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comparison equalizer.agda @ 233:4bba19bc71be

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e is now explict parameter
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 08 Sep 2013 01:37:24 +0900
parents b0fe61882014
children c02287d3d2dc
comparison
equal deleted inserted replaced
232:b0fe61882014 233:4bba19bc71be
18 module equalizer { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } where 18 module equalizer { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } where
19 19
20 open import HomReasoning 20 open import HomReasoning
21 open import cat-utility 21 open import cat-utility
22 22
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 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
24 field 24 field
25 e : Hom A c a
26 fe=ge : A [ A [ f o e ] ≈ A [ g o e ] ] 25 fe=ge : A [ A [ f o e ] ≈ A [ g o e ] ]
27 k : {d : Obj A} (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d c 26 k : {d : Obj A} (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d c
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 ] 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 ]
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 } → 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 } →
30 A [ A [ e o k' ] ≈ h ] → A [ k {d} h eq ≈ k' ] 29 A [ A [ e o k' ] ≈ h ] → A [ k {d} h eq ≈ k' ]
73 ≈↑⟨ assoc ⟩ 72 ≈↑⟨ assoc ⟩
74 g o ( e o h ) 73 g o ( e o h )
75 ∎ 74 ∎
76 75
77 -- 76 --
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
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 ] →
84 A [ k eqa (id1 A a) (f1=g1 eq (id1 A a)) ≈ id1 A a ]
85 equalizer-eq-k {a} {b} {f} {g} eq eqa e=1 = let open ≈-Reasoning (A) in
86 begin
87 k eqa (id1 A a) (f1=g1 eq (id1 A a))
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
98 equalizer-eq-e : { a b : Obj A } {f g : Hom A a b } → ( eqa : Equalizer A {a} f g) → (eq : A [ f ≈ g ] ) →
99 A [ k eqa (id1 A a) (f1=g1 eq (id1 A a)) ≈ id1 A a ] →
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 ⟩
107 e eqa o k eqa (id1 A a) (f1=g1 eq (id1 A a))
108 ≈⟨ ek=h eqa ⟩
109 id1 A a
110 ∎
111
112 --
113 -- 77 --
114 -- An isomorphic element c' of c makes another equalizer 78 -- An isomorphic element c' of c makes another equalizer
115 -- 79 --
116 -- e eqa f g f 80 -- e eqa f g f
117 -- c ----------> a ------->b 81 -- c ----------> a ------->b
119 -- || 83 -- ||
120 -- h || h-1 84 -- h || h-1
121 -- v| 85 -- v|
122 -- c' 86 -- c'
123 87
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' ) → 88 equalizer+iso : {a b c c' : Obj A } {f g : Hom A a b } {e : Hom A c a } { e' : Hom A c' a }
125 A [ A [ h-1 o h ] ≈ id1 A c ] → A [ A [ h o h-1 ] ≈ id1 A c' ] 89 ( fe=ge' : A [ A [ f o e' ] ≈ A [ g o e' ] ] )
126 → Equalizer A {c'} f g 90 ( eqa : Equalizer A e f g ) → (h-1 : Hom A c' c ) → (h : Hom A c c' ) →
127 equalizer+iso {a} {b} {c} {c'} {f} {g} eqa h-1 h h-1-id h-id = record { 91 A [ A [ e o h-1 ] ≈ e' ] → A [ A [ e' o h ] ≈ e ]
128 e = A [ e eqa o h-1 ] ; 92 → Equalizer A e' f g
93 equalizer+iso {a} {b} {c} {c'} {f} {g} {e} {e'} fe=ge' eqa h-1 h e→e' e'→e = record {
129 fe=ge = fe=ge1 ; 94 fe=ge = fe=ge1 ;
130 k = λ j eq → A [ h o k eqa j eq ] ; 95 k = λ j eq → A [ h o k eqa j eq ] ;
131 ek=h = ek=h1 ; 96 ek=h = ek=h1 ;
132 uniqueness = uniqueness1 97 uniqueness = uniqueness1
133 } where 98 } where
