
78 -- For e f f, we need e eqa = id1 A a, but it is equal to say k eqa (id a) is id | |
79 -- | |
80 -- Equalizer has free choice of c up to isomorphism, we cannot prove eqa = id a | |
219 | 81 |
82 equalizer-eq-k : { a b : Obj A } {f g : Hom A a b } → (eq : A [ f ≈ g ] ) → ( eqa : Equalizer A {a} f g) → | |
83 A [ e eqa ≈ id1 A a ] → | |
224 | 84 A [ k eqa (id1 A a) (f1=g1 eq (id1 A a)) ≈ id1 A a ] |
219 | 85 equalizer-eq-k {a} {b} {f} {g} eq eqa e=1 = let open ≈-Reasoning (A) in |
86 begin | |
224 | 87 k eqa (id1 A a) (f1=g1 eq (id1 A a)) |
219 | 88 ≈⟨ uniqueness eqa ( begin |
89 e eqa o id1 A a | |
90 ≈⟨ idR ⟩ | |
91 e eqa | |
92 ≈⟨ e=1 ⟩ | |
93 id1 A a | |
94 ∎ )⟩ | |
95 id1 A a | |
96 ∎ | |
97 | |
222 | 98 equalizer-eq-e : { a b : Obj A } {f g : Hom A a b } → ( eqa : Equalizer A {a} f g) → (eq : A [ f ≈ g ] ) → |
224 | 99 A [ k eqa (id1 A a) (f1=g1 eq (id1 A a)) ≈ id1 A a ] → |
222 | 100 A [ e eqa ≈ id1 A a ] |
101 equalizer-eq-e {a} {b} {f} {g} eqa eq k=1 = let open ≈-Reasoning (A) in | |
102 begin | |
103 e eqa | |
104 ≈↑⟨ idR ⟩ | |
105 e eqa o id1 A a | |
106 ≈↑⟨ cdr k=1 ⟩ | |
224 | 107 e eqa o k eqa (id1 A a) (f1=g1 eq (id1 A a)) |
222 | 108 ≈⟨ ek=h eqa ⟩ |
109 id1 A a | |
110 ∎ | |
111 | |
225 | 112 -- |
113 -- | |
114 -- An isomorphic element c' of c makes another equalizer | |
115 -- | |
222 | 116 -- e eqa f g f |
117 -- c ----------> a ------->b | |
118 -- |^ | |
119 -- || | |
120 -- h || h-1 | |
121 -- v| | |
122 -- c' | |
123 | |
124 equalizer+iso : {a b c c' : Obj A } {f g : Hom A a b } ( eqa : Equalizer A {c} f g) → (h-1 : Hom A c' c ) → (h : Hom A c c' ) → | |
125 A [ A [ h-1 o h ] ≈ id1 A c ] → A [ A [ h o h-1 ] ≈ id1 A c' ] | |
126 → Equalizer A {c'} f g | |
127 equalizer+iso {a} {b} {c} {c'} {f} {g} eqa h-1 h h-1-id h-id = record { | |
128 e = A [ e eqa o h-1 ] ; | |
129 fe=ge = fe=ge1 ; | |
130 k = λ j eq → A [ h o k eqa j eq ] ; | |
131 ek=h = ek=h1 ; | |
132 uniqueness = uniqueness1 | |
133 } where | |
134 fe=ge1 : A [ A [ f o A [ e eqa o h-1 ] ] ≈ A [ g o A [ e eqa o h-1 ] ] ] | |
135 fe=ge1 = let open ≈-Reasoning (A) in | |
136 begin | |
137 f o ( e eqa o h-1 ) | |
138 ≈⟨ assoc ⟩ | |
139 (f o e eqa ) o h-1 | |
140 ≈⟨ car (fe=ge eqa) ⟩ | |
141 (g o e eqa ) o h-1 | |
142 ≈↑⟨ assoc ⟩ | |
143 g o ( e eqa o h-1 ) | |
144 ∎ | |
145 ek=h1 : {d : Obj A} {j : Hom A d a} {eq : A [ A [ f o j ] ≈ A [ g o j ] ]} → | |
146 A [ A [ A [ e eqa o h-1 ] o A [ h o k eqa j eq ] ] ≈ j ] | |
147 ek=h1 {d} {j} {eq} = let open ≈-Reasoning (A) in | |
148 begin | |
