Mercurial > hg > Members > kono > Proof > category
view SetsCompleteness.agda @ 594:db76b73b526c
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 22 May 2017 12:04:34 +0900 |
| parents | a158ebb391f2 |
| children | 0386e82f0dd8 |
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open import Category -- https://github.com/konn/category-agda open import Level open import Category.Sets renaming ( _o_ to _*_ ) module SetsCompleteness where open import cat-utility open import Relation.Binary.Core open import Function import Relation.Binary.PropositionalEquality -- Extensionality a b = {A : Set a} {B : A → Set b} {f g : (x : A) → B x} → (∀ x → f x ≡ g x) → f ≡ g → ( λ x → f x ≡ λ x → g x ) postulate extensionality : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → Relation.Binary.PropositionalEquality.Extensionality c₂ c₂ ≡cong = Relation.Binary.PropositionalEquality.cong ≈-to-≡ : { c₂ : Level } {a b : Obj (Sets { c₂})} {f g : Hom Sets a b} → Sets [ f ≈ g ] → (x : a ) → f x ≡ g x ≈-to-≡ refl x = refl record Σ {a} (A : Set a) (B : Set a) : Set a where constructor _,_ field proj₁ : A proj₂ : B open Σ public SetsProduct : { c₂ : Level} → CreateProduct ( Sets { c₂} ) SetsProduct { c₂ } = record { product = λ a b → Σ a b ; π1 = λ a b → λ ab → (proj₁ ab) ; π2 = λ a b → λ ab → (proj₂ ab) ; isProduct = λ a b → record { _×_ = λ f g x → record { proj₁ = f x ; proj₂ = g x } -- ( f x , g x ) ; π1fxg=f = refl ; π2fxg=g = refl ; uniqueness = refl ; ×-cong = λ {c} {f} {f'} {g} {g'} f=f g=g → prod-cong a b f=f g=g } } where prod-cong : ( a b : Obj (Sets {c₂}) ) {c : Obj (Sets {c₂}) } {f f' : Hom Sets c a } {g g' : Hom Sets c b } → Sets [ f ≈ f' ] → Sets [ g ≈ g' ] → Sets [ (λ x → f x , g x) ≈ (λ x → f' x , g' x) ] prod-cong a b {c} {f} {.f} {g} {.g} refl refl = refl record sproduct {a} (I : Set a) ( Product : I → Set a ) : Set a where field proj : ( i : I ) → Product i open sproduct iproduct1 : { c₂ : Level} → (I : Obj (Sets { c₂}) ) (fi : I → Obj Sets ) {q : Obj Sets} → ((i : I) → Hom Sets q (fi i)) → Hom Sets q (sproduct I fi) iproduct1 I fi {q} qi x = record { proj = λ i → (qi i) x } ipcx : { c₂ : Level} → (I : Obj (Sets { c₂})) (fi : I → Obj Sets ) {q : Obj Sets} {qi qi' : (i : I) → Hom Sets q (fi i)} → ((i : I) → Sets [ qi i ≈ qi' i ]) → (x : q) → iproduct1 I fi qi x ≡ iproduct1 I fi qi' x ipcx I fi {q} {qi} {qi'} qi=qi x = begin record { proj = λ i → (qi i) x } ≡⟨ ≡cong ( λ qi → record { proj = qi } ) ( extensionality Sets (λ i → ≈-to-≡ (qi=qi i) x )) ⟩ record { proj = λ i → (qi' i) x } ∎ where open import Relation.Binary.PropositionalEquality open ≡-Reasoning ip-cong : { c₂ : Level} → (I : Obj (Sets { c₂}) ) (fi : I → Obj Sets ) {q : Obj Sets} {qi qi' : (i : I) → Hom Sets q (fi i)} → ((i : I) → Sets [ qi i ≈ qi' i ]) → Sets [ iproduct1 I fi qi ≈ iproduct1 I fi qi' ] ip-cong I fi {q} {qi} {qi'} qi=qi = extensionality Sets ( ipcx I fi qi=qi ) SetsIProduct : { c₂ : Level} → (I : Obj Sets) (fi : I → Obj Sets ) → IProduct ( Sets { c₂} ) I SetsIProduct I fi = record { ai = fi ; iprod = sproduct I fi ; pi = λ i prod → proj prod i ; isIProduct = record { iproduct = iproduct1 I fi ; pif=q = pif=q ; ip-uniqueness = ip-uniqueness ; ip-cong = ip-cong I fi } } where pif=q : {q : Obj Sets} (qi : (i : I) → Hom Sets q (fi i)) {i : I} → Sets [ Sets [ (λ prod → proj prod i) o iproduct1 I fi qi ] ≈ qi i ] pif=q {q} qi {i} = refl ip-uniqueness : {q : Obj Sets} {h : Hom Sets q (sproduct I fi)} → Sets [ iproduct1 I fi (λ i → Sets [ (λ prod → proj prod i) o h ]) ≈ h ] ip-uniqueness = refl -- -- e f -- c -------→ a ---------→ b f ( f' -- ^ . ---------→ -- | . g -- |k . -- | . h --y : d -- cf. https://github.com/danr/Agda-projects/blob/master/Category-Theory/Equalizer.agda data sequ {c : Level} (A B : Set c) ( f g : A → B ) : Set c where elem : (x : A ) → (eq : f x ≡ g x) → sequ A B f g equ : { c₂ : Level} {a b : Obj (Sets {c₂}) } { f g : Hom (Sets {c₂}) a b } → ( sequ a b f g ) → a equ (elem x eq) = x fe=ge0 : { c₂ : Level} {a b : Obj (Sets {c₂}) } { f g : Hom (Sets {c₂}) a b } → (x : sequ a b f g) → (Sets [ f o (λ e → equ e) ]) x ≡ (Sets [ g o (λ e → equ e) ]) x fe=ge0 (elem x eq ) = eq irr : { c₂ : Level} {d : Set c₂ } { x y : d } ( eq eq' : x ≡ y ) → eq ≡ eq' irr refl refl = refl elm-cong : { c₂ : Level} → {a b : Obj (Sets {c₂}) } {f g : Hom (Sets {c₂}) a b} → (x y : sequ a b f g) → equ x ≡ equ y → x ≡ y elm-cong ( elem x eq ) (elem .x eq' ) refl = ≡cong ( λ ee → elem x ee ) ( irr eq eq' ) fe=ge : { c₂ : Level} → {a b : Obj (Sets {c₂}) } {f g : Hom (Sets {c₂}) a b} → Sets [ Sets [ f o (λ e → equ {_} {a} {b} {f} {g} e ) ] ≈ Sets [ g o (λ e → equ e ) ] ] fe=ge = extensionality Sets (fe=ge0 ) k : { c₂ : Level} → {a b : Obj (Sets {c₂}) } {f g : Hom (Sets {c₂}) a b} → {d : Obj Sets} (h : Hom Sets d a) → Sets [ Sets [ f o h ] ≈ Sets [ g o h ] ] → Hom Sets d (sequ a b f g) k {_} {_} {_} {_} {_} {d} h eq = λ x → elem (h x) ( ≈-to-≡ eq x ) ek=h : { c₂ : Level} → {a b : Obj (Sets {c₂}) } {f g : Hom (Sets {c₂}) a b} → {d : Obj Sets} {h : Hom Sets d a} {eq : Sets [ Sets [ f o h ] ≈ Sets [ g o h ] ]} → Sets [ Sets [ (λ e → equ {_} {a} {b} {f} {g} e ) o k h eq ] ≈ h ] ek=h {_} {_} {_} {_} {_} {d} {h} {eq} = refl open sequ -- equalizer-c = sequ a b f g -- ; equalizer = λ e → equ e SetsIsEqualizer : { c₂ : Level} → (a b : Obj (Sets {c₂}) ) (f g : Hom (Sets {c₂}) a b) → IsEqualizer Sets (λ e → equ e )f g SetsIsEqualizer {c₂} a b f g = record { fe=ge = fe=ge { c₂ } {a} {b} {f} {g} ; k = λ {d} h eq → k { c₂ } {a} {b} {f} {g} {d} h eq ; ek=h = λ {d} {h} {eq} → ek=h {c₂} {a} {b} {f} {g} {d} {h} {eq} ; uniqueness = uniqueness } where injection : { c₂ : Level } {a b : Obj (Sets { c₂})} (f : Hom Sets a b) → Set