134 fe=ge1 : A [ A [ f o A [ e eqa o h-1 ] ] ≈ A [ g o A [ e eqa o h-1 ] ] ] 99 fe=ge1 : A [ A [ f o e' ] ≈ A [ g o e' ] ]
135 fe=ge1 = let open ≈-Reasoning (A) in 100 fe=ge1 = let open ≈-Reasoning (A) in
136 begin 101 begin
137 f o ( e eqa o h-1 ) 102 f o e'
103 ≈↑⟨ cdr e→e' ⟩
104 f o ( e o h-1 )
138 ≈⟨ assoc ⟩ 105 ≈⟨ assoc ⟩
139 (f o e eqa ) o h-1 106 (f o e ) o h-1
140 ≈⟨ car (fe=ge eqa) ⟩ 107 ≈⟨ car (fe=ge eqa) ⟩
141 (g o e eqa ) o h-1 108 (g o e ) o h-1
142 ≈↑⟨ assoc ⟩ 109 ≈↑⟨ assoc ⟩
143 g o ( e eqa o h-1 ) 110 g o ( e o h-1 )
111 ≈⟨ cdr e→e' ⟩
112 g o e'
144 ∎ 113 ∎
145 ek=h1 : {d : Obj A} {j : Hom A d a} {eq : A [ A [ f o j ] ≈ A [ g o j ] ]} → 114 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 ] 115 A [ A [ e' o A [ h o k eqa j eq ] ] ≈ j ]
147 ek=h1 {d} {j} {eq} = let open ≈-Reasoning (A) in 116 ek=h1 {d} {j} {eq} = let open ≈-Reasoning (A) in
148 begin 117 begin
149 (e eqa o h-1 ) o ( h o k eqa j eq ) 118 e' o ( h o k eqa j eq )
150 ≈↑⟨ assoc ⟩ 119 ≈⟨ assoc ⟩
151 e eqa o ( h-1 o ( h o k eqa j eq )) 120 ( e' o h) o k eqa j eq
152 ≈⟨ cdr assoc ⟩ 121 ≈⟨ car e'→e ⟩
153 e eqa o (( h-1 o h ) o k eqa j eq ) 122 e 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 ⟩ 123 ≈⟨ ek=h eqa ⟩
159 j 124 j
160 ∎ 125 ∎
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'} → 126 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' ] → 127 A [ A [ e' o j ] ≈ h' ] →
163 A [ A [ h o k eqa h' eq ] ≈ j ] 128 A [ A [ h o k eqa h' eq ] ≈ j ]
164 uniqueness1 {d} {h'} {eq} {j} ej=h = let open ≈-Reasoning (A) in 129 uniqueness1 {d} {h'} {eq} {j} ej=h = let open ≈-Reasoning (A) in
165 begin 130 begin
166 h o k eqa h' eq 131 h o k eqa h' eq
167 ≈⟨ cdr (uniqueness eqa ( 132 ≈⟨ {!!} ⟩
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 133 j
183 ∎ 134 ∎
184 135
185 -- If we have equalizer f g, e fh gh is also equalizer if we have isomorphic pair (same as above) 136
186 --
187 -- e eqa f g f
188 -- c ----------> a ------->b
189 -- ^ ---> d --->
190 -- | i h
191 -- j| k' (d' → d)
192 -- | k (d' → a)
193 -- d'
194
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 ]
197 → Equalizer A {c} (A [ f o h ]) (A [ g o h ] )
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
200 fe=ge = fe=ge1 ;
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 ∎
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
220 begin
221 ( f o h ) o i
222 ≈↑⟨ assoc ⟩
223 f o (h o i )
224 ≈⟨ cdr eq ⟩
225 f o (e eqa)
226 ≈⟨ fe=ge eqa ⟩
227 g o (e eqa)
228 ≈↑⟨ cdr eq ⟩
229 g o (h o i )
230 ≈⟨ assoc ⟩
231 ( g o h ) o i
232 ∎
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
236 begin
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'
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 ∎
273
274 -- If we have equalizer f g, e hf hg is also equalizer if we have isomorphic pair
275
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 ;
281 fe=ge = fe=ge1 ;
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 ∎
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
311 begin
312 ( h o f ) o e eqa
313 ≈↑⟨ assoc ⟩
314 h o (f o e eqa )
315 ≈⟨ cdr (fe=ge eqa) ⟩
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
327 -- If we have equalizer f g, e (ef) (eg) is also an equalizer and e = id c
328
329 eefeg : {a b c : 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} {f} {g} eqa = record {
332 e = id1 A c ; -- i ; -- A [ h-1 o e eqa ] ; -- Hom A a d
333 fe=ge = fe=ge1 ;
