Mercurial > hg > Members > kono > Proof > category
diff equalizer.agda @ 233:4bba19bc71be
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 |
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--- a/equalizer.agda Sat Sep 07 23:29:13 2013 +0900 +++ b/equalizer.agda Sun Sep 08 01:37:24 2013 +0900 @@ -20,9 +20,8 @@ open import HomReasoning open import cat-utility -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 +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 field - e : Hom A c a fe=ge : A [ A [ f o e ] ≈ A [ g o e ] ] k : {d : Obj A} (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d c 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 ] @@ -75,41 +74,6 @@ ∎ -- --- For e f f, we need e eqa = id1 A a, but it is equal to say k eqa (id a) is id --- --- Equalizer has free choice of c up to isomorphism, we cannot prove eqa = id a - -equalizer-eq-k : { a b : Obj A } {f g : Hom A a b } → (eq : A [ f ≈ g ] ) → ( eqa : Equalizer A {a} f g) → - A [ e eqa ≈ id1 A a ] → - A [ k eqa (id1 A a) (f1=g1 eq (id1 A a)) ≈ id1 A a ] -equalizer-eq-k {a} {b} {f} {g} eq eqa e=1 = let open ≈-Reasoning (A) in - begin - k eqa (id1 A a) (f1=g1 eq (id1 A a)) - ≈⟨ uniqueness eqa ( begin - e eqa o id1 A a - ≈⟨ idR ⟩ - e eqa - ≈⟨ e=1 ⟩ - id1 A a - ∎ )⟩ - id1 A a - ∎ - -equalizer-eq-e : { a b : Obj A } {f g : Hom A a b } → ( eqa : Equalizer A {a} f g) → (eq : A [ f ≈ g ] ) → - A [ k eqa (id1 A a) (f1=g1 eq (id1 A a)) ≈ id1 A a ] → - A [ e eqa ≈ id1 A a ] -equalizer-eq-e {a} {b} {f} {g} eqa eq k=1 = let open ≈-Reasoning (A) in - begin - e eqa - ≈↑⟨ idR ⟩ - e eqa o id1 A a - ≈↑⟨ cdr k=1 ⟩ - e eqa o k eqa (id1 A a) (f1=g1 eq (id1 A a)) - ≈⟨ ek=h eqa ⟩ - id1 A a - ∎ - --- -- -- An isomorphic element c' of c makes another equalizer -- @@ -121,282 +85,72 @@ -- v| -- c' -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' ) → - A [ A [ h-1 o h ] ≈ id1 A c ] → A [ A [ h o h-1 ] ≈ id1 A c' ] - → Equalizer A {c'} f g -equalizer+iso {a} {b} {c} {c'} {f} {g} eqa h-1 h h-1-id h-id = record { - e = A [ e eqa o h-1 ] ; +equalizer+iso : {a b c c' : Obj A } {f g : Hom A a b } {e : Hom A c a } { e' : Hom A c' a } + ( fe=ge' : A [ A [ f o e' ] ≈ A [ g o e' ] ] ) + ( eqa : Equalizer A e f g ) → (h-1 : Hom A c' c ) → (h : Hom A c c' ) → + A [ A [ e o h-1 ] ≈ e' ] → A [ A [ e' o h ] ≈ e ] + → Equalizer A e' f g +equalizer+iso {a} {b} {c} {c'} {f} {g} {e} {e'} fe=ge' eqa h-1 h e→e' e'→e = record { fe=ge = fe=ge1 ; k = λ j eq → A [ h o k eqa j eq ] ; ek=h = ek=h1 ; uniqueness = uniqueness1 } where - fe=ge1 : A [ A [ f o A [ e eqa o h-1 ] ] ≈ A [ g o A [ e eqa o h-1 ] ] ] + fe=ge1 : A [ A [ f o e' ] ≈ A [ g o e' ] ] fe=ge1 = let open ≈-Reasoning (A) in begin - f o ( e eqa o h-1 ) + f o e' + ≈↑⟨ cdr e→e' ⟩ + f o ( e o h-1 ) ≈⟨ assoc ⟩ - (f o e eqa ) o h-1 + (f o e ) o h-1 ≈⟨ car (fe=ge eqa) ⟩ - (g o e eqa ) o h-1 + (g o e ) o h-1 ≈↑⟨ assoc ⟩ - g o ( e eqa o h-1 ) + g o ( e o h-1 ) + ≈⟨ cdr e→e' ⟩ + g o e' ∎ ek=h1 : {d : Obj A} {j : Hom A d a} {eq : A [ A [ f o j ] ≈ A [ g o j ] ]} → - A [ A [ A [ e eqa o h-1 ] o A [ h o k eqa j eq ] ] ≈ j ] + A [ A [ e' o A [ h o k eqa j eq ] ] ≈ j ] ek=h1 {d} {j} {eq} = let open ≈-Reasoning (A) in begin - (e eqa o h-1 ) o ( h o k eqa j eq ) - ≈↑⟨ assoc ⟩ - e eqa o ( h-1 o ( h o k eqa j eq )) - ≈⟨ cdr assoc ⟩ - e eqa o (( h-1 o h ) o k eqa j eq ) - ≈⟨ cdr (car (h-1-id )) ⟩ - e eqa o (id1 A c o k eqa j eq ) - ≈⟨ cdr idL ⟩ - e eqa o (k eqa j eq ) + e' o ( h o k eqa j eq ) + ≈⟨ assoc ⟩ + ( e' o h) o k eqa j eq + ≈⟨ car e'→e ⟩ + e o k eqa j eq ≈⟨ ek=h eqa ⟩ j ∎ uniqueness1 : {d : Obj A} {h' : Hom A d a} {eq : A [ A [ f o h' ] ≈ A [ g o h' ] ]} {j : Hom A d c'} → - A [ A [ A [ e eqa o h-1 ] o j ] ≈ h' ] → + A [ A [ e' o j ] ≈ h' ] → A [ A [ h o k eqa h' eq ] ≈ j ] uniqueness1 {d} {h'} {eq} {j} ej=h = let open ≈-Reasoning (A) in begin h o k eqa h' eq - ≈⟨ cdr (uniqueness eqa ( - begin - e eqa o ( h-1 o j ) - ≈⟨ assoc ⟩ - (e eqa o h-1 ) o j - ≈⟨ ej=h ⟩ - h' - ∎ - )) ⟩ - h o ( h-1 o j ) - ≈⟨ assoc ⟩ - (h o h-1 ) o j - ≈⟨ car h-id ⟩ - id1 A c' o j - ≈⟨ idL ⟩ + ≈⟨ {!!} ⟩ j ∎ --- If we have equalizer f g, e fh gh is also equalizer if we have isomorphic pair (same as above) --- --- e eqa f g f --- c ----------> a ------->b --- ^ ---> d ---> --- | i h --- j| k' (d' → d) --- | k (d' → a) --- d' -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 ) - → A [ A [ h o i ] ≈ e eqa ] → A [ A [ h-1 o h ] ≈ id1 A d ] - → Equalizer A {c} (A [ f o h ]) (A [ g o h ] ) -equalizer+h {a} {b} {c} {d} {f} {g} eqa i h h-1 eq h-1-id = record { - e = i ; -- A [ h-1 o e eqa ] ; -- Hom A a d - fe=ge = fe=ge1 ; - k = λ j eq' → k eqa (A [ h o j ]) (fhj=ghj j eq' ) ; - ek=h = ek=h1 ; - uniqueness = uniqueness1 - } where - fhj=ghj : {d' : Obj A } → (j : Hom A d' d ) → - A [ A [ A [ f o h ] o j ] ≈ A [ A [ g o h ] o j ] ] → - A [ A [ f o A [ h o j ] ] ≈ A [ g o A [ h o j ] ] ] - fhj=ghj j eq' = let open ≈-Reasoning (A) in - begin - f o ( h o j ) - ≈⟨ assoc ⟩ - (f o h ) o j - ≈⟨ eq' ⟩ - (g o h ) o j - ≈↑⟨ assoc ⟩ - g o ( h o j ) - ∎ - fe=ge1 : A [ A [ A [ f o h ] o i ] ≈ A [ A [ g o h ] o i ] ] - fe=ge1 = let open ≈-Reasoning (A) in - begin - ( f o h ) o i - ≈↑⟨ assoc ⟩ - f o (h o i ) - ≈⟨ cdr eq ⟩ - f o (e eqa) - ≈⟨ fe=ge eqa ⟩ - g o (e eqa) - ≈↑⟨ cdr eq ⟩ - g o (h o i ) - ≈⟨ assoc ⟩ - ( g o h ) o i - ∎ - 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' ] ]} → - A [ A [ i o k eqa (A [ h o k' ]) (fhj=ghj k' eq') ] ≈ k' ] - ek=h1 {d'} {k'} {eq'} = let open ≈-Reasoning (A) in - begin - 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') - ≈↑⟨ idL ⟩ - (id1 A d ) o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) - ≈↑⟨ car h-1-id ⟩ - ( h-1 o h ) o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) - ≈↑⟨ assoc ⟩ - h-1 o ( h o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) ) - ≈⟨ cdr assoc ⟩ - h-1 o ( (h o i ) o k eqa (h o k' ) (fhj=ghj k' eq')) - ≈⟨ cdr (car eq ) ⟩ - h-1 o ( (e eqa) o k eqa (h o k' ) (fhj=ghj k' eq')) - ≈⟨ cdr (ek=h eqa) ⟩ - h-1 o ( h o k' ) - ≈⟨ assoc ⟩ - ( h-1 o h ) o k' - ≈⟨ car h-1-id ⟩ - id1 A d o k' - ≈⟨ idL ⟩ - k' - ∎ - 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} → - A [ A [ i o k' ] ≈ h' ] → A [ k eqa (A [ h o h' ]) (fhj=ghj h' eq') ≈ k' ] - uniqueness1 {d'} {h'} {eq'} {k'} ik=h = let open ≈-Reasoning (A) in - begin - k eqa (A [ h o h' ]) (fhj=ghj h' eq') - ≈⟨ uniqueness eqa ( begin - e eqa o k' - ≈↑⟨ car eq ⟩ - (h o i ) o k' - ≈↑⟨ assoc ⟩ - h o (i o k') - ≈⟨ cdr ik=h ⟩ - h o h' - ∎ ) ⟩ - k' - ∎ - --- If we have equalizer f g, e hf hg is also equalizer if we have isomorphic pair - -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 ) - → (h-1 : Hom A d b ) → A [ A [ h-1 o h ] ≈ id1 A b ] - → Equalizer A {c} (A [ h o f ]) (A [ h o g ] ) -h+equalizer {a} {b} {c} {d} {f} {g} eqa h h-1 h-1-id = record { - e = e eqa ; - fe=ge = fe=ge1 ; - k = λ j eq' → k eqa j (fj=gj j eq') ; - ek=h = ek=h1 ; - uniqueness = uniqueness1 - } where - 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 ] ] - fj=gj j eq = let open ≈-Reasoning (A) in - begin - f o j - ≈↑⟨ idL ⟩ - id1 A b o ( f o j ) - ≈↑⟨ car h-1-id ⟩ - (h-1 o h ) o ( f o j ) - ≈↑⟨ assoc ⟩ - h-1 o (h o ( f o j )) - ≈⟨ cdr assoc ⟩ - h-1 o ((h o f) o j ) - ≈⟨ cdr eq ⟩ - h-1 o ((h o g) o j ) - ≈↑⟨ cdr assoc ⟩ - h-1 o (h o ( g o j )) - ≈⟨ assoc ⟩ - (h-1 o h) o ( g o j ) - ≈⟨ car h-1-id ⟩ - id1 A b o ( g o j ) - ≈⟨ idL ⟩ - g o j - ∎ - fe=ge1 : A [ A [ A [ h o f ] o e eqa ] ≈ A [ A [ h o g ] o e eqa ] ] - fe=ge1 = let open ≈-Reasoning (A) in - begin - ( h o f ) o e eqa - ≈↑⟨ assoc ⟩ - h o (f o e eqa ) - ≈⟨ cdr (fe=ge eqa) ⟩ - h o (g o e eqa ) - ≈⟨ assoc ⟩ - ( h o g ) o e eqa - ∎ - 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 ] ]} → - A [ A [ e eqa o k eqa j (fj=gj j eq) ] ≈ j ] - ek=h1 {d₁} {j} {eq} = ek=h eqa - 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} → - A [ A [ e eqa o k' ] ≈ j ] → A [ k eqa j (fj=gj j eq) ≈ k' ] - uniqueness1 = uniqueness eqa - --- If we have equalizer f g, e (ef) (eg) is also an equalizer and e = id c - -eefeg : {a b c : Obj A } {f g : Hom A a b } ( eqa : Equalizer A {c} f g) - → Equalizer A {c} (A [ f o e eqa ]) (A [ g o e eqa ] ) -eefeg {a} {b} {c} {f} {g} eqa = record { - e = id1 A c ; -- i ; -- A [ h-1 o e eqa ] ; -- Hom A a d - fe=ge = fe=ge1 ; - k = λ j eq' → k eqa (A [ h o j ]) (fhj=ghj j eq' ) ; - ek=h = ek=h1 ; - uniqueness = uniqueness1 - } where - i = id1 A c - h = e eqa - fhj=ghj : {d' : Obj A } → (j : Hom A d' c ) → - A [ A [ A [ f o h ] o j ] ≈ A [ A [ g o h ] o j ] ] → - A [ A [ f o A [ h o j ] ] ≈ A [ g o A [ h o j ] ] ] - fhj=ghj j eq' = let open ≈-Reasoning (A) in - begin - f o ( h o j ) - ≈⟨ assoc ⟩ - (f o h ) o j - ≈⟨ eq' ⟩ - (g o h ) o j - ≈↑⟨ assoc ⟩ - g o ( h o j ) - ∎ - fe=ge1 : A [ A [ A [ f o h ] o i ] ≈ A [ A [ g o h ] o i ] ] - fe=ge1 = let open ≈-Reasoning (A) in - begin - ( f o h ) o i - ≈⟨ car ( fe=ge eqa ) ⟩ - ( g o h ) o i - ∎ - 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' ] ]} → - A [ A [ i o k eqa (A [ h o k' ]) (fhj=ghj k' eq') ] ≈ k' ] - ek=h1 {d'} {k'} {eq'} = let open ≈-Reasoning (A) in - begin - 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') - ≈⟨ idL ⟩ - k eqa (e eqa o k' ) (fhj=ghj k' eq') - ≈⟨ uniqueness eqa refl-hom ⟩ - k' - ∎ - 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} → - A [ A [ i o k' ] ≈ h' ] → A [ k eqa (A [ h o h' ]) (fhj=ghj h' eq') ≈ k' ] - uniqueness1 {d'} {h'} {eq'} {k'} ik=h = let open ≈-Reasoning (A) in - begin - k eqa ( e eqa o h') (fhj=ghj h' eq') - ≈⟨ uniqueness eqa ( begin - e