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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 07 Sep 2013 23:29:13 +0900 |
| parents | 1dc1c697145f |
| children | 4bba19bc71be |
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--- -- -- Equalizer -- -- e f -- c --------> a ----------> b -- ^ . ----------> -- | . g -- |k . -- | . h -- d -- -- Shinji KONO <kono@ie.u-ryukyu.ac.jp> ---- open import Category -- https://github.com/konn/category-agda open import Level module equalizer { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } where 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 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 ] 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 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 ] -- 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 ]) ] open Equalizer open Burroni -- -- Some obvious conditions for k (fe = ge) → ( fh = gh ) -- 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 ] ] 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=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 begin f o ( e o h ) ≈⟨ assoc ⟩ (f o e ) o h ≈⟨ car eq ⟩ (g o e ) o h ≈↑⟨ assoc ⟩ g o ( e o h ) ∎ -- -- 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 -- -- e eqa f g f -- c ----------> a ------->b -- |^ -- || -- h || h-1 -- 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 ] ; 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 = let open ≈-Reasoning (A) in begin f o ( e eqa o h-1 ) ≈⟨ assoc ⟩ (f o e eqa ) o h-1 ≈⟨ car (fe=ge eqa) ⟩ (g o e eqa ) o h-1 ≈↑⟨ assoc ⟩ g o ( e eqa o h-1 ) ∎ 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 ] 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 ) ≈⟨ 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 [ 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 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' 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 ) → ( 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 = 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 -- | 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 -- 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' ]) ) → 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 ∎ ---- -- -- An equalizer satisfies Burroni equations -- -- b4 is not yet done ---- 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 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 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 ; b3 = lemma-b3 ; b4 = lemma-b4 } where -- -- e eqa f g f -- c ----------> a ------->b -- ^ 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 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 ) → 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 f o ( h o e (eqa (f o h) ( g o h ))) ≈⟨ assoc ⟩ (f o h) o e (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 )) ≈↑⟨ assoc ⟩ g o ( h o e (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 ])) ] ] 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) ≈⟨ ek=h (eqa f g) ⟩ h o e (eqa (f o h ) ( g o h )) ∎ ------- 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 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 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 ) ))) ) ≈⟨ {!!} ⟩ j ∎ -- end