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annotate equalizer.agda @ 259:c442322d22b3
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Tue, 17 Sep 2013 11:27:57 +0900 |
parents | 281b8962abbe |
children | a87d3ea9efe4 |
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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
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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 |
253 | 33 |
230 | 34 -- |
251 | 35 -- Burroni's Flat Equational Definition of Equalizer |
230 | 36 -- |
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37 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 | 38 field |
245 | 39 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 | 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 |
245 | 41 δ : {a b c : Obj A } → (e : Hom A c a ) → (f : Hom A a b) → Hom A a c |
242 | 42 cong-α : {a b c : Obj A } → { e : Hom A c a } |
245 | 43 → {f g g' : Hom A a b } → A [ g ≈ g' ] → A [ α f g e ≈ α f g' e ] |
242 | 44 cong-γ : {a _ c d : Obj A } → {f g : Hom A a b} {h h' : Hom A d a } → A [ h ≈ h' ] |
243 | 45 → A [ γ {a} {b} {c} {d} f g h ≈ γ f g h' ] |
245 | 46 cong-δ : {a b c : Obj A } → {e : Hom A c a} → {f f' : Hom A a b} → A [ f ≈ f' ] → A [ δ e f ≈ δ e f' ] |
47 b1 : A [ A [ f o α {a} {b} {c} f g e ] ≈ A [ g o α {a} {b} {c} f g e ] ] | |
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) ] ] | |
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
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50 -- b4 : {c d : Obj A } {k : Hom A c a} → A [ β f g ( A [ α f g o k ] ) ≈ k ] |
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51 b4 : {d : Obj A } {k : Hom A d c} → |
245 | 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
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53 -- A [ α f g o β f g h ] ≈ h |
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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 |
245 | 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
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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 |
255
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83 ------------------------------- |
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84 -- |
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85 -- Every equalizer is monic |
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86 -- |
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87 -- e i = e j → i = j |
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88 -- |
259 | 89 -- e eqa f g f |
90 -- c ----------> a ------->b | |
91 -- ^^ | |
92 -- || | |
93 -- i||j | |
94 -- || | |
95 -- d | |
255
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96 |
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97 monoic : { c a b d : Obj A } {f g : Hom A a b } → {e : Hom A c a } ( eqa : Equalizer A e f g) |
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98 → { i j : Hom A d c } |
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99 → A [ A [ equalizer eqa o i ] ≈ A [ equalizer eqa o j ] ] → A [ i ≈ j ] |
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100 monoic {c} {a} {b} {d} {f} {g} {e} eqa {i} {j} ei=ej = let open ≈-Reasoning (A) in begin |
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101 i |