eqa o k' - ≈↑⟨ cdr idL ⟩ - e eqa o (id1 A c o k' ) - ≈⟨ cdr ik=h ⟩ - e eqa o h' - ∎ ) ⟩ - k' - ∎ -- -- If we have two equalizers on c and c', there are isomorphic pair h, h' -- -- h : c → c' h' : c' → c --- h h' = 1 and h' h = 1 --- not yet done +-- e' = h o e +-- e = h' o e' -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 ) - → ( keqa : Equalizer A {c} (A [ f o e eqa' ]) (A [ g o e eqa' ]) ) +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 } + ( eqa : Equalizer A e f g) → ( eqa' : Equalizer A e' f g ) + → ( keqa : Equalizer A (k eqa' e {!!} ) (A [ f o e' ]) (A [ g o e' ]) ) → Hom A c c' -c-iso-l {c} {c'} eqa eqa' keqa = e keqa +c-iso-l {c} {c'} eqa eqa' eff = {!!} -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 ) - → ( keqa : Equalizer A {c} (A [ f o e eqa' ]) (A [ g o e eqa' ]) ) +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 ) + → ( keqa : Equalizer A (k eqa' e {!!} ) (A [ f o e' ]) (A [ g o e' ]) ) → Hom A c' c c-iso-r {c} {c'} eqa eqa' keqa = k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) @@ -413,56 +167,15 @@ -- h = j e f -> j = j' -- -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 ) - → ( keqa : Equalizer A {c} (A [ f o e eqa' ]) (A [ g o e eqa' ]) ) +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 ) + → ( keqa : Equalizer A (k eqa' e {!!} ) (A [ f o e' ]) (A [ g o e' ]) ) → A [ A [ c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa ] ≈ id1 A c' ] c-iso {c} {c'} {a} {b} {f} {g} eqa eqa' keqa = let open ≈-Reasoning (A) in begin c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa - ≈⟨ ek=h keqa ⟩ + ≈⟨ {!!} ⟩ id1 A c' ∎ --- To prove c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa --- ke = e' k'e' = e → k k' = 1 , k' k = 1 --- ke = e' --- k'ke = k'e' = e kk' = 1 --- x e = e -> x = id? - ------ --- reverse arrow of e (eqa f g) --- --- e eqa f g f --- c ----------> a ------->b --- <--------- --- k (eff) id1a --- (e eqa f g) o k (eff) id1 A a = id1 A a --- --- eqa (f (e eqa f g) ) (g (e eqa f g) ) --- e (eqa (f (e eqa f g) ) (g (e eqa f g) ) ) = k (eff) id1 a --- --- (e α) o k α (id1 A c) = id1 A c --- c a c --- ((k (eff) id1a )) o k α e = id1 A c - - -reverse-e' : {a b c : Obj A} (f g : Hom A a b) → (h i : Hom A c b ) → - ( eqa : {a b c : Obj A} → (f g : Hom A a b) → Equalizer A {c} f g ) → - 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) ]))) ] -reverse-e' = ? - -reverse-e : {a b c : Obj A} (f g : Hom A a b) → (h i : Hom A c b ) → - ( eqa : {a b c : Obj A} → (f g : Hom A a b) → Equalizer A {c} f g ) → - A [ - 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)) ] - ≈ id1 A c ] -reverse-e {a} {b} {c} f g h i eqa = let open ≈-Reasoning (A) in - begin - 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) {!!} - ≈⟨ car {!!