Mercurial > hg > Members > kono > Proof > category
annotate equalizer.agda @ 241:9e4dc349831e
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sun, 08 Sep 2013 12:34:35 +0900 |
parents | 964e258e08fb |
children | 5d1ecfec6f41 |
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 | |
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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 -- |
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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 |
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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 |
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40 δ : {a b c : Obj A } → {e : Hom A c a } → (f : Hom A a b) → Hom A a c |
240 | 41 cong-α : {a b c : Obj A } → {g g' : Hom A a b } → A [ g ≈ g' ] → { α α' : Hom A c a } → A [ α ≈ α' ] |
42 cong-γ : {a _ c d : Obj A } → {h h' : Hom A d a } → A [ h ≈ h' ] → { γ γ' : Hom A d c } → A [ γ ≈ γ' ] | |
43 cong-δ : {a b c : Obj A } → {f f' : Hom A a b} → A [ f ≈ f' ] → { δ δ' : Hom A a c } → A [ δ ≈ δ' ] | |
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44 b1 : A [ A [ f o α {a} {b} {c} f g {e} ] ≈ A [ g o α {a} {b} {c} f g {e} ] ] |
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45 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} ] ] |
240 | 46 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 ] |
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47 -- 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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48 b4 : {d : Obj A } {k : Hom A d c} → |
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49 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 ] |
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50 -- A [ α f g o β f g h ] ≈ h |
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51 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 |
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52 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 ]) ] |
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53 |
209 | 54 open Equalizer |
225 | 55 open Burroni |
209 | 56 |
225 | 57 -- |
58 -- Some obvious conditions for k (fe = ge) → ( fh = gh ) | |
59 -- | |
219 | 60 |
224 | 61 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 ] ] |
62 f1=g1 eq h = let open ≈-Reasoning (A) in (resp refl-hom eq ) | |
63 | |
226 | 64 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 | 65 f1=f1 f = let open ≈-Reasoning (A) in refl-hom |
226 | 66 |
224 | 67 f1=gh : { a b c d : Obj A } {f g : Hom A a b } → { e : Hom A c a } → { h : Hom A d c } → |
68 (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 | 69 f1=gh {a} {b} {c} {d} {f} {g} {e} {h} eq = let open ≈-Reasoning (A) in |
224 | 70 begin |
71 f o ( e o h ) | |
72 ≈⟨ assoc ⟩ | |
230 | 73 (f o e ) o h |
224 | 74 ≈⟨ car eq ⟩ |
230 | 75 (g o e ) o h |
224 | 76 ≈↑⟨ assoc ⟩ |
77 g o ( e o h ) | |
78 ∎ | |
219 | 79 |
225 | 80 -- |
81 -- | |
82 -- An isomorphic element c' of c makes another equalizer | |
83 -- | |
230 | 84 -- e eqa f g f |
222 | 85 -- c ----------> a ------->b |
230 | 86 -- |^ |
87 -- || | |
222 | 88 -- h || h-1 |
230 | 89 -- v| |
90 -- c' | |
222 | 91 |
234 | 92 equalizer+iso : {a b c c' : Obj A } {f g : Hom A a b } {e : Hom A c a } |
93 (h-1 : Hom A c' c ) → (h : Hom A c c' ) → | |
94 A [ A [ h o h-1 ] ≈ id1 A c' ] → A [ A [ h-1 o h ] ≈ id1 A c ] → | |
95 ( fe=ge' : A [ A [ f o A [ e o h-1 ] ] ≈ A [ g o A [ e o h-1 ] ] ] ) | |
96 ( eqa : Equalizer A e f g ) | |
97 → Equalizer A (A [ e o h-1 ] ) f g | |
98 equalizer+iso {a} {b} {c} {c'} {f} {g} {e} h-1 h hh-1=1 h-1h=1 fe=ge' eqa = record { | |
222 | 99 fe=ge = fe=ge1 ; |
100 k = λ j eq → A [ h o k eqa j eq ] ; | |
230 | 101 ek=h = ek=h1 ; |
222 | 102 uniqueness = uniqueness1 |
103 } where | |
