Mercurial > hg > Members > kono > Proof > category
changeset 443:f526f4b68565
fix IsEqualizer
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 04 Sep 2016 21:05:39 +0900 |
| parents | 87600d338337 |
| children | 59e47e75188f c2616de4d208 |
| files | cat-utility.agda equalizer.agda freyd.agda limit-to.agda pullback.agda |
| diffstat | 5 files changed, 122 insertions(+), 116 deletions(-) [+] |
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--- a/cat-utility.agda Thu Sep 01 17:34:28 2016 +0900 +++ b/cat-utility.agda Sun Sep 04 21:05:39 2016 +0900 @@ -170,21 +170,21 @@ -- | . h -- d - 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 + record IsEqualizer { 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 + equalizer1 : Hom A c a + equalizer1 = e - record CreateEqualizer { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where + record Equalizer { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a b : Obj A} (f g : Hom A a b) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where field - equalizer : {a b : Obj A} (f g : Hom A a b) -> Obj A - equalizer-e : {a b : Obj A} (f g : Hom A a b) -> Hom A (equalizer f g ) a - isEqualizer : {a b : Obj A} (f g : Hom A a b) -> Equalizer A (equalizer-e f g ) f g + equalizer-c : Obj A + equalizer : Hom A equalizer-c a + isEqualizer : IsEqualizer A equalizer f g -- -- Product @@ -303,4 +303,4 @@ equalizer-p : {a b : Obj A} (f g : Hom A a b) -> Obj A equalizer-e : {a b : Obj A} (f g : Hom A a b) -> Hom A (equalizer-p f g) a - isEqualizer : {a b : Obj A} (f g : Hom A a b) -> Equalizer A (equalizer-e f g) f g + isEqualizer : {a b : Obj A} (f g : Hom A a b) -> IsEqualizer A (equalizer-e f g) f g
--- a/equalizer.agda Thu Sep 01 17:34:28 2016 +0900 +++ b/equalizer.agda Sun Sep 04 21:05:39 2016 +0900 @@ -41,21 +41,23 @@ 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 cong-α : {a b c : Obj A } → { e : Hom A c a } - → {f g g' : Hom A a b } → A [ g ≈ g' ] → A [ α f g e ≈ α f g' e ] - cong-γ : {a _ c d : Obj A } → {f g : Hom A a b} {h h' : Hom A d a } → A [ h ≈ h' ] - → A [ γ {a} {b} {c} {d} f g h ≈ γ f g h' ] - cong-δ : {a b c : Obj A } → {e : Hom A c a} → {f f' : Hom A a b} → A [ f ≈ f' ] → A [ δ e f ≈ δ e f' ] + → {f g g' : Hom A a b } → A [ g ≈ g' ] → A [ α f g e ≈ α f g' e ] + cong-γ : {a _ c d : Obj A } → {f g : Hom A a b} {h h' : Hom A d a } → A [ h ≈ h' ] + → A [ γ {a} {b} {c} {d} f g h ≈ γ f g h' ] + cong-δ : {a b c : Obj A } → {e : Hom A c a} → {f f' : Hom A a b} → A [ f ≈ f' ] → A [ δ e f ≈ δ e f' ] b1 : A [ A [ f o α {a} {b} {c} f g e ] ≈ A [ g o α {a} {b} {c} f g e ] ] 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 b d : Obj A} → (f : Hom A a b ) → {h : Hom A d a } → A [ A [ α {a} {b} {d} f f h o δ {a} {b} {d} h 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} → + 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 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} {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 