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 ∎
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
355 begin
356 ( f o h ) o i
357 ≈⟨ car ( fe=ge eqa ) ⟩
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 ∎
384 137
385 -- 138 --
386 -- If we have two equalizers on c and c', there are isomorphic pair h, h' 139 -- If we have two equalizers on c and c', there are isomorphic pair h, h'
387 -- 140 --
388 -- h : c → c' h' : c' → c 141 -- h : c → c' h' : c' → c
389 -- h h' = 1 and h' h = 1 142 -- e' = h o e
390 -- not yet done 143 -- e = h' o e'
391 144
392 145
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 ) 146 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 }
394 → ( keqa : Equalizer A {c} (A [ f o e eqa' ]) (A [ g o e eqa' ]) ) 147 ( eqa : Equalizer A e f g) → ( eqa' : Equalizer A e' f g )
148 → ( keqa : Equalizer A (k eqa' e {!!} ) (A [ f o e' ]) (A [ g o e' ]) )
395 → Hom A c c' 149 → Hom A c c'
396 c-iso-l {c} {c'} eqa eqa' keqa = e keqa 150 c-iso-l {c} {c'} eqa eqa' eff = {!!}
397 151
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 ) 152 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 )
399 → ( keqa : Equalizer A {c} (A [ f o e eqa' ]) (A [ g o e eqa' ]) ) 153 → ( keqa : Equalizer A (k eqa' e {!!} ) (A [ f o e' ]) (A [ g o e' ]) )
400 → Hom A c' c 154 → Hom A c' c
401 c-iso-r {c} {c'} eqa eqa' keqa = k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) 155 c-iso-r {c} {c'} eqa eqa' keqa = k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') )
402 156
403 157
404 -- e(eqa') f 158 -- e(eqa') f
411 -- 165 --
412 -- h j e f = h j e g → h = 'j e f 166 -- h j e f = h j e g → h = 'j e f
413 -- h = j e f -> j = j' 167 -- h = j e f -> j = j'
414 -- 168 --
415 169
416 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 ) 170 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 )
417 → ( keqa : Equalizer A {c} (A [ f o e eqa' ]) (A [ g o e eqa' ]) ) 171 → ( keqa : Equalizer A (k eqa' e {!!} ) (A [ f o e' ]) (A [ g o e' ]) )
418 → A [ A [ c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa ] ≈ id1 A c' ] 172 → A [ A [ c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa ] ≈ id1 A c' ]
419 c-iso {c} {c'} {a} {b} {f} {g} eqa eqa' keqa = let open ≈-Reasoning (A) in begin 173 c-iso {c} {c'} {a} {b} {f} {g} eqa eqa' keqa = let open ≈-Reasoning (A) in begin
420 c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa 174 c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa
421 ≈⟨ ek=h keqa ⟩ 175 ≈⟨ {!!} ⟩
422 id1 A c' 176 id1 A c'
423 ∎ 177 ∎
424 178
425 -- To prove c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa
426 -- ke = e' k'e' = e → k k' = 1 , k' k = 1
427 -- ke = e'
428 -- k'ke = k'e' = e kk' = 1
429 -- x e = e -> x = id?
430
431 -----
432 -- reverse arrow of e (eqa f g)
433 --
434 -- e eqa f g f
435 -- c ----------> a ------->b
436 -- <---------
437 -- k (eff) id1a
438 -- (e eqa f g) o k (eff) id1 A a = id1 A a
439 --
440 -- eqa (f (e eqa f g) ) (g (e eqa f g) )
441 -- e (eqa (f (e eqa f g) ) (g (e eqa f g) ) ) = k (eff) id1 a
442 --
443 -- (e α) o k α (id1 A c) = id1 A c
444 -- c a c
445 -- ((k (eff) id1a )) o k α e = id1 A c
446
447
448 reverse-e' : {a b c : Obj A} (f g : Hom A a b) → (h i : Hom A c b ) →
449 ( eqa : {a b c : Obj A} → (f g : Hom A a b) → Equalizer A {c} f g ) →
450 A [ k (eqa f f ) (id1 A a ) ( f1=f1 f ) ≈ (e (eqa (A [ f o e (eqa f g) ]) (A [ g o e (eqa f g) ]))) ]
451 reverse-e' = ?