234 | 104 fe=ge1 : A [ A [ f o A [ e o h-1 ] ] ≈ A [ g o A [ e o h-1 ] ] ] |
105 fe=ge1 = fe=ge' | |
222 | 106 ek=h1 : {d : Obj A} {j : Hom A d a} {eq : A [ A [ f o j ] ≈ A [ g o j ] ]} → |
234 | 107 A [ A [ A [ e o h-1 ] o A [ h o k eqa j eq ] ] ≈ j ] |
222 | 108 ek=h1 {d} {j} {eq} = let open ≈-Reasoning (A) in |
109 begin | |
234 | 110 ( e o h-1 ) o ( h o k eqa j eq ) |
111 ≈↑⟨ assoc ⟩ | |
112 e o ( h-1 o ( h o k eqa j eq ) ) | |
113 ≈⟨ cdr assoc ⟩ | |
114 e o (( h-1 o h) o k eqa j eq ) | |
115 ≈⟨ cdr (car h-1h=1 ) ⟩ | |
116 e o (id1 A c o k eqa j eq ) | |
117 ≈⟨ cdr idL ⟩ | |
118 e o k eqa j eq | |
222 | 119 ≈⟨ ek=h eqa ⟩ |
120 j | |
121 ∎ | |
122 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 | 123 A [ A [ A [ e o h-1 ] o j ] ≈ h' ] → |
222 | 124 A [ A [ h o k eqa h' eq ] ≈ j ] |
125 uniqueness1 {d} {h'} {eq} {j} ej=h = let open ≈-Reasoning (A) in | |
126 begin | |
127 h o k eqa h' eq | |
234 | 128 ≈⟨ cdr (uniqueness eqa ( begin |
129 e o ( h-1 o j ) | |
130 ≈⟨ assoc ⟩ | |
131 (e o h-1 ) o j | |
132 ≈⟨ ej=h ⟩ | |
133 h' | |
134 ∎ )) ⟩ | |
135 h o ( h-1 o j ) | |
136 ≈⟨ assoc ⟩ | |
137 (h o h-1 ) o j | |
138 ≈⟨ car hh-1=1 ⟩ | |
139 id1 A c' o j | |
140 ≈⟨ idL ⟩ | |
222 | 141 j |
142 ∎ | |
143 | |
225 | 144 -- |
145 -- If we have two equalizers on c and c', there are isomorphic pair h, h' | |
146 -- | |
147 -- h : c → c' h' : c' → c | |
233
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148 -- e' = h o e |
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149 -- e = h' o e' |
225 | 150 |
151 | |
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152 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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153 ( eqa : Equalizer A e f g) → ( eqa' : Equalizer A e' f g ) |
234 | 154 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) |
241 | 155 → Hom A c c' -- should be e' = c-sio-l o e |
234 | 156 c-iso-l {c} {c'} eqa eqa' keqa = equalizer keqa |
226 | 157 |
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158 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 | 159 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) |
241 | 160 → Hom A c' c -- e = c-sio-r o e' |
230 | 161 c-iso-r {c} {c'} eqa eqa' keqa = k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) |
223 | 162 |
234 | 163 -- e' f |
230 | 164 -- c'----------> a ------->b f e j = g e j |
165 -- ^ g | |
166 -- |k h | |
229
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167 -- | h = e(eqaj) o k jhek = jh (uniqueness) |
230 | 168 -- | |
169 -- c j o (k (eqa ef ef) j ) = id c h = e(eqaj) | |
170 -- | |
171 -- h j e f = h j e g → h = 'j e f | |
229
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172 -- h = j e f -> j = j' |
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173 -- |
228 | 174 |
241 | 175 -- e = c-iso-l o e' is assumed by equalizer's degree of freedom |
234 | 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' ]) ) | |
229
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178 → A [ A [ c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa ] ≈ id1 A c' ] |
234 | 179 c-iso→ {c} {c'} {a} {b} {f} {g} eqa eqa' keqa = let open ≈-Reasoning (A) in begin |
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180 c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa |
234 | 181 ≈⟨⟩ |
182 equalizer keqa o k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) | |
183 ≈⟨ ek=h keqa ⟩ | |
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184 id1 A c' |
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185 ∎ |
226 | 186 |
234 | 187 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 ) |
188 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa )) (A [ f o e' ]) (A [ g o e' ]) ) | |
189 → ( keqa' : Equalizer A (k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') )) (A [ f o e ]) (A [ g o e ]) ) | |