IsEqualizer open Burroni -- @@ -95,24 +97,24 @@ -- || -- d -monoic : { c a b d : Obj A } {f g : Hom A a b } → {e : Hom A c a } ( eqa : Equalizer A e f g) - → { i j : Hom A d c } +monoic : { c a b d : Obj A } {f g : Hom A a b } → ( eqa : Equalizer A f g) + → { i j : Hom A d (equalizer-c eqa) } → A [ A [ equalizer eqa o i ] ≈ A [ equalizer eqa o j ] ] → A [ i ≈ j ] -monoic {c} {a} {b} {d} {f} {g} {e} eqa {i} {j} ei=ej = let open ≈-Reasoning (A) in begin +monoic {c} {a} {b} {d} {f} {g} eqa {i} {j} ei=ej = let open ≈-Reasoning (A) in begin i - ≈↑⟨ uniqueness eqa ( begin + ≈↑⟨ uniqueness (isEqualizer eqa) ( begin equalizer eqa o i ≈⟨ ei=ej ⟩ equalizer eqa o j ∎ )⟩ - k eqa (equalizer eqa o j) ( f1=gh (fe=ge eqa ) ) - ≈⟨ uniqueness eqa ( begin + k (isEqualizer eqa) (equalizer eqa o j) ( f1=gh (fe=ge (isEqualizer eqa) ) ) + ≈⟨ uniqueness (isEqualizer eqa) ( begin equalizer eqa o j ≈⟨⟩ equalizer eqa o j ∎ )⟩ j - ∎ + ∎ -------------------------------- -- @@ -127,42 +129,43 @@ -- 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 ] → - ( 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 eqa = record { +equalizer+iso : {a b c' : Obj A } {f g : Hom A a b } → + ( eqa : Equalizer A f g ) → + (h-1 : Hom A c' (equalizer-c eqa) ) → (h : Hom A (equalizer-c eqa) c' ) → + A [ A [ h o h-1 ] ≈ id1 A c' ] → A [ A [ h-1 o h ] ≈ id1 A (equalizer-c eqa) ] + → IsEqualizer A (A [ equalizer eqa o h-1 ] ) f g +equalizer+iso {a} {b} {c'} {f} {g} eqa h-1 h hh-1=1 h-1h=1 = record { fe=ge = fe=ge1 ; - k = λ j eq → A [ h o k eqa j eq ] ; + k = λ j eq → A [ h o k (isEqualizer eqa) j eq ] ; ek=h = ek=h1 ; uniqueness = uniqueness1 } where + e = equalizer eqa fe=ge1 : A [ A [ f o A [ e o h-1 ] ] ≈ A [ g o A [ e o h-1 ] ] ] - fe=ge1 = f1=gh ( fe=ge eqa ) + fe=ge1 = f1=gh ( fe=ge (isEqualizer eqa) ) 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 ] + A [ A [ A [ e o h-1 ] o A [ h o k (isEqualizer 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 ) + ( e o h-1 ) o ( h o k (isEqualizer eqa) j eq ) ≈↑⟨ assoc ⟩ - e o ( h-1 o ( h o k eqa j eq ) ) + e o ( h-1 o ( h o k (isEqualizer eqa) j eq ) ) ≈⟨ cdr assoc ⟩ - e o (( h-1 o h) o k eqa j eq ) + e o (( h-1 o h) o k (isEqualizer eqa) j eq ) ≈⟨ cdr (car h-1h=1 ) ⟩ - e o (id c o k eqa j eq ) + e o (id1 A (equalizer-c eqa) o k (isEqualizer eqa) j eq ) ≈⟨ cdr idL ⟩ - e o k eqa j eq - ≈⟨ ek=h eqa ⟩ + e o k (isEqualizer eqa) j eq + ≈⟨ ek=h (isEqualizer 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 ] + A [ A [ h o k (isEqualizer 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 + h o k (isEqualizer eqa) h' eq + ≈⟨ cdr (uniqueness (isEqualizer eqa) ( begin e o ( h-1 o j ) ≈⟨ assoc ⟩ (e o h-1 ) o j @@ -198,28 +201,28 @@ -- e eqa' f g 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 ) + ( eqa : IsEqualizer A e f g) → ( eqa' : IsEqualizer A e' f g ) → Hom A c c' -c-iso-l {c} {c'} eqa eqa' = k eqa' (equalizer eqa) ( fe=ge eqa ) +c-iso-l {c} {c'} {a} {b} {f} {g} {e} eqa eqa' = k eqa' e ( fe=ge 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 ) + ( eqa : IsEqualizer A e f g) → ( eqa' : IsEqualizer A e' f g ) → Hom A c' c -c-iso-r {c} {c'} eqa eqa' = k eqa (equalizer eqa') ( fe=ge eqa' ) +c-iso-r {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' = k eqa e' ( fe=ge eqa' ) c-iso-lr : { 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 ) → + ( eqa : IsEqualizer A e f g) → ( eqa' : IsEqualizer A e' f g ) → A [ A [ c-iso-l eqa eqa' o c-iso-r eqa eqa' ] ≈ id1 A c' ] c-iso-lr {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' = let open ≈-Reasoning (A) in begin c-iso-l eqa eqa' o c-iso-r eqa eqa' ≈⟨⟩ - k eqa' (equalizer eqa) ( fe=ge eqa ) o k eqa (equalizer eqa') ( fe=ge eqa' ) + k eqa' e ( fe=ge eqa ) o k eqa e' ( fe=ge eqa' ) ≈↑⟨ uniqueness eqa' ( begin - e' o ( k eqa' (equalizer eqa) (fe=ge eqa) o k eqa (equalizer eqa') (fe=ge eqa') ) + e' o ( k eqa' e (fe=ge eqa) o k eqa e' (fe=ge eqa') ) ≈⟨ assoc ⟩ - ( e' o k eqa' (equalizer eqa) (fe=ge eqa) ) o k eqa (equalizer eqa') (fe=ge eqa') + ( e' o k eqa' e (fe=ge eqa) ) o k eqa e' (fe=ge eqa') ≈⟨ car (ek=h eqa') ⟩ - e o k eqa (equalizer eqa') (fe=ge eqa') + e o k eqa e' (fe=ge eqa') ≈⟨ ek=h eqa ⟩ e' ∎ )⟩ @@ -235,6 +238,7 @@ -- c-iso-rl is obvious from the symmetry + -------------------------------- ---- -- @@ -243,18 +247,18 @@ ---- 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 ) + ( eqa : {a b c : Obj A} → (f g : Hom A a b) → {e : Hom A c a } → IsEqualizer 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 {a} {b} {c} f g ) {d} ( A [ h o (equalizer ( eqa (A [ f o h ] ) (A [ g o h ] ))) ] ) + α = λ {a} {b} {c} f g e → equalizer1 (eqa {a} {b} {c} f g {e} ) ; -- Hom A c a + γ = λ {a} {b} {c} {d} f g h → k (eqa {a} {b} {c} f g ) {d} ( A [ h o (equalizer1 ( 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) (f1=f1 f); -- Hom A a c + δ = λ {a} {b} {c} e f → k (eqa {a} {b} {c} f f {e} ) {a} (id1 A a) (f1=f1 f); -- Hom A a c cong-α = λ {a b c e f g g'} eq → cong-α1 {a} {b} {c} {e} {f} {g} {g'} eq ; cong-γ = λ {a} {_} {c} {d} {f} {g} {h} {h'} eq → cong-γ1 {a} {c} {d} {f} {g} {h} {h'} eq ; cong-δ = λ {a b c e f f'} f=f' → cong-δ1 {a} {b} {c} {e} {f} {f'} f=f' ; b1 = fe=ge (eqa {a} {b} {c} f g {e}) ; - b2 = lemma-b2 ; + b2 = λ {d} {h} → lemma-b2 {d} {h}; b3 = lemma-b3 ; b4 = lemma-b4 } where @@ -270,41 +274,41 @@ -- -- e o id1 ≈ e → k e ≈ id - lemma-b3 : {a b d : Obj A} (f : Hom A a b ) { h : Hom A d a } → A [ A [ equalizer (eqa f f ) o k (eqa f f) (id1 A a) (f1=f1 f) ] ≈ id1 A a ] + lemma-b3 : {a b d : Obj A} (f : Hom A a b ) { h : Hom A d a } → A [ A [ equalizer1 (eqa f f ) o k (eqa f f) (id1 A a) (f1=f1 f) ] ≈ id1 A a ] lemma-b3 {a} {b} {d} f {h} = let open ≈-Reasoning (A) in begin - equalizer (eqa f f) o k (eqa f f) (id a) (f1=f1 f) + equalizer1 (eqa f f) o