452
453 reverse-e : {a b c : Obj A} (f g : Hom A a b) → (h i : Hom A c b ) →
454 ( eqa : {a b c : Obj A} → (f g : Hom A a b) → Equalizer A {c} f g ) →
455 A [
456 A [ k (eqa f f ) (id1 A a ) ( f1=f1 f ) o k (eqa ( A [ f o (e (eqa f g)) ] ) (A [ g o (e (eqa f g )) ] )) (id1 A c) (f1=g1 (fe=ge (eqa f g)) (id1 A c)) ]
457 ≈ id1 A c ]
458 reverse-e {a} {b} {c} f g h i eqa = let open ≈-Reasoning (A) in
459 begin
460 k (eqa f f ) (id1 A a ) (f1=f1 f) o k (eqa ( A [ f o (e (eqa f g)) ] ) (A [ g o (e (eqa f g )) ] )) (id1 A c) {!!}
461 ≈⟨ car {!!} ⟩
462 e (eqa ( A [ f o (e (eqa f g)) ] ) (A [ g o (e (eqa f g )) ] )) o k (eqa ( A [ f o (e (eqa f g)) ] ) (A [ g o (e (eqa f g )) ] )) (id1 A c) (f1=g1 (fe=ge (eqa f g)) (id1 A c))
463 ≈⟨ ek=h (eqa ( A [ f o (e (eqa f g)) ] ) (A [ g o (e (eqa f g )) ] )) ⟩
464 id1 A c
465 ∎
466 179
467 ---- 180 ----
468 -- 181 --
469 -- An equalizer satisfies Burroni equations 182 -- An equalizer satisfies Burroni equations
470 -- 183 --
471 -- b4 is not yet done 184 -- b4 is not yet done
472 ---- 185 ----
473 186
474 lemma-equ1 : {a b c : Obj A} (f g : Hom A a b) → 187 lemma-equ1 : {a b c : Obj A} (f g : Hom A a b) →
475 ( {a b c : Obj A} → (f g : Hom A a b) → Equalizer A {c} f g ) → Burroni A {c} f g 188 ( eqa : {a b c : Obj A} → (f g : Hom A a b) → {e : Hom A c a } { fe=ge1 : A [ A [ f o e ] ≈ A [ g o e ] ] } → Equalizer A e f g )
189 → Burroni A {c} f g
476 lemma-equ1 {a} {b} {c} f g eqa = record { 190 lemma-equ1 {a} {b} {c} f g eqa = record {
477 α = λ f g → e (eqa f g ) ; -- Hom A c a 191 α = λ f g → equalizer (eqa f g ) ; -- Hom A c a
478 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 192 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 ] ))) ] ) (lemma-equ4 {a} {b} {c} {d} f g h ) ; -- Hom A c d
479 δ = λ {a} f → k (eqa f f) (id1 A a) (lemma-equ2 f); -- Hom A a c 193 δ = λ {a} f → k (eqa f f) (id1 A a) (lemma-equ2 f); -- Hom A a c
480 b1 = fe=ge (eqa f g) ; 194 b1 = fe=ge (eqa f g) ;
481 b2 = lemma-b2 ; 195 b2 = lemma-b2 ;
482 b3 = lemma-b3 ; 196 b3 = lemma-b3 ;
483 b4 = lemma-b4 197 b4 = lemma-b4
494 -- 208 --
495 -- e o id1 ≈ e → k e ≈ id 209 -- e o id1 ≈ e → k e ≈ id
496 210
497 lemma-equ2 : {a b : Obj A} (f : Hom A a b) → A [ A [ f o id1 A a ] ≈ A [ f o id1 A a ] ] 211 lemma-equ2 : {a b : Obj A} (f : Hom A a b) → A [ A [ f o id1 A a ] ≈ A [ f o id1 A a ] ]
498 lemma-equ2 f = let open ≈-Reasoning (A) in refl-hom 212 lemma-equ2 f = let open ≈-Reasoning (A) in refl-hom
499 lemma-b3 : A [ A [ e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) ] ≈ id1 A a ] 213 lemma-b3 : A [ A [ equalizer (eqa f f ) o k (eqa f f) (id1 A a) (lemma-equ2 f) ] ≈ id1 A a ]
500 lemma-b3 = let open ≈-Reasoning (A) in 214 lemma-b3 = let open ≈-Reasoning (A) in
501 begin 215 begin
502 e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) 216 equalizer (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f)
503 ≈⟨ ek=h (eqa f f ) ⟩ 217 ≈⟨ ek=h (eqa f f ) ⟩
504 id1 A a 218 id1 A a
505 ∎ 219 ∎
506 lemma-equ4 : {a b c d : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → 220 lemma-equ4 : {a b c d : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) →
507 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 ])) ] ] ] 221 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 ])) ] ] ]