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102 ≈↑⟨ uniqueness eqa ( begin |
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103 equalizer eqa o i |
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104 ≈⟨ ei=ej ⟩ |
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105 equalizer eqa o j |
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106 ∎ )⟩ |
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107 k eqa (equalizer eqa o j) ( f1=gh (fe=ge eqa ) ) |
257 | 108 ≈⟨ uniqueness eqa ( begin |
109 equalizer eqa o j | |
110 ≈⟨⟩ | |
111 equalizer eqa o j | |
112 ∎ )⟩ | |
255
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113 j |
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114 ∎ |
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115 |
251 | 116 -------------------------------- |
225 | 117 -- |
118 -- | |
259 | 119 -- Isomorphic arrows from c' to c makes another equalizer |
225 | 120 -- |
230 | 121 -- e eqa f g f |
222 | 122 -- c ----------> a ------->b |
230 | 123 -- |^ |
124 -- || | |
222 | 125 -- h || h-1 |
230 | 126 -- v| |
127 -- c' | |
222 | 128 |
234 | 129 equalizer+iso : {a b c c' : Obj A } {f g : Hom A a b } {e : Hom A c a } |
130 (h-1 : Hom A c' c ) → (h : Hom A c c' ) → | |
131 A [ A [ h o h-1 ] ≈ id1 A c' ] → A [ A [ h-1 o h ] ≈ id1 A c ] → | |
132 ( eqa : Equalizer A e f g ) | |
133 → Equalizer A (A [ e o h-1 ] ) f g | |
254 | 134 equalizer+iso {a} {b} {c} {c'} {f} {g} {e} h-1 h hh-1=1 h-1h=1 eqa = record { |
222 | 135 fe=ge = fe=ge1 ; |
136 k = λ j eq → A [ h o k eqa j eq ] ; | |
230 | 137 ek=h = ek=h1 ; |
222 | 138 uniqueness = uniqueness1 |
139 } where | |
234 | 140 fe=ge1 : A [ A [ f o A [ e o h-1 ] ] ≈ A [ g o A [ e o h-1 ] ] ] |
254 | 141 fe=ge1 = f1=gh ( fe=ge eqa ) |
222 | 142 ek=h1 : {d : Obj A} {j : Hom A d a} {eq : A [ A [ f o j ] ≈ A [ g o j ] ]} → |
234 | 143 A [ A [ A [ e o h-1 ] o A [ h o k eqa j eq ] ] ≈ j ] |
222 | 144 ek=h1 {d} {j} {eq} = let open ≈-Reasoning (A) in |
145 begin | |
234 | 146 ( e o h-1 ) o ( h o k eqa j eq ) |
147 ≈↑⟨ assoc ⟩ | |
148 e o ( h-1 o ( h o k eqa j eq ) ) | |
149 ≈⟨ cdr assoc ⟩ | |
150 e o (( h-1 o h) o k eqa j eq ) | |
151 ≈⟨ cdr (car h-1h=1 ) ⟩ | |
253 | 152 e o (id c o k eqa j eq ) |
234 | 153 ≈⟨ cdr idL ⟩ |
154 e o k eqa j eq | |
222 | 155 ≈⟨ ek=h eqa ⟩ |
156 j | |
157 ∎ | |
158 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 | 159 A [ A [ A [ e o h-1 ] o j ] ≈ h' ] → |
222 | 160 A [ A [ h o k eqa h' eq ] ≈ j ] |
161 uniqueness1 {d} {h'} {eq} {j} ej=h = let open ≈-Reasoning (A) in | |
162 begin | |
163 h o k eqa h' eq | |
234 | 164 ≈⟨ cdr (uniqueness eqa ( begin |
165 e o ( h-1 o j ) | |
166 ≈⟨ assoc ⟩ | |
167 (e o h-1 ) o j | |
168 ≈⟨ ej=h ⟩ | |
169 h' | |
170 ∎ )) ⟩ | |
171 h o ( h-1 o j ) | |
172 ≈⟨ assoc ⟩ | |
173 (h o h-1 ) o j | |
174 ≈⟨ car hh-1=1 ⟩ | |
253 | 175 id c' o j |
234 | 176 ≈⟨ idL ⟩ |
222 | 177 j |
178 ∎ | |
179 | |
251 | 180 -------------------------------- |
225 | 181 -- |
182 -- If we have two equalizers on c and c', there are isomorphic pair h, h' | |
183 -- | |
184 -- h : c → c' h' : c' → c | |
233
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185 -- e' = h o e |
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186 -- e = h' o e' |
259 | 187 -- |
188 -- | |
189 -- | |
190 -- e eqa f g f | |
191 -- c ---------->a ------->b | |
192 -- ^ ^ g | |
193 -- | | | |