k (eqa f f) (id a) (f1=f1 f) ≈⟨ ek=h (eqa f f ) ⟩ id 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 ])) ] ] ] + A [ A [ f o A [ h o equalizer1 (eqa (A [ f o h ]) (A [ g o h ])) ] ] ≈ A [ g o A [ h o equalizer1 (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 ))) + f o ( h o equalizer1 (eqa (f o h) ( g o h ))) ≈⟨ assoc ⟩ - (f o h) o equalizer (eqa (f o h) ( g o h )) + (f o h) o equalizer1 (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 )) + (g o h) o equalizer1 (eqa (f o h) ( g o h )) ≈↑⟨ assoc ⟩ - g o ( h o equalizer (eqa (f o h) ( g o h ))) + g o ( h o equalizer1 (eqa (f o h) ( g o h ))) ∎ cong-α1 : {a b c : Obj A } → { e : Hom A c a } - → {f g g' : Hom A a b } → A [ g ≈ g' ] → A [ equalizer (eqa {a} {b} {c} f g {e} )≈ equalizer (eqa {a} {b} {c} f g' {e} ) ] + → {f g g' : Hom A a b } → A [ g ≈ g' ] → A [ equalizer1 (eqa {a} {b} {c} f g {e} )≈ equalizer1 (eqa {a} {b} {c} f g' {e} ) ] cong-α1 {a} {b} {c} {e} {f} {g} {g'} eq = let open ≈-Reasoning (A) in refl-hom cong-γ1 : {a c d : Obj A } → {f g : Hom A a b} {h h' : Hom A d a } → A [ h ≈ h' ] → { e : Hom A c a} → - A [ k (eqa f g {e} ) {d} ( A [ h o (equalizer ( eqa (A [ f o h ] ) (A [ g o h ] ) {id1 A d} )) ] ) (lemma-equ4 {a} {b} {c} {d} f g h ) - ≈ k (eqa f g {e} ) {d} ( A [ h' o (equalizer ( eqa (A [ f o h' ] ) (A [ g o h' ] ) {id1 A d} )) ] ) (lemma-equ4 {a} {b} {c} {d} f g h' ) ] + A [ k (eqa f g {e} ) {d} ( A [ h o (equalizer1 ( eqa (A [ f o h ] ) (A [ g o h ] ) {id1 A d} )) ] ) (lemma-equ4 {a} {b} {c} {d} f g h ) + ≈ k (eqa f g {e} ) {d} ( A [ h' o (equalizer1 ( eqa (A [ f o h' ] ) (A [ g o h' ] ) {id1 A d} )) ] ) (lemma-equ4 {a} {b} {c} {d} f g h' ) ] cong-γ1 {a} {c} {d} {f} {g} {h} {h'} h=h' {e} = let open ≈-Reasoning (A) in begin - 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 ) + k (eqa f g ) {d} ( A [ h o (equalizer1 ( eqa (A [ f o h ] ) (A [ g o h ] ))) ] ) (lemma-equ4 {a} {b} {c} {d} f g h ) ≈⟨ uniqueness (eqa f g) ( begin - e o 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' ) + e o k (eqa f g ) {d} ( A [ h' o (equalizer1 ( eqa (A [ f o h' ] ) (A [ g o h' ] ))) ] ) (lemma-equ4 {a} {b} {c} {d} f g h' ) ≈⟨ ek=h (eqa f g ) ⟩ - h' o (equalizer ( eqa (A [ f o h' ] ) (A [ g o h' ] ))) + h' o (equalizer1 ( eqa (A [ f o h' ] ) (A [ g o h' ] ))) ≈↑⟨ car h=h' ⟩ - h o (equalizer ( eqa (A [ f o h' ] ) (A [ g o h' ] ))) + h o (equalizer1 ( eqa (A [ f o h' ] ) (A [ g o 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' ) + k (eqa f g ) {d} ( A [ h' o (equalizer1 ( eqa (A [ f o h' ] ) (A [ g o h' ] ))) ] ) (lemma-equ4 {a} {b} {c} {d} f g h' ) ∎ cong-δ1 : {a b c : Obj A} {e : Hom A c a } {f f' : Hom A a b} → A [ f ≈ f' ] → A [ k (eqa {a} {b} {c} f f {e} ) (id1 A a) (f1=f1 f) ≈ k (eqa {a} {b} {c} f' f' {e} ) (id1 A a) (f1=f1 f') ] @@ -320,39 +324,39 @@ ∎ 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 ])) ] ] + A [ equalizer1 (eqa f g) o k (eqa f g) (A [ h o