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 lemma1 : { c₂ : Level } {a b : Obj (Sets { c₂})} {f g : Hom Sets a b} → Sets [ f ≈ g ] → (x : a ) → f x ≡ g x lemma1 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 iproduct {a} (I : Set a) ( pi0 : I → Set a ) : Set a where field pi1 : ( i : I ) → pi0 i open iproduct SetsIProduct : { c₂ : Level} → (I : Obj Sets) (ai : I → Obj Sets ) → IProduct ( Sets { c₂} ) I SetsIProduct I fi = record { ai = fi ; iprod = iproduct I fi ; pi = λ i prod → pi1 prod i ; isIProduct = record { iproduct = iproduct1 ; pif=q = pif=q ; ip-uniqueness = ip-uniqueness ; ip-cong = ip-cong } } where iproduct1 : {q : Obj Sets} → ((i : I) → Hom Sets q (fi i)) → Hom Sets q (iproduct I fi) iproduct1 {q} qi x = record { pi1 = λ i → (qi i) x } pif=q : {q : Obj Sets} (qi : (i : I) → Hom Sets q (fi i)) {i : I} → Sets [ Sets [ (λ prod → pi1 prod i) o iproduct1 qi ] ≈ qi i ] pif=q {q} qi {i} = refl ip-uniqueness : {q : Obj Sets} {h : Hom Sets q (iproduct I fi)} → Sets [ iproduct1 (λ i → Sets [ (λ prod → pi1 prod i) o h ]) ≈ h ] ip-uniqueness = refl ipcx : {q : Obj Sets} {qi qi' : (i : I) → Hom Sets q (fi i)} → ((i : I) → Sets [ qi i ≈ qi' i ]) → (x : q) → iproduct1 qi x ≡ iproduct1 qi' x ipcx {q} {qi} {qi'} qi=qi x = begin record { pi1 = λ i → (qi i) x } ≡⟨ ≡cong ( λ QIX → record { pi1 = QIX } ) ( extensionality Sets (λ i → ≡cong ( λ f → f x ) (qi=qi i) )) ⟩ record { pi1 = λ i → (qi' i) x } ∎ where open import Relation.Binary.PropositionalEquality open ≡-Reasoning ip-cong : {q : Obj Sets} {qi qi' : (i : I) → Hom Sets q (fi i)} → ((i : I) → Sets [ qi i ≈ qi' i ]) → Sets [ iproduct1 qi ≈ iproduct1 qi' ] ip-cong {q} {qi} {qi'} qi=qi = extensionality Sets ( ipcx qi=qi ) -- -- 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 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) ( ≡cong ( λ y → y x ) eq ) 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 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 (k h fh=gh x) ≡ equ (k' x) lemma5 refl x = refl -- somehow this is not equal to lemma1 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 ≡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 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 ) record snproj { c₂ } { I OC : Set c₂ } ( sobj : OC → Set c₂ ) ( smap : { i j : OC } → (f : I → I )→ sobj i → sobj j ) : Set c₂ where field snmap : ( i : OC ) → sobj i open snproj record snequ { c₂ } { I OC : Set c₂ } ( sobj : OC → Set c₂ ) ( smap : { i j : OC } → (f : I → I )→ sobj i → sobj j ) : Set c₂ where field snequ1 : { i j : OC } → ( f : I → I ) → sequ ( snproj sobj smap ) (sobj j) ( λ x → smap f ( snmap x i ) ) ( λ x → snmap x j ) open snequ 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 (snequ (ΓObj s Γ) (ΓMap s Γ) )) Γ Cone C I s Γ = record { TMap = λ i → λ se → snmap ( equ ( snequ1 se {i} {i} (λ x → x )) ) i ; isNTrans = record { commute = comm1 } } where comm1 : {a b : Obj C} {f : Hom C a b} → Sets [ Sets [ FMap Γ f o (λ se → snmap (equ (snequ1 se (λ x → x))) a) ] ≈ Sets [ (λ se → snmap (equ (snequ1 se (λ x → x))) b) o FMap (K Sets C (snequ (ΓObj s Γ) (ΓMap s Γ))) f ] ] comm1 {a} {b} {f} = begin FMap Γ f o (λ se → snmap (equ (snequ1 se (λ x → x))) a) ≈⟨⟩ ( λ se → FMap Γ f (snmap se a )) o (λ se → equ (snequ1 se (λ x → x)) ) ≈⟨ {!!