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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Thu, 06 Apr 2017 03:24:44 +0900 |
| parents | adef39d19884 |
| children |
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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 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 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 ; k = k ; ek=h = λ {d} {h} {eq} → ek=h {d} {h} {eq} ; uniqueness = uniqueness } where fe=ge : Sets [ Sets [ f o (λ e → equ e ) ] ≈ Sets [ g o (λ e → equ e ) ] ] fe=ge = extensionality Sets (fe=ge0 ) k : {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 : {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 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 elm-cong : (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' ) 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 hom← : {i j : Obj C } → ( f : 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 ) → ΓObj s Γ i → ΓObj s Γ j ΓMap s Γ {i} {j} f = FMap Γ ( hom← s f ) record snat { c₂ } { I OC : Set c₂ } ( sobj : OC → Set c₂ ) ( smap : { i j : OC } → (f : I )→ sobj i → sobj j ) : Set c₂ where field snmap : ( i : OC ) → sobj i sncommute : { i j : OC } → ( f : I ) → smap f ( snmap i ) ≡ snmap j open snat snmeq : { c₂ : Level } { I OC : Set c₂ } { SO : OC → Set c₂ } { SM : { i j : OC } → (f : I )→ SO i → SO j } ( s t : snat SO SM ) → ( i : OC ) → { snmapsi snmapti : SO i } → snmapsi ≡ snmapti → SO i snmeq s t i {pi} {.pi} refl = pi snmc : { c₂ : Level } { I OC : Set c₂ } { SO : OC → Set c₂ } { SM : { i j : OC } → (f : I )→ SO i → SO j } ( s t : snat SO SM ) → { i j : OC } → { f : I } → { snmapsi snmapti : SO i } → (eqi : snmapsi ≡ snmapti ) → { snmapsj snmaptj : SO j } → (eqj : snmapsj ≡ snmaptj ) → ( SM f ( snmapsi ) ≡ snmapsj ) → ( SM f ( snmapti ) ≡ snmaptj ) → SM f (snmeq s t i (eqi)) ≡ snmeq s t j (eqj) snmc s t refl refl refl refl = refl snat1 : { c₂ : Level } { I OC : Set c₂ } ( SO : OC → Set c₂ ) ( SM : { i j : OC } → (f : I )→ SO i → SO j ) → ( s t : snat SO SM ) → ( eq1 : ( i : OC ) → snmap s i ≡ snmap t i ) → ( eq2 : ( i j : OC ) ( f : I ) → SM {i} {j} f ( snmap s i ) ≡ snmap s j ) → ( eq3 : ( i j : OC ) ( f : I ) → SM {i} {j} f ( snmap t i ) ≡ snmap t j ) → snat SO SM snat1 SO SM s t eq1 eq2 eq3 = record { snmap = λ i → snmeq s t i (eq1 i ) ; sncommute = λ {i} {j} f → snmc s t (eq1 i) (eq1 j) (eq2 i j f ) (eq3 i j f ) } ≡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 subst2 : { c c' : Level } { A B : Set c } { C : Set c' } { a a' : A } { b b' : B } ( f : A → C ) ( g : B → C ) → f a ≡ g b → a ≡ a' → b ≡ b' → f a' ≡ g b' subst2 {_} {_} {A} {B} {C} { a} {.a} {b} {.b} f g f=g refl refl = f=g snmeqeqs : { c₂ : Level } { I OC : Set c₂ } ( SO : OC → Set c₂ ) ( SM : { i j : OC } → (f : I )→ SO i → SO j ) ( s t : snat SO SM ) → ( i : OC ) → ( eq1 : ( i : OC ) → snmap s i ≡ snmap t i ) → snmap s i ≡ snmeq s t i (eq1 i ) snmeqeqs SO SM s t i eq1 = lemma (eq1 i) refl where lemma : { snmapsi snmapti sm : SO i } → ( eq1 : snmapsi ≡ snmapti ) → ( snmapsi ≡ sm ) → sm ≡ snmeq s t i eq1 lemma refl refl = refl snmeqeqt : { c₂ : Level } { I OC : Set c₂ } ( SO : OC → Set c₂ ) ( SM : { i j : OC } → (f : I )→ SO i → SO j ) ( s t : snat SO SM ) → ( i : OC ) → ( eq1 : ( i : OC ) → snmap s i ≡ snmap t i ) → snmap t i ≡ snmeq s t i (eq1 i ) snmeqeqt SO SM s t i eq1 = lemma (eq1 i) refl where lemma : { snmapsi snmapti sm : SO i } → ( eq1 : snmapsi ≡ snmapti ) → ( snmapti ≡ sm ) → sm ≡ snmeq s t i eq1 lemma refl refl = refl scomm2 : { c₂ : Level } { I OC : Set c₂ } ( SO : OC → Set c₂ ) ( SM : { i j : OC } → (f : I )→ SO i → SO j ) ( s t : snat SO SM ) → ( eq1 : ( i : OC ) → snmap s i ≡ snmap t i ) → ( i j : OC ) → ( f : I ) → SM f ( snmap s i ) ≡ snmap s j → {x : ( i : OC ) → SO i } → (x ≡ λ i → snmeq s t i (eq1 i ) ) → SM f (x i) ≡ x j scomm2 SO