Mercurial > hg > Members > kono > Proof > category
view equalizer.agda @ 217:306f07bece85
add equalizer+h
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Wed, 04 Sep 2013 12:13:27 +0900 |
| parents | 0135419f375c |
| children | 749a1ecbc0b5 |
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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 ef=eg : 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 record EqEqualizer { 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 EqEqualizer -- Equalizer is unique up to iso equalizer-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 ) → Hom A c c' --- != id1 A c equalizer-iso {c} eqa eqa' = k eqa' (e eqa) (ef=eg eqa) -- e eqa f g f -- c ----------> a ------->b -- ---> d ---> -- i h 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 ) → A [ A [ h o i ] ≈ e eqa ] → Equalizer A {c} (A [ f o h ]) (A [ g o h ] ) equalizer+h {a} {b} {c} {d} {f} {g} eqa i h eq = record { e = i ; -- Hom A a d ef=eg = ef=eg1 ; 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 ) ∎ ef=eg1 : A [ A [ A [ f o h ] o i ] ≈ A [ A [ g o h ] o i ] ] ef=eg1 = let open ≈-Reasoning (A) in begin ( f o h ) o i ≈↑⟨ assoc ⟩ f o (h o i ) ≈⟨ cdr eq ⟩ f o (e eqa) ≈⟨ ef=eg eqa ⟩ g o (e eqa) ≈↑⟨ cdr eq ⟩ g o (h o i ) ≈⟨ assoc ⟩ ( g o h ) o i ∎ ek=h1 : {d' : Obj A} {h' : Hom A d' d} {eq' : A [ A [ A [ f o h ] o h' ] ≈ A [ A [ g o h ] o h' ] ]} → A [ A [ i o k eqa (A [ h o h' ]) (fhj=ghj h' eq') ] ≈ h' ] ek=h1 {d'} {h'} {eq'} = let open ≈-Reasoning (A) in begin i o k eqa (h o h' ) (fhj=ghj h' eq') ≈⟨ {!!} ⟩ h' ∎ 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' ∎ lemma-equ1 : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → {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 ) → EqEqualizer A {c} f g lemma-equ1 A {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 = ef=eg (eqa f g) ; b2 = lemma-equ5 ; b3 = lemma-equ3 ; b4 = lemma-equ6 } 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-equ3 : A [ A [ e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) ] ≈ id1 A a ] lemma-equ3 = 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 )) ≈⟨ ef=eg (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-equ5 : {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-equ5 {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 )) ∎ lemma-equ6 : {d : Obj A} {k₁ : Hom A d c} → A [ A [ k (eqa f g) (A [ A [ e (eqa f g) o k₁ ] o e (eqa (A [ f o A [ e (eqa f g) o k₁ ] ]) (A [ g o A [ e (eqa f g) o k₁ ] ])) ]) (lemma-equ4 {a} {b} {c} f g (A [ e (eqa f g) o k₁ ])) o k (eqa (A [ f o A [ e (eqa f g) o k₁ ] ]) (A [ f o A [ e (eqa f g) o k₁ ] ])) (id1 A d) (lemma-equ2 (A [ f o A [ e (eqa f g) o k₁ ] ])) ] ≈ k₁ ] lemma-equ6 {d} {k₁} = let open ≈-Reasoning (A) in begin ( k (eqa f g) (( ( e (eqa f g) o k₁ ) o e (eqa (( f o ( e (eqa f g) o k₁ ) )) (( g o ( e (eqa f g) o k₁ ) ))) )) (lemma-equ4 {a} {b} {c} f g (( e (eqa f g) o k₁ ))) o k (eqa (( f o ( e (eqa f g) o k₁ ) )) (( f o ( e (eqa f g) o k₁ ) ))) (id1 A d) (lemma-equ2 (( f o ( e (eqa f g) o k₁ ) ))) ) ≈⟨ car ( uniqueness (eqa f g) ( begin e (eqa f g) o k₁ ≈⟨ {!!} ⟩ (e (eqa f g) o k₁) o e (eqa (f o e (eqa f g) o k₁) (g o e (eqa f g) o k₁)) ∎ )) ⟩ k₁ o k (eqa (( f o ( e (eqa f g) o k₁ ) )) (( f o ( e (eqa f g) o k₁ ) ))) (id1 A d) (lemma-equ2 (( f o ( e (eqa f g) o k₁ ) ))) ≈⟨ cdr ( uniqueness (eqa (( f o ( e (eqa f g) o k₁ ) )) (( f o ( e (eqa f g) o k₁ ) ))) ( begin e (eqa (f o e (eqa f g) o k₁) (f o e (eqa f g) o k₁)) o id1 A d ≈⟨ {!!} ⟩ id1 A d ∎ )) ⟩ k₁ o id1 A d ≈⟨ idR ⟩ k₁ ∎