Mercurial > hg > Members > kono > Proof > category
annotate equalizer.agda @ 250:a1e2228c2a6b
equalizer iso done.
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Mon, 09 Sep 2013 15:56:34 +0900 |
parents | bdeed843f8b1 |
children | 40947f08bab6 |
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 |
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 |
230 | 33 -- |
225 | 34 -- Flat Equational Definition of Equalizer |
230 | 35 -- |
236
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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 |
245 | 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 |
245 | 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 } |
245 | 42 → {f g g' : Hom A a b } → A [ g ≈ g' ] → A [ α f g e ≈ α f g' e ] |
242 | 43 cong-γ : {a _ c d : Obj A } → {f g : Hom A a b} {h h' : Hom A d a } → A [ h ≈ h' ] |
243 | 44 → A [ γ {a} {b} {c} {d} f g h ≈ γ f g h' ] |
245 | 45 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' ] |
46 b1 : A [ A [ f o α {a} {b} {c} f g e ] ≈ A [ g o α {a} {b} {c} f g e ] ] | |
47 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) ] ] | |
48 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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49 -- 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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50 b4 : {d : Obj A } {k : Hom A d c} → |
245 | 51 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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52 -- A [ α f g o β f g h ] ≈ h |
238
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53 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 | 54 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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55 |
209 | 56 open Equalizer |
225 | 57 open Burroni |
209 | 58 |
225 | 59 -- |
60 -- Some obvious conditions for k (fe = ge) → ( fh = gh ) | |
61 -- | |
219 | 62 |
224 | 63 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 ] ] |
64 f1=g1 eq h = let open ≈-Reasoning (A) in (resp refl-hom eq ) | |
65 | |
226 | 66 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 | 67 f1=f1 f = let open ≈-Reasoning (A) in refl-hom |
226 | 68 |
224 | 69 f1=gh : { a b c d : Obj A } {f g : Hom A a b } → { e : Hom A c a } → { h : Hom A d c } → |
70 (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 | 71 f1=gh {a} {b} {c} {d} {f} {g} {e} {h} eq = let open ≈-Reasoning (A) in |
224 | 72 begin |
73 f o ( e o h ) | |
74 ≈⟨ assoc ⟩ | |
230 | 75 (f o e ) o h |
224 | 76 ≈⟨ car eq ⟩ |
230 | 77 (g o e ) o h |
224 | 78 ≈↑⟨ assoc ⟩ |
79 g o ( e o h ) | |
80 ∎ | |
219 | 81 |
225 | 82 -- |
83 -- | |
249 | 84 -- An isomorphic arrow c' to c makes another equalizer |
225 | 85 -- |
230 | 86 -- e eqa f g f |
222 | 87 -- c ----------> a ------->b |
230 | 88 -- |^ |
89 -- || | |
222 | 90 -- h || h-1 |
230 | 91 -- v| |
92 -- c' | |
222 | 93 |
234 | 94 equalizer+iso : {a b c c' : Obj A } {f g : Hom A a b } {e : Hom A c a } |
95 (h-1 : Hom A c' c ) → (h : Hom A c c' ) → | |
96 A [ A [ h o h-1 ] ≈ id1 A c' ] → A [ A [ h-1 o h ] ≈ id1 A c ] → | |
97 ( fe=ge' : A [ A [ f o A [ e o h-1 ] ] ≈ A [ g o A [ e o h-1 ] ] ] ) | |
98 ( eqa : Equalizer A e f g ) | |
99 → Equalizer A (A [ e o h-1 ] ) f g | |
100 equalizer+iso {a} {b} {c} {c'} {f} {g} {e} h-1 h hh-1=1 h-1h=1 fe=ge' eqa = record { | |
222 | 101 fe=ge = fe=ge1 ; |
102 k = λ j eq → A [ h o k eqa j eq ] ; | |
230 | 103 ek=h = ek=h1 ; |
222 | 104 uniqueness = uniqueness1 |
105 } where | |
234 | 106 fe=ge1 : A [ A [ f o A [ e o h-1 ] ] ≈ A [ g o A [ e o h-1 ] ] ] |
107 fe=ge1 = fe=ge' | |
222 | 108 ek=h1 : {d : Obj A} {j : Hom A d a} {eq : A [ A [ f o j ] ≈ A [ g o j ] ]} → |
234 | 109 A [ A [ A [ e o h-1 ] o A [ h o k eqa j eq ] ] ≈ j ] |
222 | 110 ek=h1 {d} {j} {eq} = let open ≈-Reasoning (A) in |
111 begin | |
234 | 112 ( e o h-1 ) o ( h o k eqa j eq ) |
113 ≈↑⟨ assoc ⟩ | |
114 e o ( h-1 o ( h o k eqa j eq ) ) | |
115 ≈⟨ cdr assoc ⟩ | |
116 e o (( h-1 o h) o k eqa j eq ) | |
117 ≈⟨ cdr (car h-1h=1 ) ⟩ | |
118 e o (id1 A c o k eqa j eq ) | |
119 ≈⟨ cdr idL ⟩ | |
120 e o k eqa j eq | |
222 | 121 ≈⟨ ek=h eqa ⟩ |
122 j | |
123 ∎ | |