241 | 190 -- e' = c-iso-r o e is assumed by equalizer's degree of freedom |
191 → { e'->e : A [ e' ≈ A [ e o equalizer keqa' ] ] } -- implicit assumption , which should be refl | |
234 | 192 → A [ A [ c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa ] ≈ id1 A c ] |
193 c-iso← {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' keqa keqa' {e'->e} = let open ≈-Reasoning (A) in begin | |
194 c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa | |
195 ≈⟨⟩ | |
196 k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) o k eqa' e (fe=ge eqa ) | |
197 ≈⟨⟩ | |
198 equalizer keqa' o k eqa' e (fe=ge eqa ) | |
199 ≈⟨ cdr ( begin | |
200 k eqa' e (fe=ge eqa ) | |
201 ≈⟨ uniqueness eqa' ( begin | |
202 e' o k keqa' (id1 A c) (f1=g1 (fe=ge eqa) (id1 A c)) | |
203 ≈⟨ car e'->e ⟩ | |
204 ( e o equalizer keqa' ) o k keqa' (id1 A c) (f1=g1 (fe=ge eqa) (id1 A c)) | |
205 ≈↑⟨ assoc ⟩ | |
206 e o ( equalizer keqa' o k keqa' (id1 A c) (f1=g1 (fe=ge eqa) (id1 A c))) | |
207 ≈⟨ cdr ( ek=h keqa' ) ⟩ | |
208 e o id1 A c | |
209 ≈⟨ idR ⟩ | |
210 e | |
211 ∎ )⟩ | |
212 k keqa' (id1 A c) ( f1=g1 (fe=ge eqa) (id1 A c) ) | |
213 ∎ )⟩ | |
214 equalizer keqa' o k keqa' (id1 A c) ( f1=g1 (fe=ge eqa) (id1 A c) ) | |
215 ≈⟨ ek=h keqa' ⟩ | |
216 id1 A c | |
217 ∎ | |
218 | |
219 | |
230 | 220 |
225 | 221 ---- |
222 -- | |
223 -- An equalizer satisfies Burroni equations | |
224 -- | |
241 | 225 -- congs are not yet done |
225 | 226 ---- |
227 | |
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228 lemma-equ1 : {a b c : Obj A} (f g : Hom A a b) → (e : Hom A c a ) → |
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229 ( 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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230 → Burroni A {c} {a} {b} f g e |
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231 lemma-equ1 {a} {b} {c} f g e eqa = record { |
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232 α = λ {a} {b} {c} f g {e} → equalizer (eqa {a} {b} {c} f g {e} ) ; -- Hom A c a |
233
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233 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 |
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234 δ = λ {a} {b} {c} {e} f → k (eqa {a} {b} {c} f f {e} ) (id1 A a) (lemma-equ2 f); -- Hom A a c |
240 | 235 cong-α = cong-α1 ; |
236 cong-γ = cong-γ1 ; | |
237 cong-δ = cong-δ1 ; | |
238
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238 b1 = fe=ge (eqa {a} {b} {c} f g {e}) ; |
226 | 239 b2 = lemma-b2 ; |
240 b3 = lemma-b3 ; | |
230 | 241 b4 = lemma-b4 |
211 | 242 } where |
240 | 243 cong-α1 : {a b c : Obj A } → {g g' : Hom A a b } → A [ g ≈ g' ] → { α α' : Hom A c a } → A [ α ≈ α' ] |
244 cong-α1 {a} {b} {c} {g} {g'} eq = let open ≈-Reasoning (A) in {!!} | |
245 cong-γ1 : {a _ c d : Obj A } → {h h' : Hom A d a } → A [ h ≈ h' ] → { γ γ' : Hom A d c } → A [ γ ≈ γ' ] | |
246 cong-γ1 = {!!} | |
247 cong-δ1 : {a b c : Obj A } → {f f' : Hom A a b} → A [ f ≈ f' ] → { δ δ' : Hom A a c } → A [ δ ≈ δ' ] | |
248 cong-δ1 = {!!} | |
216 | 249 -- |
250 -- e eqa f g f | |
251 -- c ----------> a ------->b | |
230 | 252 -- ^ g |
253 -- | | |
216 | 254 -- |k₁ = e eqa (f o (e (eqa f g))) (g o (e (eqa f g)))) |
230 | 255 -- | |
216 | 256 -- d |
230 | 257 -- |
258 -- | |
216 | 259 -- e o id1 ≈ e → k e ≈ id |
260 | |
211 | 261 lemma-equ2 : {a b : Obj A} (f : Hom A a b) → A [ A [ f o id1 A a ] ≈ A [ f o id1 A a ] ] |
262 lemma-equ2 f = let open ≈-Reasoning (A) in refl-hom | |
240 | 263 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) (lemma-equ2 f) ] ≈ id1 A a ] |
264 lemma-b3 {a} {b} {d} f {h} = let open ≈-Reasoning (A) in | |
230 | 265 begin |
233
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266 equalizer (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) |
215 | 267 ≈⟨ ek=h (eqa f f ) ⟩ |
211 | 268 id1 A a |
269 ∎ | |