194 -- |k = id c' | | |
195 -- v | | |
196 -- c'-----------+ | |
197 -- e eqa' f g | |
225 | 198 |
233
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199 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 } |
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200 ( eqa : Equalizer A e f g) → ( eqa' : Equalizer A e' f g ) |
258 | 201 → Hom A c c' |
202 c-iso-l {c} {c'} eqa eqa' = k eqa' (equalizer eqa) ( fe=ge eqa ) | |
223 | 203 |
258 | 204 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 } |
205 ( eqa : Equalizer A e f g) → ( eqa' : Equalizer A e' f g ) | |
206 → Hom A c' c | |
207 c-iso-r {c} {c'} eqa eqa' = k eqa (equalizer eqa') ( fe=ge eqa' ) | |
228 | 208 |
258 | 209 c-iso-lr : { c c' a b : Obj A } {f g : Hom A a b } → {e : Hom A c a } { e' : Hom A c' a } |
210 ( eqa : Equalizer A e f g) → ( eqa' : Equalizer A e' f g ) → | |
211 A [ A [ c-iso-l eqa eqa' o c-iso-r eqa eqa' ] ≈ id1 A c' ] | |
212 c-iso-lr {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' = let open ≈-Reasoning (A) in begin | |
213 c-iso-l eqa eqa' o c-iso-r eqa eqa' | |
250 | 214 ≈⟨⟩ |
258 | 215 k eqa' (equalizer eqa) ( fe=ge eqa ) o k eqa (equalizer eqa') ( fe=ge eqa' ) |
216 ≈↑⟨ uniqueness eqa' ( begin | |
217 e' o ( k eqa' (equalizer eqa) (fe=ge eqa) o k eqa (equalizer eqa') (fe=ge eqa') ) | |
218 ≈⟨ assoc ⟩ | |
219 ( e' o k eqa' (equalizer eqa) (fe=ge eqa) ) o k eqa (equalizer eqa') (fe=ge eqa') | |
220 ≈⟨ car (ek=h eqa') ⟩ | |
221 e o k eqa (equalizer eqa') (fe=ge eqa') | |
222 ≈⟨ ek=h eqa ⟩ | |
223 e' | |
224 ∎ )⟩ | |
225 k eqa' e' ( fe=ge eqa' ) | |
226 ≈⟨ uniqueness eqa' ( begin | |
227 e' o id c' | |
250 | 228 ≈⟨ idR ⟩ |
258 | 229 e' |
230 ∎ )⟩ | |
253 | 231 id c' |
229
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232 ∎ |
226 | 233 |
258 | 234 -- c-iso-rl is obvious from the symmetry |
234 | 235 |
230 | 236 |
251 | 237 -------------------------------- |
225 | 238 ---- |
239 -- | |
254 | 240 -- Existence of equalizer satisfies Burroni equations |
225 | 241 -- |
242 ---- | |
243 | |
236
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244 lemma-equ1 : {a b c : Obj A} (f g : Hom A a b) → (e : Hom A c a ) → |
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245 ( eqa : {a b c : Obj A} → (f g : Hom A a b) → {e : Hom A c a } → Equalizer A e f g ) |
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246 → Burroni A {c} {a} {b} f g e |
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247 lemma-equ1 {a} {b} {c} f g e eqa = record { |
245 | 248 α = λ {a} {b} {c} f g e → equalizer (eqa {a} {b} {c} f g {e} ) ; -- Hom A c a |
242 | 249 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 ] ))) ] ) |
250 (lemma-equ4 {a} {b} {c} {d} f g h ) ; -- Hom A c d | |
249 | 251 δ = λ {a} {b} {c} e f → k (eqa {a} {b} {c} f f {e} ) (id1 A a) (f1=f1 f); -- Hom A a c |
246
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252 cong-α = λ {a b c e f g g'} eq → cong-α1 {a} {b} {c} {e} {f} {g} {g'} eq ; |
247 | 253 cong-γ = λ {a} {_} {c} {d} {f} {g} {h} {h'} eq → cong-γ1 {a} {c} {d} {f} {g} {h} {h'} eq ; |
245 | 254 cong-δ = λ {a b c e f f'} f=f' → cong-δ1 {a} {b} {c} {e} {f} {f'} f=f' ; |
238
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255 b1 = fe=ge (eqa {a} {b} {c} f g {e}) ; |
226 | 256 b2 = lemma-b2 ; |
257 b3 = lemma-b3 ; | |
230 | 258 b4 = lemma-b4 |
211 | 259 } where |
216 | 260 -- |