SM s t eq1 i j f eq2 refl = lemma eq2 (snmeqeqs SO SM s t i eq1) (snmeqeqs SO SM s t j eq1) where lemma : { si sni : SO i} { sj snj : SO j } → ( SM f si ≡ sj ) → (si ≡ sni ) → (sj ≡ snj ) → ( SM f sni ≡ snj ) lemma eq1 eq2 eq3 = subst2 (λ x → SM f x) (λ y → y ) eq1 eq2 eq3 tcomm2 : { c₂ : Level } { I OC : Set c₂ } ( SO : OC → Set c₂ ) ( SM : { i j : OC } → (f : I )→ SO i → SO j ) ( s t : snat SO SM ) → ( eq1 : ( i : OC ) → snmap s i ≡ snmap t i ) → ( i j : OC ) → ( f : I ) → SM f ( snmap t i ) ≡ snmap t j → SM f (snmeq s t i (eq1 i)) ≡ snmeq s t j (eq1 j) tcomm2 SO SM s t eq1 i j f eq2 = lemma eq2 (snmeqeqt SO SM s t i eq1) (snmeqeqt SO SM s t j eq1) where lemma : { si sni : SO i} { sj snj : SO j } → ( SM f si ≡ sj ) → (si ≡ sni ) → (sj ≡ snj ) → ( SM f sni ≡ snj ) lemma eq1 eq2 eq3 = subst2 (λ x → SM f x) (λ y → y ) eq1 eq2 eq3 snat-cong : { c : Level } { I OC : Set c } ( SObj : OC → Set c ) ( SMap : { i j : OC } → (f : I )→ SObj i → SObj j) { s t : snat SObj SMap } → (( i : OC ) → snmap s i ≡ snmap t i ) → ( ( i j : OC ) ( f : I ) → SMap {i} {j} f ( snmap s i ) ≡ snmap s j ) → ( ( i j : OC ) ( f : I ) → SMap {i} {j} f ( snmap t i ) ≡ snmap t j ) → s ≡ t snat-cong {_} {I} {OC} SO SM {s} {t} eq1 eq2 eq3 = begin record { snmap = λ i → snmap s i ; sncommute = λ {i} {j} f → sncommute s f } ≡⟨ ≡cong2 ( λ x y → record { snmap = λ i → x i ; sncommute = λ {i} {j} f → y ? i j f } ) ( extensionality Sets ( λ i → snmeqeqs SO SM s t i eq1 ) ) ( extensionality Sets ( λ x → ( extensionality Sets ( λ i → ( extensionality Sets ( λ j → ( extensionality Sets ( λ f → scomm2 SO SM s t eq1 i j f (eq2 i j f ) x )))))))) ⟩ record { snmap = λ i → snmeq s t i (eq1 i ) ; sncommute = λ {i} {j} f → snmc s t (eq1 i) (eq1 j) (eq2 i j f ) (eq3 i j f ) } ≡⟨ {!!} ⟩ record { snmap = λ i → snmap t i ; sncommute = λ {i} {j} f → sncommute t f } ∎ where open import Relation.Binary.PropositionalEquality open ≡-Reasoning 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 (snat (ΓObj s Γ) (ΓMap s Γ) ) ) Γ Cone C I s Γ = record { TMap = λ i → λ sn → snmap sn i ; isNTrans = record { commute = comm1 } } where comm1 : {a b : Obj C} {f : Hom C a b} → Sets [ Sets [ FMap Γ f o (λ sn → snmap sn a) ] ≈ Sets [ (λ sn → (snmap sn b)) o FMap (K Sets C (snat (ΓObj s Γ) (ΓMap s Γ))) f ] ] comm1 {a} {b} {f} = extensionality Sets ( λ sn → begin FMap Γ f (snmap sn a ) ≡⟨ ≡cong ( λ f → ( FMap Γ f (snmap sn a ))) (sym ( hom-iso s )) ⟩ FMap Γ ( hom← s ( hom→ s f)) (snmap sn a ) ≡⟨⟩ ΓMap s Γ (hom→ s f) (snmap sn a ) ≡⟨ sncommute sn (hom→ s f) ⟩ snmap sn b ∎ ) 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 = snat (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 = snat (ΓObj s Γ) (ΓMap s Γ) comm2 : { a : Obj Sets } {x : a } {i j : Obj C} (t : NTrans C Sets (K Sets C a) Γ) (f : 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 ) ) limit1 : (a : Obj Sets) → NTrans C Sets (K Sets C a) Γ → Hom Sets a (snat (ΓObj s Γ) (ΓMap s Γ)) limit1 a t = λ x → record { snmap = λ i → ( TMap t i ) x ; sncommute = λ f → comm2 t f } 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 -- ≡⟨⟩ -- snmap ( record { snmap = λ i → ( TMap t i ) x ; sncommute = λ {i j} f → comm2 {a} {x} {i} {j} t f } ) i ≡⟨⟩ 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 (snat (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 { snmap = λ i → ( TMap t i ) x ; sncommute = λ f → comm2 t f } ≡⟨ snat-cong (ΓObj s Γ) (ΓMap s Γ) (eq1 x) (eq2 x ) (eq3 x ) ⟩ record { snmap = λ i → snmap (f x ) i ; sncommute = sncommute (f x ) } ≡⟨⟩ f x ∎ ) where open import Relation.Binary.PropositionalEquality open ≡-Reasoning eq1 : (x : a ) (i : Obj C) → TMap t i x ≡ snmap (f x) i eq1 x i = sym ( ≡cong ( λ f → f x ) cif=t ) eq2 : (x : a ) (i j : Obj C) (f : 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) → ΓMap s Γ k (snmap (f x) i) ≡ snmap (f x) j eq3 x i j k = sncommute (f x ) k