124 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 | 125 A [ A [ A [ e o h-1 ] o j ] ≈ h' ] → |
222 | 126 A [ A [ h o k eqa h' eq ] ≈ j ] |
127 uniqueness1 {d} {h'} {eq} {j} ej=h = let open ≈-Reasoning (A) in | |
128 begin | |
129 h o k eqa h' eq | |
234 | 130 ≈⟨ cdr (uniqueness eqa ( begin |
131 e o ( h-1 o j ) | |
132 ≈⟨ assoc ⟩ | |
133 (e o h-1 ) o j | |
134 ≈⟨ ej=h ⟩ | |
135 h' | |
136 ∎ )) ⟩ | |
137 h o ( h-1 o j ) | |
138 ≈⟨ assoc ⟩ | |
139 (h o h-1 ) o j | |
140 ≈⟨ car hh-1=1 ⟩ | |
141 id1 A c' o j | |
142 ≈⟨ idL ⟩ | |
222 | 143 j |
144 ∎ | |
145 | |
225 | 146 -- |
147 -- If we have two equalizers on c and c', there are isomorphic pair h, h' | |
148 -- | |
149 -- h : c → c' h' : c' → c | |
233
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150 -- e' = h o e |
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151 -- e = h' o e' |
225 | 152 |
233
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153 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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154 ( eqa : Equalizer A e f g) → ( eqa' : Equalizer A e' f g ) |
234 | 155 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) |
241 | 156 → Hom A c c' -- should be e' = c-sio-l o e |
234 | 157 c-iso-l {c} {c'} eqa eqa' keqa = equalizer keqa |
226 | 158 |
233
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159 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 | 160 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) |
241 | 161 → Hom A c' c -- e = c-sio-r o e' |
230 | 162 c-iso-r {c} {c'} eqa eqa' keqa = k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) |
223 | 163 |
234 | 164 -- e' f |
230 | 165 -- c'----------> a ------->b f e j = g e j |
166 -- ^ g | |
167 -- |k h | |
229
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168 -- | h = e(eqaj) o k jhek = jh (uniqueness) |
230 | 169 -- | |
170 -- c j o (k (eqa ef ef) j ) = id c h = e(eqaj) | |
171 -- | |
172 -- h j e f = h j e g → h = 'j e f | |
229
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173 -- h = j e f -> j = j' |
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174 -- |
228 | 175 |
250 | 176 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 ) |
177 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) | |
178 → A [ A [ e' o c-iso-l eqa eqa' keqa ] ≈ e ] | |
179 c-iso→' {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' keqa = let open ≈-Reasoning (A) in begin | |
180 e' o c-iso-l eqa eqa' keqa | |
181 ≈⟨⟩ | |
182 e' o k eqa' e (fe=ge eqa) | |
183 ≈⟨⟩ | |
184 equalizer eqa' o k eqa' e (fe=ge eqa) | |
185 ≈⟨ ek=h eqa' ⟩ | |
186 e | |
187 ∎ | |
188 | |
189 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 ) | |
190 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) | |
191 → A [ A [ e o c-iso-r eqa eqa' keqa ] ≈ e' ] | |
192 c-iso←' {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' keqa = let open ≈-Reasoning (A) in begin | |
193 e o c-iso-r eqa eqa' keqa | |
194 ≈⟨⟩ | |
195 e o k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) | |
196 ≈↑⟨ car (ek=h eqa' ) ⟩ | |
197 ( equalizer eqa' o k eqa' e (fe=ge eqa) ) o k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) | |
198 ≈⟨⟩ | |
199 ( e' o k eqa' e (fe=ge eqa) ) o k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) | |
200 ≈↑⟨ assoc ⟩ | |
201 e' o (( k eqa' e (fe=ge eqa) ) o k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) ) | |
202 ≈⟨⟩ | |
203 e' o (equalizer keqa o k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) ) | |
204 ≈⟨ cdr ( ek=h keqa ) ⟩ | |
205 e' o id1 A c' | |
206 ≈⟨ idR ⟩ | |
207 e' | |
208 ∎ | |
209 | |
234 | 210 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 ) |
211 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) | |
229
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212 → A [ A [ c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa ] ≈ id1 A c' ] |
234 | 213 c-iso→ {c} {c'} {a} {b} {f} {g} eqa eqa' keqa = let open ≈-Reasoning (A) in begin |
229
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214 c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa |
234 | 215 ≈⟨⟩ |
216 equalizer keqa o k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) | |