149 (e eqa o h-1 ) o ( h o k eqa j eq ) | |
150 ≈↑⟨ assoc ⟩ | |
151 e eqa o ( h-1 o ( h o k eqa j eq )) | |
152 ≈⟨ cdr assoc ⟩ | |
153 e eqa o (( h-1 o h ) o k eqa j eq ) | |
154 ≈⟨ cdr (car (h-1-id )) ⟩ | |
155 e eqa o (id1 A c o k eqa j eq ) | |
156 ≈⟨ cdr idL ⟩ | |
157 e eqa o (k eqa j eq ) | |
158 ≈⟨ ek=h eqa ⟩ | |
159 j | |
160 ∎ | |
161 uniqueness1 : {d : Obj A} {h' : Hom A d a} {eq : A [ A [ f o h' ] ≈ A [ g o h' ] ]} {j : Hom A d c'} → | |
162 A [ A [ A [ e eqa o h-1 ] o j ] ≈ h' ] → | |
163 A [ A [ h o k eqa h' eq ] ≈ j ] | |
164 uniqueness1 {d} {h'} {eq} {j} ej=h = let open ≈-Reasoning (A) in | |
165 begin | |
166 h o k eqa h' eq | |
167 ≈⟨ cdr (uniqueness eqa ( | |
168 begin | |
169 e eqa o ( h-1 o j ) | |
170 ≈⟨ assoc ⟩ | |
171 (e eqa o h-1 ) o j | |
172 ≈⟨ ej=h ⟩ | |
173 h' | |
174 ∎ | |
175 )) ⟩ | |
176 h o ( h-1 o j ) | |
177 ≈⟨ assoc ⟩ | |
178 (h o h-1 ) o j | |
179 ≈⟨ car h-id ⟩ | |
180 id1 A c' o j | |
181 ≈⟨ idL ⟩ | |
182 j | |
183 ∎ | |
184 | |
225 | 185 -- If we have equalizer f g, e fh gh is also equalizer if we have isomorphic pair (same as above) |
186 -- | |
217 | 187 -- e eqa f g f |
188 -- c ----------> a ------->b | |
218 | 189 -- ^ ---> d ---> |
190 -- | i h | |
191 -- j| k' (d' → d) | |
192 -- | k (d' → a) | |
193 -- d' | |
217 | 194 |
218 | 195 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 ) |
196 → A [ A [ h o i ] ≈ e eqa ] → A [ A [ h-1 o h ] ≈ id1 A d ] | |
217 | 197 → Equalizer A {c} (A [ f o h ]) (A [ g o h ] ) |
218 | 198 equalizer+h {a} {b} {c} {d} {f} {g} eqa i h h-1 eq h-1-id = record { |
199 e = i ; -- A [ h-1 o e eqa ] ; -- Hom A a d | |
221 | 200 fe=ge = fe=ge1 ; |
217 | 201 k = λ j eq' → k eqa (A [ h o j ]) (fhj=ghj j eq' ) ; |
202 ek=h = ek=h1 ; | |
203 uniqueness = uniqueness1 | |
204 } where | |
205 fhj=ghj : {d' : Obj A } → (j : Hom A d' d ) → | |
206 A [ A [ A [ f o h ] o j ] ≈ A [ A [ g o h ] o j ] ] → | |
207 A [ A [ f o A [ h o j ] ] ≈ A [ g o A [ h o j ] ] ] | |
208 fhj=ghj j eq' = let open ≈-Reasoning (A) in | |
209 begin | |
210 f o ( h o j ) | |
211 ≈⟨ assoc ⟩ | |
212 (f o h ) o j | |
213 ≈⟨ eq' ⟩ | |
214 (g o h ) o j | |
215 ≈↑⟨ assoc ⟩ | |
216 g o ( h o j ) | |
217 ∎ | |
221 | 218 fe=ge1 : A [ A [ A [ f o h ] o i ] ≈ A [ A [ g o h ] o i ] ] |
219 fe=ge1 = let open ≈-Reasoning (A) in | |
217 | 220 begin |
221 ( f o h ) o i | |
222 ≈↑⟨ assoc ⟩ | |
223 f o (h o i ) | |
224 ≈⟨ cdr eq ⟩ | |
225 f o (e eqa) | |
221 | 226 ≈⟨ fe=ge eqa ⟩ |
217 | 227 g o (e eqa) |
228 ≈↑⟨ cdr eq ⟩ | |
229 g o (h o i ) | |
230 ≈⟨ assoc ⟩ | |
231 ( g o h ) o i | |