230 | 270 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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271 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 | 272 lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in |
212 | 273 begin |
233
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274 f o ( h o equalizer (eqa (f o h) ( g o h ))) |
212 | 275 ≈⟨ assoc ⟩ |
233
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276 (f o h) o equalizer (eqa (f o h) ( g o h )) |
221 | 277 ≈⟨ fe=ge (eqa (A [ f o h ]) (A [ g o h ])) ⟩ |
233
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278 (g o h) o equalizer (eqa (f o h) ( g o h )) |
212 | 279 ≈↑⟨ assoc ⟩ |
233
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280 g o ( h o equalizer (eqa (f o h) ( g o h ))) |
212 | 281 ∎ |
230 | 282 lemma-b2 : {d : Obj A} {h : Hom A d a} → A [ |
233
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283 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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284 ≈ A [ h o equalizer (eqa (A [ f o h ]) (A [ g o h ])) ] ] |
226 | 285 lemma-b2 {d} {h} = let open ≈-Reasoning (A) in |
212 | 286 begin |
233
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287 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 | 288 ≈⟨ ek=h (eqa f g) ⟩ |
233
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289 h o equalizer (eqa (f o h ) ( g o h )) |
212 | 290 ∎ |
230 | 291 |
292 lemma-b4 : {d : Obj A} {j : Hom A d c} → A [ | |
236
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293 A [ k (eqa f g) (A [ A [ equalizer (eqa f g) o j ] o |
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294 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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295 (lemma-equ4 {a} {b} {c} f g (A [ equalizer (eqa f g) o j ])) o |
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296 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 ] ])) ] |
222 | 297 ≈ j ] |
230 | 298 lemma-b4 {d} {j} = let open ≈-Reasoning (A) in |
215 | 299 begin |
236
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300 ( 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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301 (lemma-equ4 {a} {b} {c} f g (( equalizer (eqa f g) o j ))) o |
236
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302 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 ) ))) ) |
235
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303 ≈⟨ car ((uniqueness (eqa f g) ( begin |
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304 equalizer (eqa f g) o j |
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305 ≈↑⟨ idR ⟩ |
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306 (equalizer (eqa f g) o j ) o id1 A d |
241 | 307 ≈⟨⟩ -- here we decide e (fej) (gej)' is id1 A d |
236
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308 ((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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309 ∎ ))) ⟩ |
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310 j o 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 ) ))) |
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311 ≈⟨ cdr ((uniqueness (eqa (( f o ( equalizer (eqa f g) o j ) )) (( f o ( equalizer (eqa f g) o j ) ))) ( begin |
236
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312 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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313 ≈⟨ idR ⟩ |
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314 equalizer (eqa (f o equalizer (eqa f g {e}) o j) (f o equalizer (eqa f g {e}) o j)) |
241 | 315 ≈⟨⟩ -- here we decide e (fej) (fej)' is id1 A d |
235
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316 id1 A d |
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317 ∎ ))) ⟩ |
