Mercurial > hg > Members > kono > Proof > category
changeset 230:1ef8c70c7054
fix
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 07 Sep 2013 18:56:46 +0900 |
| parents | 68b2681ea9df |
| children | 1dc1c697145f |
| files | equalizer.agda |
| diffstat | 1 files changed, 117 insertions(+), 145 deletions(-) [+] |
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--- a/equalizer.agda Fri Sep 06 11:52:41 2013 +0900 +++ b/equalizer.agda Sat Sep 07 18:56:46 2013 +0900 @@ -6,14 +6,14 @@ -- c --------> a ----------> b -- ^ . ----------> -- | . g --- |k . --- | . h --- d +-- |k . +-- | . h +-- d -- -- Shinji KONO <kono@ie.u-ryukyu.ac.jp> ---- -open import Category -- https://github.com/konn/category-agda +open import Category -- https://github.com/konn/category-agda open import Level module equalizer { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } where @@ -22,31 +22,31 @@ 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 field - e : Hom A c a + 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 ] - uniqueness : {d : Obj A} → ∀ {h : Hom A d a} → {eq : A [ A [ f o h ] ≈ A [ g o h ] ] } → {k' : Hom A d c } → + uniqueness : {d : Obj A} → ∀ {h : Hom A d a} → {eq : A [ A [ f o h ] ≈ A [ g o h ] ] } → {k' : Hom A d c } → A [ A [ e o k' ] ≈ h ] → A [ k {d} h eq ≈ k' ] equalizer : Hom A c a equalizer = e --- +-- -- Flat Equational Definition of Equalizer --- +-- record Burroni { 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 field α : {a b c : Obj A } → (f : Hom A a b) → (g : Hom A a b ) → Hom A c a 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 - δ : {a b c : Obj A } → (f : Hom A a b) → Hom A a c + δ : {a b c : Obj A } → (f : Hom A a b) → Hom A a c b1 : A [ A [ f o α {a} {b} {a} f g ] ≈ A [ g o α f g ] ] b2 : {d : Obj A } → {h : Hom A d a } → A [ A [ ( α f g) o (γ {a} {b} {c} f g h) ] ≈ A [ h o α (A [ f o h ]) (A [ g o h ]) ] ] b3 : A [ A [ α f f o δ {a} {b} {a} f ] ≈ id1 A a ] -- b4 : {c d : Obj A } {k : Hom A c a} → A [ β f g ( A [ α f g o k ] ) ≈ k ] - b4 : {d : Obj A } {k : Hom A d c} → A [ A [ γ {a} {b} {c} {d} f g ( A [ α {a} {b} {c} f g o k ] ) o δ (A [ f o A [ α f g o k ] ] ) ] ≈ k ] + b4 : {d : Obj A } {k : Hom A d c} → A [ A [ γ {a} {b} {c} {d} f g ( A [ α {a} {b} {c} f g o k ] ) o δ (A [ f o A [ α f g o k ] ] ) ] ≈ k ] -- A [ α f g o β f g h ] ≈ h β : { d e a b : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → Hom A d c - β {d} {e} {a} {b} f g h = A [ γ {a} {b} {c} f g h o δ (A [ f o h ]) ] + β {d} {e} {a} {b} f g h = A [ γ {a} {b} {c} f g h o δ (A [ f o h ]) ] open Equalizer @@ -60,17 +60,17 @@ f1=g1 eq h = let open ≈-Reasoning (A) in (resp refl-hom eq ) f1=f1 : { a b : Obj A } (f : Hom A a b ) → A [ A [ f o (id1 A a) ] ≈ A [ f o (id1 A a) ] ] -f1=f1 f = let open ≈-Reasoning (A) in refl-hom +f1=f1 f = let open ≈-Reasoning (A) in refl-hom f1=gh : { a b c d : Obj A } {f g : Hom A a b } → { e : Hom A c a } → { h : Hom A d c } → (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 ] ] ] -f1=gh {a} {b} {c} {d} {f} {g} {e} {h} eq = let open ≈-Reasoning (A) in +f1=gh {a} {b} {c} {d} {f} {g} {e} {h} eq = let open ≈-Reasoning (A) in begin f o ( e o h ) ≈⟨ assoc ⟩ - (f o e ) o h + (f o e ) o h ≈⟨ car eq ⟩ - (g o e ) o h + (g o e ) o h ≈↑⟨ assoc ⟩ g o ( e o h ) ∎ @@ -80,28 +80,28 @@ -- -- 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) → +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 ] + 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)) + k eqa (id1 A a) (f1=g1 eq (id1 A a)) ≈⟨ uniqueness eqa ( begin e eqa o id1 A a ≈⟨ idR ⟩ - e eqa + 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 ] ) → +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 ] + 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 + e eqa ≈↑⟨ idR ⟩ e eqa o id1 A a ≈↑⟨ cdr k=1 ⟩ @@ -114,26 +114,26 @@ -- -- An isomorphic element c' of c makes another equalizer -- --- e eqa f g f +-- e eqa f g f -- c ----------> a ------->b --- |^ --- || +-- |^ +-- || -- h || h-1 --- v| --- c' +-- 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 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 ] ; fe=ge = fe=ge1 ; k = λ j eq → A [ h o k eqa j eq ] ; - ek=h = ek=h1 ; + 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 = let open ≈-Reasoning (A) in + fe=ge1 = let open ≈-Reasoning (A) in begin f o ( e eqa o h-1 ) ≈⟨ assoc ⟩ @@ -169,16 +169,16 @@ begin e eqa o ( h-1 o j ) ≈⟨ assoc ⟩ - (e eqa o h-1 ) o j + (e eqa o h-1 ) o j ≈⟨ ej=h ⟩ h' ∎ )) ⟩ h o ( h-1 o j ) ≈⟨ assoc ⟩ - (h o h-1 ) o j + (h o h-1 ) o j ≈⟨ car h-id ⟩ - id1 A c' o j + id1 A c' o j ≈⟨ idL ⟩ j ∎ @@ -187,37 +187,37 @@ -- -- e eqa f g f -- c ----------> a ------->b --- ^ ---> d ---> +-- ^ ---> d ---> -- | i h -- j| k' (d' → d) -- | k (d' → a) --- d' +-- 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 ) +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 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 ; + ek=h = ek=h1 ; uniqueness = uniqueness1 } where - fhj=ghj : {d' : Obj A } → (j : Hom A d' d ) → + 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 ] ] ] + 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 + (f o h ) o j ≈⟨ eq' ⟩ - (g o h ) o j + (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 + fe=ge1 = let open ≈-Reasoning (A) in begin ( f o h ) o i ≈↑⟨ assoc ⟩ @@ -237,9 +237,9 @@ 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')) + (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')) + ( 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 ⟩ @@ -249,9 +249,9 @@ ≈⟨ cdr (ek=h eqa) ⟩ h-1 o ( h o k' ) ≈⟨ assoc ⟩ - ( h-1 o h ) o k' + ( h-1 o h ) o k' ≈⟨ car h-1-id ⟩ - id1 A d o k' + id1 A d o k' ≈⟨ idL ⟩ k' ∎ @@ -267,21 +267,21 @@ ≈↑⟨ assoc ⟩ h o (i o k') ≈⟨ cdr ik=h ⟩ - h o h' + h o h' ∎ ) ⟩ k' ∎ --- If we have equalizer f g, e hf hg is also equalizer if we have isomorphic pair +-- 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+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 ] ) + → 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 ; + e = e eqa ; + fe=ge = fe=ge1 ; k = λ j eq' → k eqa j (fj=gj j eq') ; - ek=h = ek=h1 ; + 