} ⟩ - 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)) - ≈⟨ ek=h (eqa ( A [ f o (e (eqa f g)) ] ) (A [ g o (e (eqa f g )) ] )) ⟩ - id1 A c - ∎ ---- -- @@ -472,10 +185,11 @@ ---- lemma-equ1 : {a b c : Obj A} (f g : Hom A a b) → - ( {a b c : Obj A} → (f g : Hom A a b) → Equalizer A {c} f g ) → Burroni A {c} f g + ( 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 ) + → Burroni A {c} f g lemma-equ1 {a} {b} {c} f g eqa = record { - α = λ f g → e (eqa f g ) ; -- Hom A c a - 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 + α = λ f g → equalizer (eqa f g ) ; -- Hom A c a + 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 δ = λ {a} f → k (eqa f f) (id1 A a) (lemma-equ2 f); -- Hom A a c b1 = fe=ge (eqa f g) ; b2 = lemma-b2 ; @@ -496,33 +210,33 @@ lemma-equ2 : {a b : Obj A} (f : Hom A a b) → A [ A [ f o id1 A a ] ≈ A [ f o id1 A a ] ] lemma-equ2 f = let open ≈-Reasoning (A) in refl-hom - lemma-b3 : A [ A [ e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) ] ≈ id1 A a ] + lemma-b3 : A [ A [ equalizer (eqa f f ) o k (eqa f f) (id1 A a) (lemma-equ2 f) ] ≈ id1 A a ] lemma-b3 = let open ≈-Reasoning (A) in begin - e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) + equalizer (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) ≈⟨ ek=h (eqa f f ) ⟩ id1 A a ∎ lemma-equ4 : {a b c d : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → - 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 ])) ] ] ] + 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 ])) ] ] ] lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in begin - f o ( h o e (eqa (f o h) ( g o h ))) + f o ( h o equalizer (eqa (f o h) ( g o h ))) ≈⟨ assoc ⟩ - (f o h) o e (eqa (f o h) ( g o h )) + (f o h) o equalizer (eqa (f o h) ( g o h )) ≈⟨ fe=ge (eqa (A [ f o h ]) (A [ g o h ])) ⟩ - (g o h) o e (eqa (f o h) ( g o h )) + (g o h) o equalizer (eqa (f o h) ( g o h )) ≈↑⟨ assoc ⟩ - g o ( h o e (eqa (f o h) ( g o h ))) + g o ( h o equalizer (eqa (f o h) ( g o h ))) ∎ lemma-b2 : {d : Obj A} {h : Hom A d a} → A [ - 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) ] - ≈ A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ] ] + 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) ] + ≈ A [ h o equalizer (eqa (A [ f o h ]) (A [ g o h ])) ] ] lemma-b2 {d} {h} = let open ≈-Reasoning (A) in begin - 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) + 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) ≈⟨ ek=h (eqa f g) ⟩ - h o e (eqa (f o h ) ( g o h )) + h o equalizer (eqa (f o h ) ( g o h )) ∎ ------- α(f,g)j id d = α(f,g)j @@ -534,15 +248,15 @@ ------- γ(f,g,α(f,g)j) δ(fα(f,g)j) = j lemma-b4 : {d : Obj A} {j : Hom A d c} → A [ - 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 ] ])) ]) - (lemma-equ4 {a} {b} {c} f g (A [ e (eqa f g) o j ])) o - 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 ] ])) ] + 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 ] ])) ]) + (lemma-equ4 {a} {b} {c} f g (A [ equalizer (eqa f g) o j ])) o + 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 ] ])) ] ≈ j ] lemma-b4 {d} {j} = let open ≈-Reasoning (A) in begin - ( 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 ) ))) )) - (lemma-equ4 {a} {b} {c} f g (( e (eqa f g) o j ))) o - k (eqa (( f o ( e (eqa f g) o j ) )) (( f o ( e (eqa f g) o j ) ))) (id1 A d) (lemma-equ2 (( f o ( e (eqa f g) o j ) ))) ) + ( 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 ) ))) )) + (lemma-equ4 {a} {b} {c} f g (( equalizer (eqa f g) o 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 ) ))) ) ≈⟨ {!!} ⟩ j ∎