equalizer1 (eqa (A [ f o h ]) (A [ g o h ])) ]) (lemma-equ4 {a} {b} {c} f g h) ] + ≈ A [ h o equalizer1 (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) + equalizer1 (eqa f g) o k (eqa f g) (h o equalizer1 (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 )) + h o equalizer1 (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) (f1=f1 (A [ f o A [ equalizer (eqa f g) o j ] ])) ] + A [ k (eqa f g) (A [ A [ equalizer1 (eqa f g) o j ] o + equalizer1 (eqa (A [ f o A [ equalizer1 (eqa f g {e}) o j ] ]) (A [ g o A [ equalizer1 (eqa f g {e} ) o j ] ])) ]) + (lemma-equ4 {a} {b} {c} f g (A [ equalizer1 (eqa f g) o j ])) + o k (eqa (A [ f o A [ equalizer1 (eqa f g) o j ] ]) (A [ f o A [ equalizer1 (eqa f g) o j ] ])) + (id1 A d) (f1=f1 (A [ f o A [ equalizer1 (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) (f1=f1 (( f o ( equalizer (eqa f g) o j ) ))) ) + ( k (eqa f g) (( ( equalizer1 (eqa f g) o j ) o equalizer1 (eqa (( f o ( equalizer1 (eqa f g {e}) o j ) )) (( g o ( equalizer1 (eqa f g {e}) o j ) ))) )) + (lemma-equ4 {a} {b} {c} f g (( equalizer1 (eqa f g) o j ))) o + k (eqa (( f o ( equalizer1 (eqa f g) o j ) )) (( f o ( equalizer1 (eqa f g) o j ) ))) (id1 A d) (f1=f1 (( f o ( equalizer1 (eqa f g) o j ) ))) ) ≈⟨ car ((uniqueness (eqa f g) ( begin - equalizer (eqa f g) o j + equalizer1 (eqa f g) o j ≈↑⟨ idR ⟩ - (equalizer (eqa f g) o j ) o id d + (equalizer1 (eqa f g) o j ) o id d ≈⟨⟩ -- here we decide e (fej) (gej)' is id 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))) + ((equalizer1 (eqa f g) o j) o equalizer1 (eqa (f o equalizer1 (eqa f g {e}) o j) (g o equalizer1 (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) (f1=f1 (( 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 id d + j o k (eqa (( f o ( equalizer1 (eqa f g) o j ) )) (( f o ( equalizer1 (eqa f g) o j ) ))) (id1 A d) (f1=f1 (( f o ( equalizer1 (eqa f g) o j ) ))) + ≈⟨ cdr ((uniqueness (eqa (( f o ( equalizer1 (eqa f g) o j ) )) (( f o ( equalizer1 (eqa f g) o j ) ))) ( begin + equalizer1 (eqa (f o equalizer1 (eqa f g {e} ) o j) (f o equalizer1 (eqa f g {e}) o j)) o id d ≈⟨ idR ⟩ - equalizer (eqa (f o equalizer (eqa f g {e}) o j) (f o equalizer (eqa f g {e}) o j)) + equalizer1 (eqa (f o equalizer1 (eqa f g {e}) o j) (f o equalizer1 (eqa f g {e}) o j)) ≈⟨⟩ -- here we decide e (fej) (fej)' is id d id d ∎ ))) ⟩ @@ -367,7 +371,7 @@ -- lemma-equ2 : {a b c : Obj A} (f g : Hom A a b) (e : Hom A c a ) - → ( bur : Burroni A {c} {a} {b} f g e ) → Equalizer A {c} {a} {b} (α bur f g e) f g + → ( bur : Burroni A {c} {a} {b} f g e ) → IsEqualizer A {c} {a} {b} (α bur f g e) f g lemma-equ2 {a} {b} {c} f g e bur = record { fe=ge = fe=ge1 ; k = k1 ;