232 ∎ | |
218 | 233 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' ] ]} → |
234 A [ A [ i o k eqa (A [ h o k' ]) (fhj=ghj k' eq') ] ≈ k' ] | |
235 ek=h1 {d'} {k'} {eq'} = let open ≈-Reasoning (A) in | |
217 | 236 begin |
218 | 237 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') |
238 ≈↑⟨ idL ⟩ | |
239 (id1 A d ) o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) | |
240 ≈↑⟨ car h-1-id ⟩ | |
241 ( h-1 o h ) o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) | |
242 ≈↑⟨ assoc ⟩ | |
243 h-1 o ( h o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) ) | |
244 ≈⟨ cdr assoc ⟩ | |
245 h-1 o ( (h o i ) o k eqa (h o k' ) (fhj=ghj k' eq')) | |
246 ≈⟨ cdr (car eq ) ⟩ | |
247 h-1 o ( (e eqa) o k eqa (h o k' ) (fhj=ghj k' eq')) | |
248 ≈⟨ cdr (ek=h eqa) ⟩ | |
249 h-1 o ( h o k' ) | |
250 ≈⟨ assoc ⟩ | |
251 ( h-1 o h ) o k' | |
252 ≈⟨ car h-1-id ⟩ | |
253 id1 A d o k' | |
254 ≈⟨ idL ⟩ | |
255 k' | |
217 | 256 ∎ |
257 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} → | |
258 A [ A [ i o k' ] ≈ h' ] → A [ k eqa (A [ h o h' ]) (fhj=ghj h' eq') ≈ k' ] | |
259 uniqueness1 {d'} {h'} {eq'} {k'} ik=h = let open ≈-Reasoning (A) in | |
260 begin | |
261 k eqa (A [ h o h' ]) (fhj=ghj h' eq') | |
262 ≈⟨ uniqueness eqa ( begin | |
263 e eqa o k' | |
264 ≈↑⟨ car eq ⟩ | |
265 (h o i ) o k' | |
266 ≈↑⟨ assoc ⟩ | |
267 h o (i o k') | |
268 ≈⟨ cdr ik=h ⟩ | |
269 h o h' | |
270 ∎ ) ⟩ | |
271 k' | |
272 ∎ | |
215 | 273 |
225 | 274 -- If we have equalizer f g, e hf hg is also equalizer if we have isomorphic pair |
275 | |
218 | 276 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 ) |
277 → (h-1 : Hom A d b ) → A [ A [ h-1 o h ] ≈ id1 A b ] | |
278 → Equalizer A {c} (A [ h o f ]) (A [ h o g ] ) | |
279 h+equalizer {a} {b} {c} {d} {f} {g} eqa h h-1 h-1-id = record { | |
280 e = e eqa ; | |
221 | 281 fe=ge = fe=ge1 ; |
218 | 282 k = λ j eq' → k eqa j (fj=gj j eq') ; |
283 ek=h = ek=h1 ; | |
284 uniqueness = uniqueness1 | |
285 } where | |
286 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 ] ] | |
287 fj=gj j eq = let open ≈-Reasoning (A) in | |
288 begin | |
289 f o j | |
290 ≈↑⟨ idL ⟩ | |
291 id1 A b o ( f o j ) | |
292 ≈↑⟨ car h-1-id ⟩ | |
293 (h-1 o h ) o ( f o j ) | |
294 ≈↑⟨ assoc ⟩ | |
295 h-1 o (h o ( f o j )) | |
296 ≈⟨ cdr assoc ⟩ | |
297 h-1 o ((h o f) o j ) | |
298 ≈⟨ cdr eq ⟩ | |
299 h-1 o ((h o g) o j ) | |
300 ≈↑⟨ cdr assoc ⟩ | |
301 h-1 o (h o ( g o j )) | |
302 ≈⟨ assoc ⟩ | |
303 (h-1 o h) o ( g o j ) | |
304 ≈⟨ car h-1-id ⟩ | |
305 id1 A b o ( g o j ) | |
