Mercurial > hg > Members > kono > Proof > category
changeset 600:3e2ef72d8d2f
Set Completeness unfinished
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 03 Jun 2017 09:46:59 +0900 |
| parents | d3b669722d77 |
| children | 2e7b5a777984 |
| files | SetsCompleteness.agda |
| diffstat | 1 files changed, 177 insertions(+), 148 deletions(-) [+] |
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--- a/SetsCompleteness.agda Fri Jun 02 16:50:49 2017 +0900 +++ b/SetsCompleteness.agda Sat Jun 03 09:46:59 2017 +0900 @@ -1,4 +1,4 @@ -open import Category -- https://github.com/konn/category-agda +open import Category -- https://github.com/konn/category-agda open import Level open import Category.Sets renaming ( _o_ to _*_ ) @@ -11,28 +11,28 @@ -- 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 +≡cong = Relation.Binary.PropositionalEquality.cong -lemma1 : { c₂ : Level } {a b : Obj (Sets { c₂})} {f g : Hom Sets a b} → +≈-to-≡ : { 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 +≈-to-≡ refl x = refl record Σ {a} (A : Set a) (B : Set a) : Set a where constructor _,_ field proj₁ : A - proj₂ : B + proj₂ : B open Σ public SetsProduct : { c₂ : Level} → CreateProduct ( Sets { c₂} ) -SetsProduct { c₂ } = record { +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 ) + _×_ = λ f g x → record { proj₁ = f x ; proj₂ = g x } -- ( f x , g x ) ; π1fxg=f = refl ; π2fxg=g = refl ; uniqueness = refl @@ -45,52 +45,53 @@ 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 +record sproduct {a} (I : Set a) ( Product : I → Set a ) : Set a where field - pi1 : ( i : I ) → pi0 i + proj : ( i : I ) → Product i + +open sproduct -open iproduct +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) (ai : I → Obj Sets ) +SetsIProduct : { c₂ : Level} → (I : Obj Sets) (fi : I → Obj Sets ) → IProduct ( Sets { c₂} ) I SetsIProduct I fi = record { ai = fi - ; iprod = iproduct I fi - ; pi = λ i prod → pi1 prod i + ; iprod = sproduct I fi + ; pi = λ i prod → proj prod i ; isIProduct = record { - iproduct = iproduct1 + iproduct = iproduct1 I fi ; pif=q = pif=q ; ip-uniqueness = ip-uniqueness - ; ip-cong = ip-cong + ; ip-cong = ip-cong I fi } } 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 : 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 (iproduct I fi)} → Sets [ iproduct1 (λ i → Sets [ (λ prod → pi1 prod i) o h ]) ≈ h ] + 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 - 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' - -- ^ . ---------→ + -- c -------→ a ---------→ b + -- ^ . ---------→ -- | . g -- |k . -- | . h - --y : d + -- d -- cf. https://github.com/danr/Agda-projects/blob/master/Category-Theory/Equalizer.agda @@ -98,40 +99,44 @@ 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 +equ (elem x eq) = x -fe=ge0 : { c₂ : Level} {a b : Obj (Sets {c₂}) } { f g : Hom (Sets {c₂}) a b } → +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 - ; k = k - ; ek=h = λ {d} {h} {eq} → ek=h {d} {h} {eq} +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 - 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 + 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 @@ -151,61 +156,73 @@ 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 + 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 +open Small -ΓObj : { c₁ c₂ ℓ : Level} { C : Category c₁ c₂ ℓ } { I : Set c₁ } ( s : Small C I ) ( Γ : Functor C ( Sets { c₁} )) +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 ) +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 snat { 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 - sncommute : { i j : OC } → ( f : I → I ) → smap f ( snmap i ) ≡ snmap j +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 + slequ : { i j : OC } → ( f : I → I ) → sequ (sproduct OC sobj ) (sobj j) ( λ x → smap f ( proj x i ) ) ( λ x → proj x j ) + slobj : OC → Set c₂ + slobj i = sobj i + slmap : {i j : OC } → (f : I → I ) → sobj i → sobj j + slmap f = smap f + ipp : {i j : OC } → (f : I → I ) → sproduct OC sobj + ipp {i} {j} f = equ ( slequ {i} {j} f ) -open snat - -≡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 +open slim -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 +smap-id : { c₁ c₂ ℓ : Level} ( C : Category c₁ c₂ ℓ ) ( I : Set c₁ ) ( s : Small C I ) ( Γ : Functor C ( Sets { c₁} )) + ( se : slim (ΓObj s Γ) (ΓMap s Γ) ) → (i : Obj C ) → (x : FObj Γ i ) → slmap se (slid C I s i) x ≡ x +smap-id C I s Γ se i x = begin + slmap se (slid C I s i) x + ≡⟨⟩ + slmap se ( hom→ s (id1 C i)) x + ≡⟨⟩ + FMap Γ (hom← s (hom→ s (id1 C i))) x + ≡⟨ ≡cong ( λ ii → FMap Γ ii x ) (hom-iso s) ⟩ + FMap Γ (id1 C i) x + ≡⟨ ≡cong ( λ f → f x ) (IsFunctor.identity ( isFunctor Γ) ) ⟩ + x + ∎ where + open import Relation.Binary.PropositionalEquality + open ≡-Reasoning + -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 : snat SObj SMap } - → (( i : OC ) → snmap s i ≡ snmap t i ) - → ( ( i j : OC ) ( f : I → I ) → SMap {i} {j} f ( snmap s i ) ≡ snmap s j ) - → ( ( i j : OC ) ( f : I → 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 {i} {j} f } - ≡⟨ - ≡cong2 ( λ x y → - record { snmap = λ i → x i ; sncommute = λ {i} {j} f → y x 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 → irr (subst2 {!!} {!!} {!!} {!!} (eq2 i j f )) {!!} - )))))))) - ⟩ - record { snmap = λ i → snmap t i ; sncommute = λ {i} {j} f → sncommute t {i} {j} f } - ∎ where +product-to : { c₂ : Level } { I OC : Set c₂ } { sobj : OC → Set c₂ } + → Hom Sets (sproduct OC sobj) (sproduct OC sobj) +product-to x = record { proj = proj x } + +lemma-equ : { c₁ c₂ ℓ : Level} ( C : Category c₁ c₂ ℓ ) ( I : Set c₁ ) ( s : Small C I ) ( Γ : Functor C ( Sets { c₁} )) + {i j i' j' : Obj C } → { f f' : I → I } + → (se : slim (ΓObj s Γ) (ΓMap s Γ) ) + → proj (ipp se {i} {j} f) i ≡ proj (ipp se {i'} {j'} f' ) i +lemma-equ C I s Γ {i} {j} {i'} {j'} {f} {f'} se = ≡cong ( λ p → proj p i ) ( begin + ipp se {i} {j} f + ≡⟨⟩ + record { proj = λ x → proj (equ (slequ se f)) x } + ≡⟨ ≡cong ( λ p → record { proj = proj p i }) ( ≡cong ( λ QIX → record { proj = QIX } ) ( + extensionality Sets ( λ x → ≡cong ( λ qi → qi x ) refl + ) )) ⟩ + record { proj = λ x → proj (equ (slequ se f')) x } + ≡⟨⟩ + ipp se {i'} {j'} f' + ∎ ) where open import Relation.Binary.PropositionalEquality open ≡-Reasoning @@ -213,75 +230,87 @@ 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 } + +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 C I s Γ = record { + TMap = λ i → λ se → proj ( ipp se {i} {i} (slid C I s i) ) i + ; isNTrans = record { commute = commute1 } } 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 ) + commute1 : {a b : Obj C} {f : Hom C a b} → Sets [ Sets [ FMap Γ f o (λ se → proj ( ipp se (slid C I s a) ) a) ] ≈ + Sets [ (λ se → proj ( ipp se (slid C I s b) ) b) o FMap (K Sets C (slim (ΓObj s Γ) (ΓMap s Γ) )) f ] ] + commute1 {a} {b} {f} = extensionality Sets ( λ se → begin + (Sets [ FMap