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 ] ] @@ -316,7 +316,7 @@ ≈⟨ cdr (fe=ge eqa) ⟩ h o (g o e eqa ) ≈⟨ assoc ⟩ - ( h o g ) o e eqa + ( 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 ] @@ -324,35 +324,35 @@ 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 : 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 ; + ek=h = ek=h1 ; uniqueness = uniqueness1 } where i = id1 A c h = e eqa - fhj=ghj : {d' : Obj A } → (j : Hom A d' c ) → + 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 ] ] ] + 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 + (f o h ) o j ≈⟨ eq' ⟩ - (g o h ) o j + (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 + fe=ge1 = let open ≈-Reasoning (A) in begin ( f o h ) o i ≈⟨ car ( fe=ge eqa ) ⟩ @@ -378,7 +378,7 @@ ≈↑⟨ cdr idL ⟩ e eqa o (id1 A c o k' ) ≈⟨ cdr ik=h ⟩ - e eqa o h' + e eqa o h' ∎ ) ⟩ k' ∎ @@ -391,34 +391,30 @@ -- not yet done -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 ) +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' ]) ) - → Hom A c c' + → Hom A c c' c-iso-l {c} {c'} eqa eqa' keqa = e keqa -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 ) +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' ]) ) - → Hom A c' c -c-iso-r {c} {c'} eqa eqa' keqa = let open ≈-Reasoning (A) in k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) + → 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') ) -- e(eqa') f - -- c'----------> a ------->b f e j = g e j - -- ^ g - -- |k h + -- c'----------> a ------->b f e j = g e j + -- ^ g + -- |k h -- | h = e(eqaj) o k jhek = jh (uniqueness) - -- | - -- c j o (k (eqa ef ef) j ) = id c h = e(eqaj) - -- - -- h j e f = h j e g → h = 'j e f + -- | + -- c j o (k (eqa ef ef) j ) = id c h = e(eqaj) + -- + -- h j e f = h j e g → h = 'j e f -- h = j e f -> j = j' -- --- e k e k = 1c e e = e -> e = 1c? --- k e k e = 1c' ? - - -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 ) +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' ]) ) → 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 @@ -427,52 +423,33 @@ id1 A c' ∎ - --- Equalizer is unique up to iso - --- --- --- e eqa f g f --- c ----------> a ------->b --- -equalizer-iso-eq' : { c c' a b : Obj A } {f g : Hom A a b } → ( eqa : Equalizer A {c} f g) → ( eqa' : Equalizer A {c'} f g ) - { h : Hom A a c } → A [ A [ h o e eqa ] ≈ id1 A c ] → A [ A [ k eqa (e eqa' ) (fe=ge eqa') o k eqa' (e eqa ) (fe=ge eqa) ] ≈ id1 A c ] -equalizer-iso-eq' {c} {c'} {f} {g} eqa eqa' {h} rev = let open ≈-Reasoning (A) in - begin - k eqa (e eqa' ) (fe=ge eqa') o k eqa' (e eqa ) (fe=ge eqa) - ≈↑⟨ idL ⟩ - (id1 A c) o ( k eqa (e eqa' ) (fe=ge eqa') o k eqa' (e eqa ) (fe=ge eqa) ) - ≈↑⟨ car rev ⟩ - ( h o e eqa ) o ( k eqa (e eqa' ) (fe=ge eqa') o k eqa' (e eqa ) (fe=ge eqa) ) - ≈↑⟨ assoc ⟩ - h o ( e eqa o ( k eqa (e eqa' ) (fe=ge eqa') o k eqa' (e eqa ) (fe=ge eqa) ) ) - ≈⟨ cdr assoc ⟩ - h o (( e eqa o k eqa (e eqa' ) (fe=ge eqa')) o k eqa' (e eqa ) (fe=ge eqa) ) - ≈⟨ cdr ( car (ek=h eqa) ) ⟩ - h o ( e eqa' o k eqa' (e eqa ) (fe=ge eqa) ) - ≈⟨ cdr (ek=h eqa' ) ⟩ - h o e eqa - ≈⟨ rev ⟩ - id1 A c - ∎ - -equalizer-iso-eq : { c c' a b : Obj A } {f g : Hom A a b } → ( eqa : Equalizer A {c} f g) → ( eqa' : Equalizer A {c'} f g ) - → A [ A [ k eqa (e eqa' ) (fe=ge eqa') o k eqa' (e eqa ) (fe=ge eqa) ] ≈ id1 A c ] -equalizer-iso-eq {c} {c'} {f} {g} eqa eqa' = equalizer-iso-eq' {c} {c'} {f} {g} eqa eqa' {{!!