217 ≈⟨ ek=h keqa ⟩ | |
229
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218 id1 A c' |
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219 ∎ |
226 | 220 |
234 | 221 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 ) |
222 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa )) (A [ f o e' ]) (A [ g o e' ]) ) | |
223 → ( keqa' : Equalizer A (k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') )) (A [ f o e ]) (A [ g o e ]) ) | |
224 → A [ A [ c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa ] ≈ id1 A c ] | |
250 | 225 c-iso← {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' keqa keqa' = let open ≈-Reasoning (A) in begin |
234 | 226 c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa |
227 ≈⟨⟩ | |
228 k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) o k eqa' e (fe=ge eqa ) | |
229 ≈⟨⟩ | |
230 equalizer keqa' o k eqa' e (fe=ge eqa ) | |
231 ≈⟨ cdr ( begin | |
232 k eqa' e (fe=ge eqa ) | |
233 ≈⟨ uniqueness eqa' ( begin | |
234 e' o k keqa' (id1 A c) (f1=g1 (fe=ge eqa) (id1 A c)) | |
250 | 235 ≈↑⟨ car (c-iso←' eqa eqa' keqa ) ⟩ |
234 | 236 ( e o equalizer keqa' ) o k keqa' (id1 A c) (f1=g1 (fe=ge eqa) (id1 A c)) |
237 ≈↑⟨ assoc ⟩ | |
238 e o ( equalizer keqa' o k keqa' (id1 A c) (f1=g1 (fe=ge eqa) (id1 A c))) | |
239 ≈⟨ cdr ( ek=h keqa' ) ⟩ | |
240 e o id1 A c | |
241 ≈⟨ idR ⟩ | |
242 e | |
243 ∎ )⟩ | |
244 k keqa' (id1 A c) ( f1=g1 (fe=ge eqa) (id1 A c) ) | |
245 ∎ )⟩ | |
250 | 246 equalizer keqa' o k keqa' (id1 A c) ( f1=g1 (fe=ge eqa) (id1 A c) ) ≈⟨ ek=h keqa' ⟩ |
234 | 247 id1 A c |
248 ∎ | |
249 | |
250 | |
230 | 251 |
225 | 252 ---- |
253 -- | |
254 -- An equalizer satisfies Burroni equations | |
255 -- | |
256 ---- | |
257 | |
236
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258 lemma-equ1 : {a b c : Obj A} (f g : Hom A a b) → (e : Hom A c a ) → |
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259 ( 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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260 → Burroni A {c} {a} {b} f g e |
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261 lemma-equ1 {a} {b} {c} f g e eqa = record { |
245 | 262 α = λ {a} {b} {c} f g e → equalizer (eqa {a} {b} {c} f g {e} ) ; -- Hom A c a |
242 | 263 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 ] ))) ] ) |
264 (lemma-equ4 {a} {b} {c} {d} f g h ) ; -- Hom A c d | |
249 | 265 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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266 cong-α = λ {a b c e f g g'} eq → cong-α1 {a} {b} {c} {e} {f} {g} {g'} eq ; |
247 | 267 cong-γ = λ {a} {_} {c} {d} {f} {g} {h} {h'} eq → cong-γ1 {a} {c} {d} {f} {g} {h} {h'} eq ; |
245 | 268 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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269 b1 = fe=ge (eqa {a} {b} {c} f g {e}) ; |
226 | 270 b2 = lemma-b2 ; |
271 b3 = lemma-b3 ; | |
230 | 272 b4 = lemma-b4 |
211 | 273 } where |
216 | 274 -- |
275 -- e eqa f g f | |
276 -- c ----------> a ------->b | |
230 | 277 -- ^ g |
278 -- | | |
216 | 279 -- |k₁ = e eqa (f o (e (eqa f g))) (g o (e (eqa f g)))) |
230 | 280 -- | |
216 | 281 -- d |
230 | 282 -- |
283 -- | |
216 | 284 -- e o id1 ≈ e → k e ≈ id |
285 | |
249 | 286 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 | 287 lemma-b3 {a} {b} {d} f {h} = let open ≈-Reasoning (A) in |
230 | 288 begin |
249 | 289 equalizer (eqa f f) o k (eqa f f) (id1 A a) (f1=f1 f) |
215 | 290 ≈⟨ ek=h (eqa f f ) ⟩ |
211 | 291 id1 A a |
292 ∎ | |
230 | 293 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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294 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 | 295 lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in |
212 | 296 begin |
233
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297 f o ( h o equalizer (eqa (f o h) ( g o h ))) |
212 | 298 ≈⟨ assoc ⟩ |
233
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299 (f o h) o equalizer (eqa (f o h) ( g o h )) |
221 | 300 ≈⟨ fe=ge (eqa (A [ f o h ]) (A [ g o h ])) ⟩ |
233
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301 (g o h) o equalizer (eqa (f o h) ( g o h )) |