306 ≈⟨ idL ⟩ | |
307 g o j | |
308 ∎ | |
221 | 309 fe=ge1 : A [ A [ A [ h o f ] o e eqa ] ≈ A [ A [ h o g ] o e eqa ] ] |
310 fe=ge1 = let open ≈-Reasoning (A) in | |
218 | 311 begin |
312 ( h o f ) o e eqa | |
313 ≈↑⟨ assoc ⟩ | |
314 h o (f o e eqa ) | |
221 | 315 ≈⟨ cdr (fe=ge eqa) ⟩ |
218 | 316 h o (g o e eqa ) |
317 ≈⟨ assoc ⟩ | |
318 ( h o g ) o e eqa | |
319 ∎ | |
320 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 ] ]} → | |
321 A [ A [ e eqa o k eqa j (fj=gj j eq) ] ≈ j ] | |
322 ek=h1 {d₁} {j} {eq} = ek=h eqa | |
323 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} → | |
324 A [ A [ e eqa o k' ] ≈ j ] → A [ k eqa j (fj=gj j eq) ≈ k' ] | |
325 uniqueness1 = uniqueness eqa | |
326 | |
225 | 327 -- If we have equalizer f g, e (ef) (eg) is also an equalizer and e = id c |
328 | |
220 | 329 eefeg : {a b c d : Obj A } {f g : Hom A a b } ( eqa : Equalizer A {c} f g) |
330 → Equalizer A {c} (A [ f o e eqa ]) (A [ g o e eqa ] ) | |
331 eefeg {a} {b} {c} {d} {f} {g} eqa = record { | |
332 e = id1 A c ; -- i ; -- A [ h-1 o e eqa ] ; -- Hom A a d | |
221 | 333 fe=ge = fe=ge1 ; |
220 | 334 k = λ j eq' → k eqa (A [ h o j ]) (fhj=ghj j eq' ) ; |
335 ek=h = ek=h1 ; | |
336 uniqueness = uniqueness1 | |
337 } where | |
338 i = id1 A c | |
339 h = e eqa | |
340 fhj=ghj : {d' : Obj A } → (j : Hom A d' c ) → | |
341 A [ A [ A [ f o h ] o j ] ≈ A [ A [ g o h ] o j ] ] → | |
342 A [ A [ f o A [ h o j ] ] ≈ A [ g o A [ h o j ] ] ] | |
343 fhj=ghj j eq' = let open ≈-Reasoning (A) in | |
344 begin | |
345 f o ( h o j ) | |
346 ≈⟨ assoc ⟩ | |
347 (f o h ) o j | |
348 ≈⟨ eq' ⟩ | |
349 (g o h ) o j | |
350 ≈↑⟨ assoc ⟩ | |
351 g o ( h o j ) | |
352 ∎ | |
221 | 353 fe=ge1 : A [ A [ A [ f o h ] o i ] ≈ A [ A [ g o h ] o i ] ] |
354 fe=ge1 = let open ≈-Reasoning (A) in | |
220 | 355 begin |
356 ( f o h ) o i | |
221 | 357 ≈⟨ car ( fe=ge eqa ) ⟩ |
220 | 358 ( g o h ) o i |
359 ∎ | |
360 ek=h1 : {d' : Obj A} {k' : Hom A d' c} {eq' : A [ A [ A [ f o h ] o k' ] ≈ A [ A [ g o h ] o k' ] ]} → | |
361 A [ A [ i o k eqa (A [ h o k' ]) (fhj=ghj k' eq') ] ≈ k' ] | |
362 ek=h1 {d'} {k'} {eq'} = let open ≈-Reasoning (A) in | |
363 begin | |
364 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') | |
365 ≈⟨ idL ⟩ | |
366 k eqa (e eqa o k' ) (fhj=ghj k' eq') | |
367 ≈⟨ uniqueness eqa refl-hom ⟩ | |
368 k' | |
369 ∎ | |
370 uniqueness1 : {d' : Obj A} {h' : Hom A d' c} {eq' : A [ A [ A [ f o h ] o h' ] ≈ A [ A [ g o h ] o h' ] ]} {k' : Hom A d' c} → | |
371 A [ A [ i o k' ] ≈ h' ] → A [ k eqa (A [ h o h' ]) (fhj=ghj h' eq') ≈ k' ] | |