261 -- e eqa f g f | |
262 -- c ----------> a ------->b | |
230 | 263 -- ^ g |
264 -- | | |
216 | 265 -- |k₁ = e eqa (f o (e (eqa f g))) (g o (e (eqa f g)))) |
230 | 266 -- | |
216 | 267 -- d |
230 | 268 -- |
269 -- | |
216 | 270 -- e o id1 ≈ e → k e ≈ id |
271 | |
249 | 272 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) (f1=f1 f) ] ≈ id1 A a ] |
240 | 273 lemma-b3 {a} {b} {d} f {h} = let open ≈-Reasoning (A) in |
230 | 274 begin |
253 | 275 equalizer (eqa f f) o k (eqa f f) (id a) (f1=f1 f) |
215 | 276 ≈⟨ ek=h (eqa f f ) ⟩ |
253 | 277 id a |
211 | 278 ∎ |
230 | 279 lemma-equ4 : {a b c d : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → |
233
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280 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 | 281 lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in |
212 | 282 begin |
233
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283 f o ( h o equalizer (eqa (f o h) ( g o h ))) |
212 | 284 ≈⟨ assoc ⟩ |
233
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285 (f o h) o equalizer (eqa (f o h) ( g o h )) |
221 | 286 ≈⟨ fe=ge (eqa (A [ f o h ]) (A [ g o h ])) ⟩ |
233
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287 (g o h) o equalizer (eqa (f o h) ( g o h )) |
212 | 288 ≈↑⟨ assoc ⟩ |
233
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289 g o ( h o equalizer (eqa (f o h) ( g o h ))) |
212 | 290 ∎ |
245 | 291 cong-α1 : {a b c : Obj A } → { e : Hom A c a } |
292 → {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} ) ] | |
293 cong-α1 {a} {b} {c} {e} {f} {g} {g'} eq = let open ≈-Reasoning (A) in refl-hom | |
247 | 294 cong-γ1 : {a c d : Obj A } → {f g : Hom A a b} {h h' : Hom A d a } → A [ h ≈ h' ] → { e : Hom A c a} → |
246
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295 A [ k (eqa f g {e} ) {d} ( A [ h o (equalizer ( eqa (A [ f o h ] ) (A [ g o h ] ) {id1 A d} )) ] ) (lemma-equ4 {a} {b} {c} {d} f g h ) |
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296 ≈ k (eqa f g {e} ) {d} ( A [ h' o (equalizer ( eqa (A [ f o h' ] ) (A [ g o h' ] ) {id1 A d} )) ] ) (lemma-equ4 {a} {b} {c} {d} f g h' ) ] |
247 | 297 cong-γ1 {a} {c} {d} {f} {g} {h} {h'} h=h' {e} = let open ≈-Reasoning (A) in begin |
245 | 298 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 ) |
299 ≈⟨ uniqueness (eqa f g) ( begin | |
248 | 300 e o 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' ) |
301 ≈⟨ ek=h (eqa f g ) ⟩ | |
302 h' o (equalizer ( eqa (A [ f o h' ] ) (A [ g o h' ] ))) | |
303 ≈↑⟨ car h=h' ⟩ | |
304 h o (equalizer ( eqa (A [ f o h' ] ) (A [ g o h' ] ))) | |
245 | 305 ∎ )⟩ |
306 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' ) | |
307 ∎ | |
249 | 308 cong-δ1 : {a b c : Obj A} {e : Hom A c a } {f f' : Hom A a b} → A [ f ≈ f' ] → A [ k (eqa {a} {b} {c} f f {e} ) (id1 A a) (f1=f1 f) ≈ |
309 k (eqa {a} {b} {c} f' f' {e} ) (id1 A a) (f1=f1 f') ] | |
247 | 310 cong-δ1 {a} {b} {c} {e} {f} {f'} f=f' = let open ≈-Reasoning (A) in |
311 begin | |
253 | 312 k (eqa {a} {b} {c} f f {e} ) (id a) (f1=f1 f) |
247 | 313 ≈⟨ uniqueness (eqa f f) ( begin |
253 | 314 e o k (eqa {a} {b} {c} f' f' {e} ) (id a) (f1=f1 f') |
247 | 315 ≈⟨ ek=h (eqa {a} {b} {c} f' f' {e} ) ⟩ |
253 | 316 id a |
247 | 317 ∎ )⟩ |
253 | 318 k (eqa {a} {b} {c} f' f' {e} ) (id a) (f1=f1 f') |
247 | 319 ∎ |
320 | |