212 | 302 ≈↑⟨ assoc ⟩ |
233
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303 g o ( h o equalizer (eqa (f o h) ( g o h ))) |
212 | 304 ∎ |
245 | 305 cong-α1 : {a b c : Obj A } → { e : Hom A c a } |
306 → {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} ) ] | |
307 cong-α1 {a} {b} {c} {e} {f} {g} {g'} eq = let open ≈-Reasoning (A) in refl-hom | |
247 | 308 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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309 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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310 ≈ 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 | 311 cong-γ1 {a} {c} {d} {f} {g} {h} {h'} h=h' {e} = let open ≈-Reasoning (A) in begin |
245 | 312 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 ) |
313 ≈⟨ uniqueness (eqa f g) ( begin | |
248 | 314 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' ) |
315 ≈⟨ ek=h (eqa f g ) ⟩ | |
316 h' o (equalizer ( eqa (A [ f o h' ] ) (A [ g o h' ] ))) | |
317 ≈↑⟨ car h=h' ⟩ | |
318 h o (equalizer ( eqa (A [ f o h' ] ) (A [ g o h' ] ))) | |
245 | 319 ∎ )⟩ |
320 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' ) | |
321 ∎ | |
249 | 322 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) ≈ |
323 k (eqa {a} {b} {c} f' f' {e} ) (id1 A a) (f1=f1 f') ] | |
247 | 324 cong-δ1 {a} {b} {c} {e} {f} {f'} f=f' = let open ≈-Reasoning (A) in |
325 begin | |
249 | 326 k (eqa {a} {b} {c} f f {e} ) (id1 A a) (f1=f1 f) |
247 | 327 ≈⟨ uniqueness (eqa f f) ( begin |
249 | 328 e o k (eqa {a} {b} {c} f' f' {e} ) (id1 A a) (f1=f1 f') |
247 | 329 ≈⟨ ek=h (eqa {a} {b} {c} f' f' {e} ) ⟩ |
330 id1 A a | |
331 ∎ )⟩ | |
249 | 332 k (eqa {a} {b} {c} f' f' {e} ) (id1 A a) (f1=f1 f') |
247 | 333 ∎ |
334 | |
230 | 335 lemma-b2 : {d : Obj A} {h : Hom A d a} → A [ |
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336 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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337 ≈ A [ h o equalizer (eqa (A [ f o h ]) (A [ g o h ])) ] ] |
226 | 338 lemma-b2 {d} {h} = let open ≈-Reasoning (A) in |
212 | 339 begin |
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340 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 | 341 ≈⟨ ek=h (eqa f g) ⟩ |
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342 h o equalizer (eqa (f o h ) ( g o h )) |
212 | 343 ∎ |
230 | 344 |
345 lemma-b4 : {d : Obj A} {j : Hom A d c} → A [ | |
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346 A [ k (eqa f g) (A [ A [ equalizer (eqa f g) o j ] o |
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347 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
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348 (lemma-equ4 {a} {b} {c} f g (A [ equalizer (eqa f g) o j ])) o |
249 | 349 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) (f1=f1 (A [ f o A [ equalizer (eqa f g) o j ] ])) ] |
222 | 350 ≈ j ] |
230 | 351 lemma-b4 {d} {j} = let open ≈-Reasoning (A) in |
215 | 352 begin |
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353 ( 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
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354 (lemma-equ4 {a} {b} {c} f g (( equalizer (eqa f g) o j ))) o |
249 | 355 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 ) ))) ) |
235
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356 ≈⟨ car ((uniqueness (eqa f g) ( begin |
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357 equalizer (eqa f g) o j |
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358 ≈↑⟨ idR ⟩ |
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359 (equalizer (eqa f g) o j ) o id1 A d |
241 | 360 ≈⟨⟩ -- here we decide e (fej) (gej)' is id1 A d |
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361 ((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
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362 ∎ ))) ⟩ |
249 | 363 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 ) ))) |
235
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364 ≈⟨ cdr ((uniqueness (eqa (( f o ( equalizer (eqa f g) o j ) )) (( f o ( equalizer (eqa f g) o j ) ))) ( begin |