508 lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in 222 lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in
509 begin 223 begin
510 f o ( h o e (eqa (f o h) ( g o h ))) 224 f o ( h o equalizer (eqa (f o h) ( g o h )))
511 ≈⟨ assoc ⟩ 225 ≈⟨ assoc ⟩
512 (f o h) o e (eqa (f o h) ( g o h )) 226 (f o h) o equalizer (eqa (f o h) ( g o h ))
513 ≈⟨ fe=ge (eqa (A [ f o h ]) (A [ g o h ])) ⟩ 227 ≈⟨ fe=ge (eqa (A [ f o h ]) (A [ g o h ])) ⟩
514 (g o h) o e (eqa (f o h) ( g o h )) 228 (g o h) o equalizer (eqa (f o h) ( g o h ))
515 ≈↑⟨ assoc ⟩ 229 ≈↑⟨ assoc ⟩
516 g o ( h o e (eqa (f o h) ( g o h ))) 230 g o ( h o equalizer (eqa (f o h) ( g o h )))
517 ∎ 231 ∎
518 lemma-b2 : {d : Obj A} {h : Hom A d a} → A [ 232 lemma-b2 : {d : Obj A} {h : Hom A d a} → A [
519 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) ] 233 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) ]
520 ≈ A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ] ] 234 ≈ A [ h o equalizer (eqa (A [ f o h ]) (A [ g o h ])) ] ]
521 lemma-b2 {d} {h} = let open ≈-Reasoning (A) in 235 lemma-b2 {d} {h} = let open ≈-Reasoning (A) in
522 begin 236 begin
523 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) 237 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)
524 ≈⟨ ek=h (eqa f g) ⟩ 238 ≈⟨ ek=h (eqa f g) ⟩
525 h o e (eqa (f o h ) ( g o h )) 239 h o equalizer (eqa (f o h ) ( g o h ))
526 ∎ 240 ∎
527 241
528 ------- α(f,g)j id d = α(f,g)j 242 ------- α(f,g)j id d = α(f,g)j
529 ------- α(f,g)j id d = α(f,g)j 243 ------- α(f,g)j id d = α(f,g)j
530 ------- α(f,g)j α(fα(f,g)j,fα(f,g)j) δ(fα(f,g)j) = α(f,g)j 244 ------- α(f,g)j α(fα(f,g)j,fα(f,g)j) δ(fα(f,g)j) = α(f,g)j
532 ------- α(f,g)j α(fα(f,g)j,gα(f,g)j) δ(fα(f,g)j) = α(f,g)j 246 ------- α(f,g)j α(fα(f,g)j,gα(f,g)j) δ(fα(f,g)j) = α(f,g)j
533 ------- α(f,g) γ(f,g,α(f,g)j) δ(fα(f,g)j) = α(f,g)j 247 ------- α(f,g) γ(f,g,α(f,g)j) δ(fα(f,g)j) = α(f,g)j
534 ------- γ(f,g,α(f,g)j) δ(fα(f,g)j) = j 248 ------- γ(f,g,α(f,g)j) δ(fα(f,g)j) = j
535 249
536 lemma-b4 : {d : Obj A} {j : Hom A d c} → A [ 250 lemma-b4 : {d : Obj A} {j : Hom A d c} → A [
537 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 ] ])) ]) 251 A [ k (eqa f g) (A [ A [ equalizer (eqa f g) o j ] o equalizer (eqa (A [ f o A [ equalizer (eqa f g) o j ] ]) (A [ g o A [ equalizer (eqa f g) o j ] ])) ])
538 (lemma-equ4 {a} {b} {c} f g (A [ e (eqa f g) o j ])) o 252 (lemma-equ4 {a} {b} {c} f g (A [ equalizer (eqa f g) o j ])) o
539 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 ] ])) ] 253 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 ] ])) ]
540 ≈ j ] 254 ≈ j ]
541 lemma-b4 {d} {j} = let open ≈-Reasoning (A) in 255 lemma-b4 {d} {j} = let open ≈-Reasoning (A) in
542 begin 256 begin
543 ( 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 ) ))) )) 257 ( k (eqa f g) (( ( equalizer (eqa f g) o j ) o equalizer (eqa (( f o ( equalizer (eqa f g) o j ) )) (( g o ( equalizer (eqa f g) o j ) ))) ))
544 (lemma-equ4 {a} {b} {c} f g (( e (eqa f g) o j ))) o 258 (lemma-equ4 {a} {b} {c} f g (( equalizer (eqa f g) o j ))) o
545 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 ) ))) ) 259 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 ) ))) )
546 ≈⟨ {!!} ⟩ 260 ≈⟨ {!!} ⟩
547 j 261 j
548 ∎ 262 ∎
549 263
550 264