230 | 321 lemma-b2 : {d : Obj A} {h : Hom A d a} → A [ |
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322 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) ] |
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323 ≈ A [ h o equalizer (eqa (A [ f o h ]) (A [ g o h ])) ] ] |
226 | 324 lemma-b2 {d} {h} = let open ≈-Reasoning (A) in |
212 | 325 begin |
233
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326 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 | 327 ≈⟨ ek=h (eqa f g) ⟩ |
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328 h o equalizer (eqa (f o h ) ( g o h )) |
212 | 329 ∎ |
230 | 330 |
331 lemma-b4 : {d : Obj A} {j : Hom A d c} → A [ | |
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332 A [ k (eqa f g) (A [ A [ equalizer (eqa f g) o j ] o |
254 | 333 equalizer (eqa (A [ f o A [ equalizer (eqa f g {e}) o j ] ]) (A [ g o A [ equalizer (eqa f g {e} ) o j ] ])) ]) |
334 (lemma-equ4 {a} {b} {c} f g (A [ equalizer (eqa f g) o j ])) | |
335 o k (eqa (A [ f o A [ equalizer (eqa f g) o j ] ]) (A [ f o A [ equalizer (eqa f g) o j ] ])) | |
336 (id1 A d) (f1=f1 (A [ f o A [ equalizer (eqa f g) o j ] ])) ] | |
222 | 337 ≈ j ] |
230 | 338 lemma-b4 {d} {j} = let open ≈-Reasoning (A) in |
215 | 339 begin |
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340 ( 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 ) ))) )) |
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341 (lemma-equ4 {a} {b} {c} f g (( equalizer (eqa f g) o j ))) o |
249 | 342 k (eqa (( f o ( equalizer (eqa f g) o j ) )) (( f o ( equalizer (eqa f g) o j ) ))) (id1 A d) (f1=f1 (( f o ( equalizer (eqa f g) o j ) ))) ) |
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343 ≈⟨ car ((uniqueness (eqa f g) ( begin |
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344 equalizer (eqa f g) o j |
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345 ≈↑⟨ idR ⟩ |
253 | 346 (equalizer (eqa f g) o j ) o id d |
347 ≈⟨⟩ -- here we decide e (fej) (gej)' is id d | |
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348 ((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))) |
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349 ∎ ))) ⟩ |
249 | 350 j o k (eqa (( f o ( equalizer (eqa f g) o j ) )) (( f o ( equalizer (eqa f g) o j ) ))) (id1 A d) (f1=f1 (( f o ( equalizer (eqa f g) o j ) ))) |
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351 ≈⟨ cdr ((uniqueness (eqa (( f o ( equalizer (eqa f g) o j ) )) (( f o ( equalizer (eqa f g) o j ) ))) ( begin |
253 | 352 equalizer (eqa (f o equalizer (eqa f g {e} ) o j) (f o equalizer (eqa f g {e}) o j)) o id d |
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353 ≈⟨ idR ⟩ |
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354 equalizer (eqa (f o equalizer (eqa f g {e}) o j) (f o equalizer (eqa f g {e}) o j)) |
253 | 355 ≈⟨⟩ -- here we decide e (fej) (fej)' is id d |
356 id d | |
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357 ∎ ))) ⟩ |
253 | 358 j o id d |
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359 ≈⟨ idR ⟩ |
222 | 360 j |
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361 ∎ |
211 | 362 |
251 | 363 -------------------------------- |
364 -- | |
365 -- Bourroni equations gives an Equalizer | |
366 -- | |
211 | 367 |
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368 lemma-equ2 : {a b c : Obj A} (f g : Hom A a b) (e : Hom A c a ) |