c₂ injection f = ∀ x y → f x ≡ f y → x ≡ y lemma5 : {d : Obj Sets} {h : Hom Sets d a} {fh=gh : Sets [ Sets [ f o h ] ≈ Sets [ g o h ] ]} {k' : Hom Sets d (sequ a b f g)} → Sets [ Sets [ (λ e → equ e) o k' ] ≈ h ] → (x : d ) → equ {_} {a} {b} {f} {g} (k h fh=gh x) ≡ equ (k' x) lemma5 refl x = refl -- somehow this is not equal to ≈-to-≡ uniqueness : {d : Obj Sets} {h : Hom Sets d a} {fh=gh : Sets [ Sets [ f o h ] ≈ Sets [ g o h ] ]} {k' : Hom Sets d (sequ a b f g)} → Sets [ Sets [ (λ e → equ e) o k' ] ≈ h ] → Sets [ k h fh=gh ≈ k' ] uniqueness {d} {h} {fh=gh} {k'} ek'=h = extensionality Sets ( λ ( x : d ) → begin k h fh=gh x ≡⟨ elm-cong ( k h fh=gh x) ( k' x ) (lemma5 {d} {h} {fh=gh} {k'} ek'=h x ) ⟩ k' x ∎ ) where open import Relation.Binary.PropositionalEquality open ≡-Reasoning open Functor ---- -- C is locally small i.e. Hom C i j is a set c₁ -- record Small { c₁ c₂ ℓ : Level} ( C : Category c₁ c₂ ℓ ) ( I : Set c₁ ) : Set (suc (c₁ ⊔ c₂ ⊔ ℓ )) where field hom→ : {i j : Obj C } → Hom C i j → I → I hom← : {i j : Obj C } → ( f : I → I ) → Hom C i j hom-iso : {i j : Obj C } → { f : Hom C i j } → hom← ( hom→ f ) ≡ f -- ≈-≡ : {a b : Obj C } { x y : Hom C a b } → (x≈y : C [ x ≈ y ] ) → x ≡ y open Small ΓObj : { c₁ c₂ ℓ : Level} { C : Category c₁ c₂ ℓ } { I : Set c₁ } ( s : Small C I ) ( Γ : Functor C ( Sets { c₁} )) (i : Obj C ) → Set c₁ ΓObj s Γ i = FObj Γ i ΓMap : { c₁ c₂ ℓ : Level} { C : Category c₁ c₂ ℓ } { I : Set c₁ } ( s : Small C I ) ( Γ : Functor C ( Sets { c₁} )) {i j : Obj C } → ( f : I → I ) → ΓObj s Γ i → ΓObj s Γ j ΓMap s Γ {i} {j} f = FMap Γ ( hom← s f ) slid : { c₁ c₂ ℓ : Level} ( C : Category c₁ c₂ ℓ ) ( I : Set c₁ ) ( s : Small C I ) → (x : Obj C) → I → I slid C I s x = hom→ s ( id1 C x ) record slim { c₂ } { I OC : Set c₂ } ( sobj : OC → Set c₂ ) ( smap : { i j : OC } → (f : I → I ) → sobj i → sobj j ) : Set c₂ where field slproj : ( i : OC ) → sobj i slequ : (i j : OC) (f : I → I ) → sequ OC (sobj j) (λ x → smap f (slproj i) ) (λ x → slproj j ) slprod : sproduct OC sobj slprod = record { proj = slproj } slmap : { i j : OC } → (f : I → I ) → sobj i → sobj j slmap f x = smap f x open slim open import HomReasoning open NTrans Cone : { c₁ c₂ ℓ : Level} ( C : Category c₁ c₂ ℓ ) ( I : Set c₁ ) ( s : Small C I ) ( Γ : Functor C (Sets {c₁} ) ) → NTrans C Sets (K Sets C ( slim (ΓObj s Γ) (ΓMap s Γ) )) Γ Cone {c₁} C I s Γ = record { TMap = λ i → λ se → proj ( slprod se ) i ; isNTrans = record { commute = commute1 } } where commute1 : {a b : Obj C} {f : Hom C a b} → Sets [ Sets [ FMap Γ f o (λ se → proj (slprod se) a) ] ≈ Sets [ (λ se → proj (slprod se) b) o FMap (K Sets C (slim (ΓObj s Γ) (ΓMap s Γ))) f ] ] commute1 {a} {b} {f} = extensionality Sets ( λ se → begin FMap Γ f (proj (slprod se ) a ) ≡⟨ ≡cong ( λ g → FMap Γ g (proj (slprod se) a)) (sym ( hom-iso s ) ) ⟩ FMap Γ (hom← s ( hom→ s f)) (proj (slprod se) a) ≡⟨ fe=ge0 (slequ se a b ( hom→ s f) ) ⟩ proj (slprod se) b ∎ ) where open import Relation.Binary.PropositionalEquality open ≡-Reasoning ≡cong2 : { c c' : Level } { A B : Set c } { C : Set c' } { a a' : A } { b b' : B } ( f : A → B → C ) → a ≡ a' → b ≡ b' → f a b ≡ f a' b' ≡cong2 _ refl refl = refl snat-cong : { c : Level } { I OC : Set c } ( SObj : OC → Set c ) ( SMap : { i j : OC } → (f : I → I) → SObj i → SObj j) { s t : slim SObj SMap } → (( i : OC ) → slproj s i ≡ slproj t i ) → ( ( i j : OC ) ( f : I → I ) → SMap {i} {j} f ( slproj s i ) ≡ slproj s j ) → ( ( i j : OC ) ( f : I → I ) → SMap {i} {j} f ( slproj t i ) ≡ slproj t j ) → s ≡ t snat-cong {_} {I} {OC} SO SM {s} {t} eq1 eq2 eq3 = begin record { slproj = λ i → slproj s i ; slequ = λ i j f → slequ s i j f } ≡⟨ ≡cong2 ( λ x y → record { slproj = λ i → x i ; slequ = λ i j f → {!!} } ) ( extensionality Sets ( λ i → eq1 i ) ) ( extensionality Sets ( λ x → ( extensionality Sets ( λ i → ( extensionality Sets ( λ j → ( extensionality Sets ( λ f → {!!} )))))))) ⟩ record { slproj = λ i → slproj t i ; slequ = λ i j f → slequ t i j f } ∎ where open import Relation.Binary.PropositionalEquality open ≡-Reasoning SetsLimit : { c₁ c₂ ℓ : Level} ( C : Category c₁ c₂ ℓ ) ( I : Set c₁ ) ( small : Small C I ) ( Γ : Functor C (Sets {c₁} ) ) → Limit Sets C Γ SetsLimit { c₂} C I s Γ = record { a0 = slim (ΓObj s Γ) (ΓMap s Γ) ; t0 = Cone C I s Γ ; isLimit = record { limit = limit1 ; t0f=t = λ {a t i } → refl ; limit-uniqueness = λ {a t f } → limit-uniqueness {a} {t} {f} } } where limit1 : (a : Obj Sets) → NTrans C Sets (K Sets C a) Γ → Hom Sets a (slim (ΓObj s Γ) (ΓMap s Γ)) limit1 a t = λ x → record { slequ = λ i j f → elem i ( ≡cong ( λ f → f x ) ( IsNTrans.commute ( isNTrans t ) )) ; slproj = λ i → ( TMap t i ) x } limit-uniqueness : {a : Obj Sets} {t : NTrans C Sets (K Sets C a) Γ} {f : Hom Sets a (slim (ΓObj s Γ) (ΓMap s Γ))} → ({i : Obj C} → Sets [ Sets [ TMap (Cone C I s Γ) i o f ] ≈ TMap t i ]) → Sets [ limit1 a t ≈ f ] limit-uniqueness {a} {t} {f} cif=t = extensionality Sets ( λ x → begin limit1 a t x ≡⟨⟩ record { slproj = λ i → TMap t i x ; slequ = λ i j f → elem i ( ≡cong ( λ f → f x ) ( IsNTrans.commute ( isNTrans t ) )) } ≡⟨ snat-cong (ΓObj s Γ) (ΓMap s Γ) (eq1 x) (eq2 x ) (eq3 x ) ⟩ record { slproj = λ i → slproj (f x ) i ; slequ = slequ (f x ) } ≡⟨⟩ f x ∎ ) where open import Relation.Binary.PropositionalEquality open ≡-Reasoning eq1 : (x : a ) (i : Obj C) → TMap t i x ≡ slproj (f x) i eq1 x i = sym ( ≡cong ( λ f → f x ) cif=t ) eq2 : (x : a ) (i j : Obj C) (f : I → I) → ΓMap s Γ f (TMap t i x) ≡ TMap t j x eq2 x i j f = ≡cong ( λ f → f x ) ( IsNTrans.commute ( isNTrans t ) ) eq3 : (x : a ) (i j : Obj C) (k : I → I) → ΓMap s Γ k (slproj (f x) i) ≡ slproj (f x) j eq3 x i j k = fe=ge0 ( slequ (f x ) i j k )