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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 08 Sep 2013 05:55:56 +0900 |
| parents | e20b81102eee |
| children | c8db99cdf72a |
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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} (e : Hom A c a) (f g : Hom A a b) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where field 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) (e : Hom A c a) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where field 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 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 } → {e : Hom A c a } → (f : Hom A a b) → Hom A a c b1 : A [ A [ f o α {a} {b} {a} f g {id1 A a} ] ≈ A [ g o α {a} {b} {a} f g {id1 A a} ] ] 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} ] ] b3 : A [ A [ α {a} {b} {a} f f {id1 A a} o δ {a} {b} {a} {id1 A 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 {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 ] -- 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 {d} {b} {d} {id1 A d} (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 ) ∎ -- -- -- 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 } {e : Hom A c a } (h-1 : Hom A c' c ) → (h : Hom A c c' ) → A [ A [ h o h-1 ] ≈ id1 A c' ] → A [ A [ h-1 o h ] ≈ id1 A c ] → ( fe=ge' : A [ A [ f o A [ e o h-1 ] ] ≈ A [ g o A [ e o h-1 ] ] ] ) ( eqa : Equalizer A e f g ) → Equalizer A (A [ e o h-1 ] ) f g equalizer+iso {a} {b} {c} {c'} {f} {g} {e} h-1 h hh-1=1 h-1h=1 fe=ge' eqa = 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 o h-1 ] ] ≈ A [ g o A [ e o h-1 ] ] ] fe=ge1 = fe=ge' ek=h1 : {d : Obj A} {j : Hom A d a} {eq : A [ A [ f o j ] ≈ A [ g o j ] ]} → A [ A [ A [ e o h-1 ] o A [ h o k eqa j eq ] ] ≈ j ] ek=h1 {d} {j} {eq} = let open ≈-Reasoning (A) in begin ( e o h-1 ) o ( h o k eqa j eq ) ≈↑⟨ assoc ⟩ e o ( h-1 o ( h o k eqa j eq ) ) ≈⟨ cdr assoc ⟩ e o (( h-1 o h) o k eqa j eq ) ≈⟨ cdr (car h-1h=1 ) ⟩ e o (id1 A c o k eqa j eq ) ≈⟨ cdr idL ⟩ 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 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 o ( h-1 o j ) ≈⟨ assoc ⟩ (e o h-1 ) o j ≈⟨ ej=h ⟩ h' ∎ )) ⟩ h o ( h-1 o j ) ≈⟨ assoc ⟩ (h o h-1 ) o j ≈⟨ car hh-1=1 ⟩ id1 A c' o j ≈⟨ idL ⟩ j ∎ -- -- If we have two equalizers on c and c', there are isomorphic pair h, h' -- -- h : c → c' h' : c' → c -- e' = h o e -- e = h' o e' 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 (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) → Hom A c c' c-iso-l {c} {c'} eqa eqa' keqa = equalizer keqa 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 (fe=ge eqa)) (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') ) -- e' 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 } → {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 (fe=ge eqa)) (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 ≈⟨⟩ equalizer keqa o k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) ≈⟨ ek=h keqa ⟩ id1 A c' ∎ 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 (fe=ge eqa )) (A [ f o e' ]) (A [ g o e' ]) ) → ( keqa' : Equalizer A (k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') )) (A [ f o e ]) (A [ g o e ]) ) → { e'->e : A [ e' ≈ A [ e o equalizer keqa' ] ] } -- refl → A [ A [ c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa ] ≈ id1 A c ] c-iso← {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' keqa keqa' {e'->e} = let open ≈-Reasoning (A) in begin c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa ≈⟨⟩ k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) o k eqa' e (fe=ge eqa ) ≈⟨⟩ equalizer keqa' o k eqa' e (fe=ge eqa ) ≈⟨ cdr ( begin k eqa' e (fe=ge eqa ) ≈⟨ uniqueness eqa' ( begin e' o k keqa' (id1 A c) (f1=g1 (fe=ge eqa) (id1 A c)) ≈⟨ car e'->e ⟩ ( e o equalizer keqa' ) o k keqa' (id1 A c) (f1=g1 (fe=ge eqa) (id1 A c)) ≈↑⟨ assoc ⟩ e o ( equalizer keqa' o k keqa' (id1 A c) (f1=g1 (fe=ge eqa) (id1 A c))) ≈⟨ cdr ( ek=h keqa' ) ⟩ e o id1 A c ≈⟨ idR ⟩ e ∎ )⟩ k keqa' (id1 A c) ( f1=g1 (fe=ge eqa) (id1 A c) ) ∎ )⟩ equalizer keqa' o k keqa' (id1 A c) ( f1=g1 (fe=ge eqa) (id1 A c) ) ≈⟨ ek=h keqa' ⟩ 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) → (e : Hom A c a ) → ( eqa : {a b c : Obj A} → (f g : Hom A a b) → {e : Hom A c a } → Equalizer A e f g ) → Burroni A {c} {a} {b} f g e lemma-equ1 {a} {b} {c} f g e eqa = record { α = λ {a} {b} {c} f g {e} → equalizer (eqa {a} {b} {c} f g {e} ) ; -- 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} {b} {c} {e} f → k (eqa {a} {b} {c} f f {e} ) (id1 A a) (lemma-equ2 f); -- Hom A a c b1 = fe=ge (eqa {a} {b} {a} f g {id1 A a}) ; 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 [ 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 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 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 equalizer (eqa (f o h) ( g o h ))) ≈⟨ assoc ⟩ (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 equalizer (eqa (f o h) ( g o h )) ≈↑⟨ assoc ⟩ g o ( h o equalizer (eqa (f o h) ( g o h ))) ∎ lemma-b2 : {d : Obj A} {h : Hom A d a} → A [ 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 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 equalizer (eqa (f o h ) ( g o h )) ∎ lemma-b4 : {d : Obj A} {j : Hom A d c} → A [ A [ k (eqa f g) (A [ A [ equalizer (eqa f g) o j ] o equalizer (eqa (A [ f o A [ equalizer (eqa f g {e}) o j ] ]) (A [ g o A [ equalizer (eqa f g {e} ) 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) (( ( 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 ) ))) )) (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 ) ))) ) ≈⟨ car ((uniqueness (eqa f g) ( begin equalizer (eqa f g) o j ≈↑⟨ idR ⟩ (equalizer (eqa f g) o j ) o id1 A d ≈⟨⟩ ((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))) ∎ ))) ⟩ 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 ) ))) ≈⟨ cdr ((uniqueness (eqa (( f o ( equalizer (eqa f g) o j ) )) (( f o ( equalizer (eqa f g) o j ) ))) ( begin equalizer (eqa (f o equalizer (eqa f g {e} ) o j) (f o equalizer (eqa f g {e}) o j)) o id1 A d ≈⟨ idR ⟩ equalizer (eqa (f o equalizer (eqa f g {e}) o j) (f o equalizer (eqa f g {e}) o j)) ≈⟨⟩ id1 A d ∎ ))) ⟩ j o id1 A d ≈⟨ idR ⟩ j ∎ -- end