Γ f o (λ se₁ → proj ( ipp se (slid C I s a) ) a) ]) se ≡⟨⟩ - ΓMap s Γ (hom→ s f) (snmap sn a ) - ≡⟨ sncommute sn (hom→ s f) ⟩ - snmap sn b + FMap Γ f (proj ( ipp se {a} {a} (slid C I s a) ) a) + ≡⟨ ≡cong ( λ g → FMap Γ g (proj ( ipp se {a} {a} (slid C I s a) ) a)) (sym ( hom-iso s ) ) ⟩ + FMap Γ (hom← s ( hom→ s f)) (proj ( ipp se {a} {a} (slid C I s a) ) a) + ≡⟨ ≡cong ( λ g → FMap Γ (hom← s ( hom→ s f)) g ) ( lemma-equ C I s Γ se ) ⟩ + FMap Γ (hom← s ( hom→ s f)) (proj ( ipp se {a} {b} (hom→ s f) ) a) + ≡⟨ fe=ge0 ( slequ se (hom→ s f ) ) ⟩ + proj (ipp se {a} {b} ( hom→ s f )) b + ≡⟨ sym ( lemma-equ C I s Γ se ) ⟩ + proj (ipp se {b} {b} (slid C I s b)) b + ≡⟨⟩ + (Sets [ (λ se₁ → proj (ipp se₁ (slid C I s b)) b) o FMap (K Sets C (slim (ΓObj s Γ) (ΓMap s Γ) )) f ]) se ∎ ) 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} ) ) + + +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 +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 } → t0f=t {a} {t} {i} - ; limit-uniqueness = λ {a t i } → limit-uniqueness {a} {t} {i} + limit = limit1 + ; t0f=t = λ {a t i } → refl + ; limit-uniqueness = λ {a} {t} {f} → uniquness1 {a} {t} {f} } } 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 → 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 + limit2 : (a : Obj Sets) → ( t : NTrans C Sets (K Sets C a) Γ ) → {i j : Obj C } → ( f : I → I ) + → ( x : a ) → ΓMap s Γ f (TMap t i x) ≡ TMap t j x + limit2 a t f x = ≡cong ( λ g → g x ) ( IsNTrans.commute ( isNTrans t ) ) + limit1 : (a : Obj Sets) → ( t : NTrans C Sets (K Sets C a) Γ ) → Hom Sets a (slim (ΓObj s Γ) (ΓMap s Γ) ) + limit1 a t x = record { + slequ = λ {i} {j} f → elem ( record { proj = λ i → TMap t i x } ) ( limit2 a t f x ) + } + uniquness2 : {a : Obj Sets} {t : NTrans C Sets (K Sets C a) Γ} {f : Hom Sets a (slim (ΓObj s Γ) (ΓMap s Γ) )} + → ( i j : Obj C ) → + ({i : Obj C} → Sets [ Sets [ TMap (Cone C I s Γ) i o f ] ≈ TMap t i ]) → (f' : I → I ) → (x : a ) + → record { proj = λ i₁ → TMap t i₁ x } ≡ equ (slequ (f x) f') + uniquness2 {a} {t} {f} i j cif=t f' x = begin + record { proj = λ i → TMap t i x } + ≡⟨ ≡cong ( λ g → record { proj = λ i → g i } ) ( extensionality Sets ( λ i → sym ( ≡cong ( λ e → e x ) cif=t ) ) ) ⟩ + record { proj = λ i → (Sets [ TMap (Cone C I s Γ) i o f ]) x } + ≡⟨⟩ + record { proj = λ i → proj (ipp (f x) {i} {i} (slid C I s i) ) i } + ≡⟨ ≡cong ( λ g → record { proj = λ i' → g i' } ) ( extensionality Sets ( λ i'' → lemma-equ C I s Γ (f x))) ⟩ + record { proj = λ i → proj (ipp (f x) f') i } + ∎ 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 + uniquness1 : {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 ] + uniquness1 {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 ) } - ≡⟨⟩ + ≡⟨⟩ + record { slequ = λ {i} {j} f' → elem ( record { proj = λ i → TMap t i x } ) ( limit2 a t f' x ) } + ≡⟨ ≡cong ( λ e → record { slequ = λ {i} {j} f' → e i j f' x } ) ( + extensionality Sets ( λ i → + extensionality Sets ( λ j → + extensionality Sets ( λ f' → + extensionality Sets ( λ x → + elm-cong ( elem ( record { proj = λ i → TMap t i x } ) ( limit2 a t f' x )) (slequ (f x) f' ) (uniquness2 {a} {t} {f} i j cif=t f' x ) ) + ))) + ) ⟩ + record { slequ = λ {i} {j} f' → slequ (f x ) f' } + ≡⟨⟩ f x - ∎ ) where + ∎ ) 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 → 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 (snmap (f x) i) ≡ snmap (f x) j - eq3 x i j k = sncommute (f x ) k -