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365 equalizer (eqa (f o equalizer (eqa f g {e} ) o j) (f o equalizer (eqa f g {e}) o j)) o id1 A d |
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366 ≈⟨ idR ⟩ |
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367 equalizer (eqa (f o equalizer (eqa f g {e}) o j) (f o equalizer (eqa f g {e}) o j)) |
241 | 368 ≈⟨⟩ -- here we decide e (fej) (fej)' is id1 A d |
235
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369 id1 A d |
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370 ∎ ))) ⟩ |
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371 j o id1 A d |
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372 ≈⟨ idR ⟩ |
222 | 373 j |
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374 ∎ |
211 | 375 |
376 | |
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377 lemma-equ2 : {a b c : Obj A} (f g : Hom A a b) (e : Hom A c a ) |
245 | 378 → ( 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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379 lemma-equ2 {a} {b} {c} f g e bur = record { |
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380 fe=ge = fe=ge1 ; |
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381 k = k1 ; |
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382 ek=h = λ {d} {h} {eq} → ek=h1 {d} {h} {eq} ; |
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383 uniqueness = λ {d} {h} {eq} {k'} ek=h → uniqueness1 {d} {h} {eq} {k'} ek=h |
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384 } where |
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385 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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386 k1 {d} h fh=gh = β bur {d} {a} {b} f g h |
245 | 387 fe=ge1 : A [ A [ f o (α bur f g e) ] ≈ A [ g o (α bur f g e) ] ] |
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388 fe=ge1 = b1 bur |
245 | 389 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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390 ek=h1 {d} {h} {eq} = let open ≈-Reasoning (A) in |
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391 begin |
245 | 392 α bur f g e o k1 h eq |
239 | 393 ≈⟨⟩ |
245 | 394 α bur f g e o ( γ bur {a} {b} {c} f g h o δ bur {d} {b} {d} (id1 A d) (f o h) ) |
239 | 395 ≈⟨ assoc ⟩ |
245 | 396 ( α bur f g e o γ bur {a} {b} {c} f g h ) o δ bur {d} {b} {d} (id1 A d) (f o h) |
239 | 397 ≈⟨ car (b2 bur) ⟩ |
245 | 398 ( h o ( α bur ( f o h ) ( g o h ) (id1 A d))) o δ bur {d} {b} {d} (id1 A d) (f o h) |
239 | 399 ≈↑⟨ assoc ⟩ |
245 | 400 h o ((( α bur ( f o h ) ( g o h ) (id1 A d) )) o δ bur {d} {b} {d} (id1 A d) (f o h) ) |
240 | 401 ≈↑⟨ cdr ( car ( cong-α bur eq)) ⟩ |
245 | 402 h o ((( α bur ( f o h ) ( f o h ) (id1 A d)))o δ bur {d} {b} {d} (id1 A d) (f o h) ) |
240 | 403 ≈⟨ cdr (b3 bur {d} {b} {d} (f o h) {id1 A d} ) ⟩ |
239 | 404 h o id1 A d |
240 | 405 ≈⟨ idR ⟩ |
238
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406 h |
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407 ∎ |
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408 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 | 409 A [ A [ (α bur f g e) o k' ] ≈ h ] → A [ k1 {d} h eq ≈ k' ] |
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410 uniqueness1 {d} {h} {eq} {k'} ek=h = let open ≈-Reasoning (A) in |
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411 begin |
240 | 412 k1 {d} h eq |
239 | 413 ≈⟨⟩ |
245 | 414 γ bur {a} {b} {c} f g h o δ bur {d} {b} {d} (id1 A d) (f o h) |
240 | 415 ≈↑⟨ car (cong-γ bur {a} {b} {c} {d} ek=h ) ⟩ |
245 | 416 γ bur f g (A [ α bur f g e o k' ]) o δ bur {d} {b} {d} (id1 A d) (f o h) |
417 ≈↑⟨ cdr (cong-δ bur (resp ek=h refl-hom )) ⟩ | |
418 γ bur f g (A [ α bur f g e o k' ]) o δ bur {d} {b} {d} (id1 A d) ( f o ( α bur f g e o k') ) | |
240 | 419 ≈⟨ b4 bur ⟩ |
238
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420 k' |
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421 ∎ |
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422 |
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423 |
225 | 424 -- end |
212 | 425 |
426 | |
427 | |
215 | 428 |
429 |