372 uniqueness1 {d'} {h'} {eq'} {k'} ik=h = let open ≈-Reasoning (A) in | |
373 begin | |
374 k eqa ( e eqa o h') (fhj=ghj h' eq') | |
375 ≈⟨ uniqueness eqa ( begin | |
376 e eqa o k' | |
377 ≈↑⟨ cdr idL ⟩ | |
378 e eqa o (id1 A c o k' ) | |
379 ≈⟨ cdr ik=h ⟩ | |
380 e eqa o h' | |
381 ∎ ) ⟩ | |
382 k' | |
383 ∎ | |
384 | |
225 | 385 -- |
386 -- If we have two equalizers on c and c', there are isomorphic pair h, h' | |
387 -- | |
388 -- h : c → c' h' : c' → c | |
226 | 389 -- h h' = 1 and h' h = 1 |
225 | 390 -- not yet done |
391 | |
392 | |
226 | 393 c-iso-l : { c c' a b : Obj A } {f g : Hom A a b } → ( eqa : Equalizer A {c} f g) → ( eqa' : Equalizer A {c'} f g ) |
394 → (eff' : Equalizer A {c'} f f ) → (eff : Equalizer A {c} f f ) | |
395 → Hom A c c' | |
396 c-iso-l eqa eqa' eff' eff = let open ≈-Reasoning (A) in k eff' (e eqa) refl-hom | |
397 | |
398 c-iso-r : { c c' a b : Obj A } {f g : Hom A a b } → ( eqa : Equalizer A {c} f g) → ( eqa' : Equalizer A {c'} f g ) | |
399 → (eff' : Equalizer A {c'} f f ) → (eff : Equalizer A {c} f f ) | |
400 → Hom A c' c | |
401 c-iso-r eqa eqa' eff' eff = let open ≈-Reasoning (A) in k eff (e eqa') refl-hom | |
223 | 402 |
226 | 403 c-iso-1 : { c c' a b : Obj A } {f g : Hom A a b } → ( eqa : Equalizer A {c} f g) → ( eqa' : Equalizer A {c'} f g ) |
404 → (eff' : Equalizer A {c'} f f ) → (eff : Equalizer A {c} f f ) | |
405 → A [ A [ k eqa' (A [ e eqa' o k eff' (id1 A a ) (f1=f1 f) ] ) (f1=gh ( fe=ge eqa' ) ) o e eff' ] ≈ id1 A c' ] | |
406 c-iso-1 = {!!} | |
407 | |
408 c-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 ) | |
409 → (eff' : Equalizer A {c'} f f ) → (eff : Equalizer A {c} f f ) | |
410 → A [ A [ c-iso-l eqa eqa' eff' eff o c-iso-r eqa eqa' eff' eff ] ≈ id1 A c' ] | |
411 c-iso {c} {c'} {a} {b} {f} {g} eqa eqa' eff' eff = let open ≈-Reasoning (A) in begin | |
412 c-iso-l eqa eqa' eff' eff o c-iso-r eqa eqa' eff' eff | |
413 ≈⟨⟩ | |
414 k eff' (e eqa) refl-hom o k eff (e eqa') refl-hom | |
415 ≈⟨ {!!} ⟩ | |
416 id1 A c' | |
417 ∎ | |
418 | |
223 | 419 |
221 | 420 -- Equalizer is unique up to iso |
421 | |
422 -- | |
423 -- | |
424 -- e eqa f g f | |
425 -- c ----------> a ------->b | |
426 -- | |
224 | 427 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 ) |
222 | 428 { h : Hom A a c } → A [ A [ h o e eqa ] ≈ id1 A c ] → A [ A [ k eqa (e eqa' ) (fe=ge eqa') o k eqa' (e eqa ) (fe=ge eqa) ] ≈ id1 A c ] |
224 | 429 equalizer-iso-eq' {c} {c'} {f} {g} eqa eqa' {h} rev = let open ≈-Reasoning (A) in |