} ⟩ ( λ se → snmap se b ) o (λ se → equ (snequ1 se (λ x → x)) ) ≈⟨⟩ (( λ se → snmap se b ) o (λ se → equ (snequ1 se (λ x → x)) ) ) o id1 Sets ( snequ (ΓObj s Γ) (ΓMap s Γ) ) ≈⟨⟩ (λ se → snmap (equ (snequ1 se (λ x → x))) b) o FMap (K Sets C (snequ (ΓObj s Γ) (ΓMap s Γ))) f ∎ where open ≈-Reasoning Sets 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 = snequ (ΓObj s Γ) (ΓMap s Γ) ; t0 = Cone C I s Γ ; isLimit = record { limit = limit1 ; t0f=t = λ {a t i } → t0f=t {a} {t} {i} ; limit-uniqueness = λ {a t i } → limit-uniqueness {a} {t} {i} } } where a0 : Obj Sets a0 = snequ (ΓObj s Γ) (ΓMap s Γ) setprod : {a : Obj Sets} → NTrans C Sets (K Sets C a) Γ → (x : a ) → snproj (ΓObj s Γ) (ΓMap s Γ) setprod t x = record { snmap = λ i → TMap t i x } comm2 : { a : Obj Sets } {x : a } {i j : Obj C} (t : NTrans C Sets (K Sets C a) Γ) (f : I → I ) → ΓMap s Γ f (TMap t i x) ≡ TMap t j x comm2 {a} {x} t f = ≡cong ( λ f → f x ) ( IsNTrans.commute ( isNTrans t ) ) comm3 : { a : Obj Sets } {x : a } {i j : Obj C} (t : NTrans C Sets (K Sets C a) Γ) (f : I → I ) → Sets [ Sets [ (λ x₁ → ΓMap s Γ f (snmap x₁ i)) o setprod t ] ≈ Sets [ (λ x₁ → snmap x₁ j) o setprod t ] ] comm3 {a} {x} t f = IsNTrans.commute (isNTrans t ) limit1 : (a : Obj Sets) → NTrans C Sets (K Sets C a) Γ → Hom Sets a (snequ (ΓObj s Γ) (ΓMap s Γ)) limit1 a t = λ ( x : a ) → record { snequ1 = λ {i} {j} f → k ( setprod t ) (comm3 {a} {x} t f ) x } t0f=t : {a : Obj Sets} {t : NTrans C Sets (K Sets C a) Γ} {i : Obj C} → Sets [ Sets [ TMap (Cone C I s Γ ) i o limit1 a t ] ≈ TMap t i ] t0f=t {a} {t} {i} = extensionality Sets ( λ x → begin ( Sets [ TMap (Cone C I s Γ ) i o limit1 a t ]) x ≡⟨⟩ TMap t i x ∎ ) where open import Relation.Binary.PropositionalEquality open ≡-Reasoning limit-uniqueness : {a : Obj Sets} {t : NTrans C Sets (K Sets C a) Γ} {f : Hom Sets a (snequ (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 { snequ1 = λ {i} {j} f' → k ( setprod t ) (comm3 {a} {x} t f' ) x } ≡⟨ ≡cong ( λ ff → record { snequ1 = λ {i} {j} f' → ff i j f' }) ( extensionality Sets ( λ i → extensionality Sets ( λ j → extensionality Sets ( λ f' → k-cong {i} {j} f' x ))) ) ⟩ record { snequ1 = λ {i} {j} f' → snequ1 (f x) f' } ≡⟨⟩ f x ∎ ) where open import Relation.Binary.PropositionalEquality open ≡-Reasoning snmap-cong : ( e : snequ (ΓObj s Γ) (ΓMap s Γ) ) {i : Obj C } { f g : I → I } → snmap ( equ ( snequ1 e f)) i ≡ snmap ( equ ( snequ1 e g)) i snmap-cong e {i} = ≡cong ( λ s → snmap s i ) refl k-cong : { i j : Obj C } ( f' : I → I ) ( x : a ) → k ( setprod t ) (comm3 {a} {x} t f' ) x ≡ snequ1 (f x) f' k-cong {i} {j} f' x = begin k ( setprod t ) (comm3 {a} {x} t f' ) x ≡⟨ elm-cong (k ( setprod t ) (comm3 {a} {x} t f' ) x ) ( snequ1 (f x) f' ) ( begin equ (k (setprod t) (comm3 {a} {x} t f') x) ≡⟨⟩ record { snmap = λ i' → TMap t i' x } ≡⟨ ≡cong ( λ s → record { snmap = λ i' → s i' } ) ( extensionality Sets ( λ i' → ( sym ( begin snmap ( equ ( snequ1 (f x) f')) i' ≡⟨ snmap-cong (f x) ⟩ snmap ( equ ( snequ1 (f x) (λ x → x ))) i' ≡⟨ ≡cong ( λ f → f x ) cif=t ⟩ TMap t i' x ∎ )))) ⟩ record { snmap = λ i' → snmap (equ (snequ1 (f x) f')) i' } ≡⟨⟩ equ (snequ1 (f x) f') ∎ ) ⟩ snequ1 (f x) f' ∎