245 | 369 → ( bur : Burroni A {c} {a} {b} f g e ) → Equalizer A {c} {a} {b} (α bur f g e) f g |
238
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370 lemma-equ2 {a} {b} {c} f g e bur = record { |
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371 fe=ge = fe=ge1 ; |
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372 k = k1 ; |
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373 ek=h = λ {d} {h} {eq} → ek=h1 {d} {h} {eq} ; |
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374 uniqueness = λ {d} {h} {eq} {k'} ek=h → uniqueness1 {d} {h} {eq} {k'} ek=h |
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375 } where |
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376 k1 : {d : Obj A} (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d c |
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377 k1 {d} h fh=gh = β bur {d} {a} {b} f g h |
245 | 378 fe=ge1 : A [ A [ f o (α bur f g e) ] ≈ A [ g o (α bur f g e) ] ] |
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379 fe=ge1 = b1 bur |
245 | 380 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 e) o k1 {d} h eq ] ≈ h ] |
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381 ek=h1 {d} {h} {eq} = let open ≈-Reasoning (A) in |
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382 begin |
245 | 383 α bur f g e o k1 h eq |
239 | 384 ≈⟨⟩ |
253 | 385 α bur f g e o ( γ bur {a} {b} {c} f g h o δ bur {d} {b} {d} (id d) (f o h) ) |
239 | 386 ≈⟨ assoc ⟩ |
253 | 387 ( α bur f g e o γ bur {a} {b} {c} f g h ) o δ bur {d} {b} {d} (id d) (f o h) |
239 | 388 ≈⟨ car (b2 bur) ⟩ |
253 | 389 ( h o ( α bur ( f o h ) ( g o h ) (id d))) o δ bur {d} {b} {d} (id d) (f o h) |
239 | 390 ≈↑⟨ assoc ⟩ |
253 | 391 h o ((( α bur ( f o h ) ( g o h ) (id d) )) o δ bur {d} {b} {d} (id d) (f o h) ) |
240 | 392 ≈↑⟨ cdr ( car ( cong-α bur eq)) ⟩ |
253 | 393 h o ((( α bur ( f o h ) ( f o h ) (id d)))o δ bur {d} {b} {d} (id d) (f o h) ) |
394 ≈⟨ cdr (b3 bur {d} {b} {d} (f o h) {id d} ) ⟩ | |
395 h o id d | |
240 | 396 ≈⟨ idR ⟩ |
238
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397 h |
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398 ∎ |
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399 uniqueness1 : {d : Obj A} → ∀ {h : Hom A d a} → {eq : A [ A [ f o h ] ≈ A [ g o h ] ] } → {k' : Hom A d c } → |
245 | 400 A [ A [ (α bur f g e) o k' ] ≈ h ] → A [ k1 {d} h eq ≈ k' ] |
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401 uniqueness1 {d} {h} {eq} {k'} ek=h = let open ≈-Reasoning (A) in |
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402 begin |
240 | 403 k1 {d} h eq |
239 | 404 ≈⟨⟩ |
253 | 405 γ bur {a} {b} {c} f g h o δ bur {d} {b} {d} (id d) (f o h) |
240 | 406 ≈↑⟨ car (cong-γ bur {a} {b} {c} {d} ek=h ) ⟩ |
253 | 407 γ bur f g (A [ α bur f g e o k' ]) o δ bur {d} {b} {d} (id d) (f o h) |
245 | 408 ≈↑⟨ cdr (cong-δ bur (resp ek=h refl-hom )) ⟩ |
253 | 409 γ bur f g (A [ α bur f g e o k' ]) o δ bur {d} {b} {d} (id d) ( f o ( α bur f g e o k') ) |
240 | 410 ≈⟨ b4 bur ⟩ |
238
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411 k' |
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412 ∎ |
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413 |
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414 |
225 | 415 -- end |
212 | 416 |
417 | |
418 | |
215 | 419 |
420 |