221 | 430 begin |
222 | 431 k eqa (e eqa' ) (fe=ge eqa') o k eqa' (e eqa ) (fe=ge eqa) |
432 ≈↑⟨ idL ⟩ | |
433 (id1 A c) o ( k eqa (e eqa' ) (fe=ge eqa') o k eqa' (e eqa ) (fe=ge eqa) ) | |
434 ≈↑⟨ car rev ⟩ | |
435 ( h o e eqa ) o ( k eqa (e eqa' ) (fe=ge eqa') o k eqa' (e eqa ) (fe=ge eqa) ) | |
436 ≈↑⟨ assoc ⟩ | |
437 h o ( e eqa o ( k eqa (e eqa' ) (fe=ge eqa') o k eqa' (e eqa ) (fe=ge eqa) ) ) | |
438 ≈⟨ cdr assoc ⟩ | |
439 h o (( e eqa o k eqa (e eqa' ) (fe=ge eqa')) o k eqa' (e eqa ) (fe=ge eqa) ) | |
440 ≈⟨ cdr ( car (ek=h eqa) ) ⟩ | |
441 h o ( e eqa' o k eqa' (e eqa ) (fe=ge eqa) ) | |
442 ≈⟨ cdr (ek=h eqa' ) ⟩ | |
443 h o e eqa | |
444 ≈⟨ rev ⟩ | |
445 id1 A c | |
221 | 446 ∎ |
447 | |
225 | 448 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 ) |
449 → A [ A [ k eqa (e eqa' ) (fe=ge eqa') o k eqa' (e eqa ) (fe=ge eqa) ] ≈ id1 A c ] | |
450 equalizer-iso-eq {c} {c'} {f} {g} eqa eqa' = equalizer-iso-eq' {c} {c'} {f} {g} eqa eqa' {{!!}} {!!} | |
451 | |
221 | 452 -- ke = e' k'e' = e → k k' = 1 , k' k = 1 |
453 -- ke = e' | |
454 -- k'ke = k'e' = e kk' = 1 | |
455 -- x e = e -> x = id? | |
218 | 456 |
225 | 457 ---- |
458 -- | |
459 -- An equalizer satisfies Burroni equations | |
460 -- | |
461 -- b4 is not yet done | |
462 ---- | |
463 | |
222 | 464 lemma-equ1 : {a b c : Obj A} (f g : Hom A a b) → |
225 | 465 ( {a b c : Obj A} → (f g : Hom A a b) → Equalizer A {c} f g ) → Burroni A {c} f g |
222 | 466 lemma-equ1 {a} {b} {c} f g eqa = record { |
216 | 467 α = λ f g → e (eqa f g ) ; -- Hom A c a |
214 | 468 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 | 469 δ = λ {a} f → k (eqa f f) (id1 A a) (lemma-equ2 f); -- Hom A a c |
221 | 470 b1 = fe=ge (eqa f g) ; |
226 | 471 b2 = lemma-b2 ; |
472 b3 = lemma-b3 ; | |
473 b4 = lemma-b4 | |
211 | 474 } where |
216 | 475 -- |
476 -- e eqa f g f | |
477 -- c ----------> a ------->b | |
478 -- ^ g | |
479 -- | | |
480 -- |k₁ = e eqa (f o (e (eqa f g))) (g o (e (eqa f g)))) | |
481 -- | | |
482 -- d | |
483 -- | |
484 -- | |
485 -- e o id1 ≈ e → k e ≈ id | |
486 | |
211 | 487 lemma-equ2 : {a b : Obj A} (f : Hom A a b) → A [ A [ f o id1 A a ] ≈ A [ f o id1 A a ] ] |
488 lemma-equ2 f = let open ≈-Reasoning (A) in refl-hom | |
226 | 489 lemma-b3 : A [ A [ e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) ] ≈ id1 A a ] |
490 lemma-b3 = let open ≈-Reasoning (A) in | |
211 | 491 begin |
492 e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) | |
215 | 493 ≈⟨ ek=h (eqa f f ) ⟩ |
211 | 494 id1 A a |
495 ∎ | |