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318 j o id1 A d |
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319 ≈⟨ idR ⟩ |
222 | 320 j |
235
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321 ∎ |
211 | 322 |
323 | |
238
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324 lemma-equ2 : {a b c : Obj A} (f g : Hom A a b) (e : Hom A c a ) |
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325 → ( bur : Burroni A {c} {a} {b} f g e ) → Equalizer A {c} {a} {b} (α bur f g) f g |
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326 lemma-equ2 {a} {b} {c} f g e bur = record { |
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327 fe=ge = fe=ge1 ; |
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328 k = k1 ; |
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329 ek=h = λ {d} {h} {eq} → ek=h1 {d} {h} {eq} ; |
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330 uniqueness = λ {d} {h} {eq} {k'} ek=h → uniqueness1 {d} {h} {eq} {k'} ek=h |
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331 } where |
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332 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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333 k1 {d} h fh=gh = β bur {d} {a} {b} f g h |
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334 fe=ge1 : A [ A [ f o (α bur f g) ] ≈ A [ g o (α bur f g) ] ] |
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335 fe=ge1 = b1 bur |
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336 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) o k1 {d} h eq ] ≈ h ] |
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337 ek=h1 {d} {h} {eq} = let open ≈-Reasoning (A) in |
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338 begin |
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339 α bur f g o k1 h eq |
239 | 340 ≈⟨⟩ |
341 α bur f g o ( γ bur {a} {b} {c} f g h o δ bur {d} {b} {d} {id1 A d} (f o h) ) | |
342 ≈⟨ assoc ⟩ | |
343 ( α bur f g o γ bur {a} {b} {c} f g h ) o δ bur {d} {b} {d} {id1 A d} (f o h) | |
344 ≈⟨ car (b2 bur) ⟩ | |
345 ( h o ( α bur ( f o h ) ( g o h ))) o δ bur {d} {b} {d} {id1 A d} (f o h) | |
346 ≈↑⟨ assoc ⟩ | |
347 h o ((( α bur ( f o h ) ( g o h ))) o δ bur {d} {b} {d} {id1 A d} (f o h) ) | |
240 | 348 ≈↑⟨ cdr ( car ( cong-α bur eq)) ⟩ |
349 h o ((( α bur ( f o h ) ( f o h ))) o δ bur {d} {b} {d} {id1 A d} (f o h) ) | |
350 ≈⟨ cdr (b3 bur {d} {b} {d} (f o h) {id1 A d} ) ⟩ | |
239 | 351 h o id1 A d |
240 | 352 ≈⟨ idR ⟩ |
238
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353 h |
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354 ∎ |
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355 uniqueness1 : {d : Obj A} → ∀ {h : Hom A d a} → {eq : A [ A [ f o h ] ≈ A [ g o h ] ] } → {k' : Hom A d c } → |
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356 A [ A [ (α bur f g) o k' ] ≈ h ] → A [ k1 {d} h eq ≈ k' ] |
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357 uniqueness1 {d} {h} {eq} {k'} ek=h = let open ≈-Reasoning (A) in |
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358 begin |
240 | 359 k1 {d} h eq |
239 | 360 ≈⟨⟩ |
240 | 361 γ bur {a} {b} {c} f g h o δ bur {d} {b} {d} {id1 A d} (f o h) |
362 ≈↑⟨ car (cong-γ bur {a} {b} {c} {d} ek=h ) ⟩ | |
363 γ bur f g (A [ α bur f g o k' ]) o δ bur {d} {b} {d} {id1 A d} (f o h) | |
364 ≈↑⟨ cdr (cong-δ bur {d} {a} {d} (resp {d} {d} {a} {id1 A d} refl-hom ek=h )) ⟩ | |
365 γ bur f g (A [ α bur f g o k' ]) o δ bur (A [ f o A [ α bur f g o k' ] ]) | |
366 ≈⟨ b4 bur ⟩ | |
238
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367 k' |
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368 ∎ |
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369 |
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370 |
225 | 371 -- end |
212 | 372 |
373 | |
374 | |
215 | 375 |
376 |