}} {!!} - +-- 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 +-- <--------- +-- +reverse-e : {a b c : Obj A} (f g : Hom A a b) → + ( eqa : {a b c : Obj A} → (f g : Hom A a b) → Equalizer A {c} f g ) → + A [ A [ k (eqa (A [ f o e (eqa f g) ]) (A [ g o e (eqa f g) ]) ) (id1 A a) + (f1=g1 (fe=ge (eqa f g) ) (id1 A a) ) o e (eqa f g ) ] ≈ id1 A c ] +reverse-e {a} {b} {c} f g eqa = ? + ---- -- -- An equalizer satisfies Burroni equations -- --- b4 is not yet done +-- b4 is not yet done ---- -lemma-equ1 : {a b c : Obj A} (f g : Hom A a b) → +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 lemma-equ1 {a} {b} {c} f g eqa = record { α = λ f g → e (eqa f g ) ; -- Hom A c a @@ -481,30 +458,30 @@ b1 = fe=ge (eqa f g) ; b2 = lemma-b2 ; b3 = lemma-b3 ; - b4 = lemma-b4 + b4 = lemma-b4 } where -- -- e eqa f g f -- c ----------> a ------->b - -- ^ g - -- | + -- ^ g + -- | -- |k₁ = e eqa (f o (e (eqa f g))) (g o (e (eqa f g)))) - -- | + -- | -- d - -- - -- + -- + -- -- e o id1 ≈ e → k e ≈ id 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 = let open ≈-Reasoning (A) in - begin + begin e (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 ) → + 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 ])) ] ] ] lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in begin @@ -516,40 +493,35 @@ ≈↑⟨ assoc ⟩ g o ( h o e (eqa (f o h) ( g o h ))) ∎ - lemma-b2 : {d : Obj A} {h : Hom A d a} → A [ + 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 ])) ] ] 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) + 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) ≈⟨ ek=h (eqa f g) ⟩ h o e (eqa (f o h ) ( g o h )) ∎ - lemma-b4 : {d : Obj A} {j : Hom A d c} → A [ + + ------- α(f,g)j id d = α(f,g)j + ------- α(f,g)j id d = α(f,g)j + ------- α(f,g)j α(fα(f,g)j,fα(f,g)j) δ(fα(f,g)j) = α(f,g)j + ------ fα = gα + ------- α(f,g)j α(fα(f,g)j,gα(f,g)j) δ(fα(f,g)j) = α(f,g)j + ------- α(f,g) γ(f,g,α(f,g)j) δ(fα(f,g)j) = α(f,g)j + ------- 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 + (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 ] ])) ] ≈ j ] - lemma-b4 {d} {j} = let open ≈-Reasoning (A) in + 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 ) ))) ) - ≈⟨ car ( uniqueness (eqa f g) ( begin - e (eqa f g) o j - ≈⟨ {!!} ⟩ - (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)) - ∎ )) ⟩ - 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 ) ))) - ≈⟨ cdr ( uniqueness (eqa (( f o ( e (eqa f g) o j ) )) (( f o ( e (eqa f g) o j ) ))) ( begin - e (eqa (f o e (eqa f g) o j) (f o e (eqa f g) o j)) o id1 A d - ≈⟨ idR ⟩ - e (eqa (f o e (eqa f g) o j) (f o e (eqa f g) o j)) - ≈⟨ {!!} ⟩ - id1 A d - ∎ )) ⟩ - j o id1 A d - ≈⟨ idR ⟩ + ≈⟨ {!!} ⟩ j ∎