214 | 496 lemma-equ4 : {a b c d : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → |
212 | 497 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 | 498 lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in |
212 | 499 begin |
500 f o ( h o e (eqa (f o h) ( g o h ))) | |
501 ≈⟨ assoc ⟩ | |
502 (f o h) o e (eqa (f o h) ( g o h )) | |
221 | 503 ≈⟨ fe=ge (eqa (A [ f o h ]) (A [ g o h ])) ⟩ |
212 | 504 (g o h) o e (eqa (f o h) ( g o h )) |
505 ≈↑⟨ assoc ⟩ | |
506 g o ( h o e (eqa (f o h) ( g o h ))) | |
507 ∎ | |
226 | 508 lemma-b2 : {d : Obj A} {h : Hom A d a} → A [ |
214 | 509 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 | 510 ≈ A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ] ] |
226 | 511 lemma-b2 {d} {h} = let open ≈-Reasoning (A) in |
212 | 512 begin |
215 | 513 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) |
514 ≈⟨ ek=h (eqa f g) ⟩ | |
212 | 515 h o e (eqa (f o h ) ( g o h )) |
516 ∎ | |
226 | 517 lemma-b4 : {d : Obj A} {j : Hom A d c} → A [ |
222 | 518 A [ k (eqa f g) (A [ A [ e (eqa f g) o j ] o e (eqa (A [ f o A [ e (eqa f g) o j ] ]) (A [ g o A [ e (eqa f g) o j ] ])) ]) |
519 (lemma-equ4 {a} {b} {c} f g (A [ e (eqa f g) o j ])) o | |
520 k (eqa (A [ f o A [ e (eqa f g) o j ] ]) (A [ f o A [ e (eqa f g) o j ] ])) (id1 A d) (lemma-equ2 (A [ f o A [ e (eqa f g) o j ] ])) ] | |
521 ≈ j ] | |
226 | 522 lemma-b4 {d} {j} = let open ≈-Reasoning (A) in |
215 | 523 begin |
222 | 524 ( k (eqa f g) (( ( e (eqa f g) o j ) o e (eqa (( f o ( e (eqa f g) o j ) )) (( g o ( e (eqa f g) o j ) ))) )) |
525 (lemma-equ4 {a} {b} {c} f g (( e (eqa f g) o j ))) o | |
526 k (eqa (( f o ( e (eqa f g) o j ) )) (( f o ( e (eqa f g) o j ) ))) (id1 A d) (lemma-equ2 (( f o ( e (eqa f g) o j ) ))) ) | |
215 | 527 ≈⟨ car ( uniqueness (eqa f g) ( begin |
222 | 528 e (eqa f g) o j |
215 | 529 ≈⟨ {!!} ⟩ |
222 | 530 (e (eqa f g) o j) o e (eqa (f o e (eqa f g) o j) (g o e (eqa f g) o j)) |
215 | 531 ∎ )) ⟩ |
222 | 532 j o k (eqa (( f o ( e (eqa f g) o j ) )) (( f o ( e (eqa f g) o j ) ))) (id1 A d) (lemma-equ2 (( f o ( e (eqa f g) o j ) ))) |
533 ≈⟨ cdr ( uniqueness (eqa (( f o ( e (eqa f g) o j ) )) (( f o ( e (eqa f g) o j ) ))) ( begin | |
534 e (eqa (f o e (eqa f g) o j) (f o e (eqa f g) o j)) o id1 A d | |
535 ≈⟨ idR ⟩ | |
536 e (eqa (f o e (eqa f g) o j) (f o e (eqa f g) o j)) | |
215 | 537 ≈⟨ {!!} ⟩ |
222 | 538 id1 A d |
215 | 539 ∎ )) ⟩ |
222 | 540 j o id1 A d |
215 | 541 ≈⟨ idR ⟩ |
222 | 542 j |
215 | 543 ∎ |
211 | 544 |
545 | |
225 | 546 -- end |
212 | 547 |
548 | |
549 | |
215 | 550 |
551 |