--- a/freyd.agda Thu Sep 01 17:34:28 2016 +0900 +++ b/freyd.agda Sun Sep 04 21:05:39 2016 +0900 @@ -44,6 +44,7 @@ open PreInitial open Complete open Equalizer +open IsEqualizer initialFromPreInitialFullSubcategory : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (comp : Complete A A) @@ -59,8 +60,8 @@ FMap← = SFSFMap← SFS lim : ( Γ : Functor A A ) → Limit A A Γ (limit-c comp Γ) (limit-u comp Γ) lim Γ = isLimit comp Γ - equ : {a b : Obj A} → (f g : Hom A a b) → Equalizer A (equalizer-e comp f g ) f g - equ f g = isEqualizer comp f g + equ : {a b : Obj A} → (f g : Hom A a b) → IsEqualizer A (equalizer-e comp f g ) f g + equ f g = Complete.isEqualizer comp f g a0 : Obj A a0 = limit-c comp F u : NTrans A A (K A A a0) F @@ -74,7 +75,7 @@ initialArrow : ∀( a : Obj A ) → Hom A a0 a initialArrow a = A [ preinitialArrow PI {a} o f ] equ-fi : { a : Obj A} → {f' : Hom A a0 a} → - Equalizer A ee ( A [ preinitialArrow PI {a} o f ] ) f' + IsEqualizer A ee ( A [ preinitialArrow PI {a} o f ] ) f' equ-fi {a} {f'} = equ ( A [ preinitialArrow PI {a} o f ] ) f' e=id : {e : Hom A a0 a0} → ( {c : Obj A} → A [ A [ TMap u c o e ] ≈ TMap u c ] ) → A [ e ≈ id1 A a0 ] e=id {e} uce=uc = let open ≈-Reasoning (A) in @@ -108,7 +109,7 @@ kfuc=1 {k1} {p} = e=id ( kfuc=uc ) -- if equalizer has right inverse, f = g lemma2 : (a b : Obj A) {c : Obj A} ( f g : Hom A a b ) - {e : Hom A c a } {e' : Hom A a c } ( equ : Equalizer A e f g ) (inv-e : A [ A [ e o e' ] ≈ id1 A a ] ) + {e : Hom A c a } {e' : Hom A a c } ( equ : IsEqualizer A e f g ) (inv-e : A [ A [ e o e' ] ≈ id1 A a ] ) → A [ f ≈ g ] lemma2 a b {c} f g {e} {e'} equ inv-e = let open ≈-Reasoning (A) in let open Equalizer in
--- a/limit-to.agda Thu Sep 01 17:34:28 2016 +0900 +++ b/limit-to.agda Sun Sep 04 21:05:39 2016 +0900 @@ -448,7 +448,7 @@ (lim : ( Γ : Functor (I' {c₁} {c₂} {ℓ}) A ) → {a0 : Obj A } {u : NTrans I' A ( K A I' a0 ) Γ } → Limit A I' Γ a0 u ) -- completeness → {a b c : Obj A} (f g : Hom A a b ) - → (e : Hom A c a ) → (fe=ge : A [ A [ f o e ] ≈ A [ g o e ] ] ) → Equalizer A e f g + → (e : Hom A c a ) → (fe=ge : A [ A [ f o e ] ≈ A [ g o e ] ] ) → IsEqualizer A e f g lim-to-equ {c₁} {c₂} {ℓ } A none lim {a} {b} {c} f g e fe=ge = record { fe=ge = fe=ge ; k = λ {d} h fh=gh → k {d} h fh=gh
--- a/pullback.agda Thu Sep 01 17:34:28 2016 +0900 +++ b/pullback.agda Sun Sep 04 21:05:39 2016 +0900 @@ -27,16 +27,17 @@ -- k (π1' × π2' ) open Equalizer +open IsEqualizer open Product open Pullback pullback-from : (a b c ab d : Obj A) ( f : Hom A a c ) ( g : Hom A b c ) ( π1 : Hom A ab a ) ( π2 : Hom A ab b ) ( e : Hom A d ab ) - ( eqa : {a b c : Obj A} → (f g : Hom A a b) → {e : Hom A c a } → Equalizer A e f g ) + ( eqa : {a b c : Obj A} → (f g : Hom A a b) → {e : Hom A c a } → IsEqualizer A e f g ) ( prod : Product A a b ab π1 π2 ) → Pullback A a b c d f g - ( A [ π1 o equalizer ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ){e} ) ] ) - ( A [ π2 o equalizer ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ){e} ) ] ) + ( A [ π1 o equalizer1 ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) ) ] ) + ( A [ π2 o equalizer1 ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) ) ] ) pullback-from a b c ab d f g π1 π2 e eqa prod = record { commute = commute1 ; p = p1 ; @@ -44,17 +45,17 @@ π2p=π2 = λ {d} {π1'} {π2'} {eq} → π2p=π21 {d} {π1'} {π2'} {eq} ; uniqueness = uniqueness1 } where - commute1 : A [ A [ f o A [ π1 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] ] - ≈ A [ g o A [ π2 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] ] ] + commute1 : A [ A [ f o A [ π1 o equalizer1 (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] ] + ≈ A [ g o A [ π2 o equalizer1 (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] ] ] commute1 = let open ≈-Reasoning (A) in begin - f o ( π1 o equalizer (eqa ( f o π1 ) ( g o π2 )) ) + f o ( π1 o equalizer1 (eqa ( f o π1 ) ( g o π2 )) ) ≈⟨ assoc ⟩ - ( f o π1 ) o equalizer (eqa ( f o π1 ) ( g o π2 )) + ( f o π1 ) o equalizer1 (eqa ( f o π1 ) ( g o π2 )) ≈⟨ fe=ge (eqa (A [ f o π1 ]) (A [ g o π2 ])) ⟩ - ( g o π2 ) o equalizer (eqa ( f o π1 ) ( g o π2 )) + ( g o π2 ) o equalizer1 (eqa ( f o π1 ) ( g o π2 )) ≈↑⟨ assoc ⟩ - g o ( π2 o equalizer (eqa ( f o π1 ) ( g o π2 )) ) + g o ( π2 o equalizer1 (eqa ( f o π1 ) ( g o π2 )) ) ∎ lemma1 : {d' : Obj A} {π1' : Hom A d' a} {π2' : Hom A d' b} → A [ A [ f o π1' ] ≈ A [ g o π2' ] ] → A [ A [ A [ f o π1 ] o (prod × π1') π2' ] ≈ A [ A [ g o π2 ] o (prod × π1') π2' ] ] @@ -76,10 +77,10 @@ p1 {d'} { π1' } { π2' } eq = let open ≈-Reasoning (A) in k ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} ) (_×_ prod π1' π2' ) ( lemma1 eq ) π1p=π11 : {d₁ : Obj A} {π1' : Hom A d₁ a} {π2' : Hom A d₁ b} {eq : A [ A [ f o π1' ] ≈ A [ g o π2' ] ]} → - A [ A [ A [ π1 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ]) {e} ) ] o p1 eq ] ≈ π1' ] + A [ A [ A [ π1 o equalizer1 (eqa (A [ f o π1 ]) (A [ g o π2 ]) {e} ) ] o p1 eq ] ≈ π1' ] π1p=π11 {d'} {π1'} {π2'} {eq} = let open ≈-Reasoning (A) in begin - ( π1 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ]) {e} ) ) o p1 eq + ( π1 o equalizer1 (eqa (A [ f o π1 ]) (A [ g o π2 ]) {e} ) ) o p1 eq ≈⟨⟩ ( π1 o e) o k ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} ) (_×_ prod π1' π2' ) (lemma1 eq) ≈↑⟨ assoc ⟩ @@ -90,10 +91,10 @@ π1' ∎ π2p=π21 : {d₁ : Obj A} {π1' : Hom A d₁ a} {π2' : Hom A d₁ b} {eq : A [ A [ f o π1' ] ≈ A [ g o π2' ] ]} → - A [ A [ A [ π2 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ]) {e} ) ] o p1 eq ] ≈ π2' ] + A [ A [ A [ π2 o equalizer1 (eqa (A [ f o π1 ]) (A [ g o π2 ]) {e} ) ] o p1 eq ] ≈ π2' ] π2p=π21 {d'} {π1'} {π2'} {eq} = let open ≈-Reasoning (A) in begin - ( π2 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ]) {e} ) ) o p1 eq + ( π2 o equalizer1 (eqa (A [ f o π1 ]) (A [ g o π2 ]) {e} ) ) o p1 eq ≈⟨⟩ ( π2 o e) o k ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} ) (_×_ prod π1' π2' ) (lemma1 eq) ≈↑⟨ assoc ⟩ @@ -105,23 +106,23 @@ ∎ uniqueness1 : {d₁ : Obj A} (p' : Hom A d₁ d) {π1' : Hom A d₁ a} {π2' : Hom A d₁ b} {eq : A [ A [ f o π1' ] ≈ A [ g o π2' ] ]} → - {eq1 : A [ A [ A [ π1 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] o p' ] ≈ π1' ]} → - {eq2 : A [ A [ A [ π2 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] o p' ] ≈ π2' ]} → + {eq1 : A [ A [ A [ π1 o equalizer1 (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] o p' ] ≈ π1' ]} → + {eq2 : A [ A [ A [ π2 o equalizer1 (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] o p' ] ≈ π2' ]} → A [ p1 eq ≈ p' ] uniqueness1 {d'} p' {π1'} {π2'} {eq} {eq1} {eq2} = let open ≈-Reasoning (A) in begin p1 eq ≈⟨⟩ k ( eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e} ) (_×_ prod π1' π2' ) (lemma1 eq) - ≈⟨ Equalizer.uniqueness (eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e}) ( begin + ≈⟨ IsEqualizer.uniqueness (eqa ( A [ f o π1 ] ) ( A [ g o π2 ] ) {e}) ( begin e o p' ≈⟨⟩ - equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) o p' + equalizer1 (eqa (A [ f o π1 ]) (A [ g o π2 ])) o p' ≈↑⟨ Product.uniqueness prod ⟩ - (prod × ( π1 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) o p') ) ( π2 o (equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) o p')) + (prod × ( π1 o equalizer1 (eqa (A [ f o π1 ]) (A [ g o π2 ])) o p') ) ( π2 o (equalizer1 (eqa (A [ f o π1 ]) (A [ g o π2 ])) o p')) ≈⟨ ×-cong prod (assoc) (assoc) ⟩ - (prod × (A [ A [ π1 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] o p' ])) - (A [ A [ π2 o equalizer (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] o p' ]) + (prod × (A [ A [ π1 o equalizer1 (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] o p' ])) + (A [ A [ π2 o equalizer1 (eqa (A [ f o π1 ]) (A [ g o π2 ])) ] o p' ]) ≈⟨ ×-cong prod eq1 eq2 ⟩ ((prod × π1') π2') ∎ ) ⟩ @@ -346,7 +347,7 @@ -- limit-ε : - ( eqa : {a b c : Obj A} → (e : Hom A c a ) → (f g : Hom A a b) → Equalizer A e f g ) + ( eqa : {a b c : Obj A} → (e : Hom A c a ) → (f g : Hom A a b) → IsEqualizer A e f g ) ( lim p : Obj A ) ( e : Hom A lim p ) ( proj : (i : Obj I ) → Hom A p (FObj Γ i) ) → NTrans I A (K A I lim) Γ @@ -375,7 +376,7 @@ limit-from : ( prod : (p : Obj A) ( ai : Obj I → Obj A ) ( pi : (i : Obj I) → Hom A p ( ai i ) ) → IProduct {c₁'} A (Obj I) p ai pi ) - ( eqa : {a b c : Obj A} → (e : Hom A c a ) → (f g : Hom A a b) → Equalizer A e f g ) + ( eqa : {a b c : Obj A} → (e : Hom A c a ) → (f g : Hom A a b) → IsEqualizer A e f g ) ( lim p : Obj A ) -- limit to be made ( e : Hom A lim p ) -- existing of equalizer ( proj : (i : Obj I ) → Hom A p (FObj Γ i) ) -- existing of product ( projection actually ) @@ -407,7 +408,7 @@ limit1 a t ≈⟨⟩ k (eqa e (id1 A p) (id1 A p )) (product ( prod p (FObj Γ) proj ) (TMap t) ) refl-hom - ≈⟨ Equalizer.uniqueness (eqa e (id1 A p) (id1 A p )) ( begin + ≈⟨ IsEqualizer.uniqueness (eqa e (id1 A p) (id1 A p )) ( begin e o f ≈↑⟨ ip-uniqueness (prod p (FObj Γ) proj) ⟩ product (prod p (FObj Γ) proj) (λ i → ( proj i o ( e o f ) ) )