Notes from Andrea and some stuff about products
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@ -10,36 +10,57 @@ open import Data.Product renaming (proj₁ to fst ; proj₂ to snd)
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open import Cat.Category
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open import Cat.Category
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open import Cat.Functor
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open import Cat.Functor
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-- Use co-patterns - they help with showing more understandable types in goals.
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lift-eq : ∀ {ℓ} {A B : Set ℓ} {a a' : A} {b b' : B} → a ≡ a' → b ≡ b' → (a , b) ≡ (a' , b')
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fst (lift-eq a b i) = a i
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snd (lift-eq a b i) = b i
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--lift-eq a b = λ i → a i , b i
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open Functor
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open Category
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module _ {ℓ ℓ' : Level} {A B : Category {ℓ} {ℓ'}} where
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lift-eq-functors : {f g : Functor A B}
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→ (eq* : Functor.func* f ≡ Functor.func* g)
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→ (eq→ : PathP (λ i → ∀ {x y} → Arrow A x y → Arrow B (eq* i x) (eq* i y))
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(func→ f) (func→ g))
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-- → (eq→ : Functor.func→ f ≡ {!!}) -- Functor.func→ g)
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-- Use PathP
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-- directly to show heterogeneous equalities by using previous
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-- equalities (i.e. continuous paths) to create new continuous paths.
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→ (eqI : PathP (λ i → ∀ {c : A .Object} → eq→ i (A .𝟙 {c}) ≡ B .𝟙 {eq* i c})
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(ident f) (ident g))
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→ (eqD : PathP (λ i → { c c' c'' : A .Object} {a : A .Arrow c c'} {a' : A .Arrow c' c''}
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→ eq→ i (A ._⊕_ a' a) ≡ B ._⊕_ (eq→ i a') (eq→ i a))
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(distrib f) (distrib g))
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→ f ≡ g
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lift-eq-functors eq* eq→ eqI eqD i = record { func* = eq* i ; func→ = eq→ i ; ident = eqI i ; distrib = eqD i }
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-- The category of categories
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-- The category of categories
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module _ {ℓ ℓ' : Level} where
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module _ {ℓ ℓ' : Level} where
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private
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private
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_⊛_ = functor-comp
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_⊛_ = functor-comp
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module _ {A B C D : Category {ℓ} {ℓ'}} {f : Functor A B} {g : Functor B C} {h : Functor C D} where
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module _ {A B C D : Category {ℓ} {ℓ'}} {f : Functor A B} {g : Functor B C} {h : Functor C D} where
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assc : h ⊛ (g ⊛ f) ≡ (h ⊛ g) ⊛ f
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postulate assc : h ⊛ (g ⊛ f) ≡ (h ⊛ g) ⊛ f
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assc = {!!}
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-- assc = lift-eq-functors refl refl {!refl!} λ i j → {!!}
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module _ {A B : Category {ℓ} {ℓ'}} where
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lift-eq : (f g : Functor A B)
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→ (eq* : Functor.func* f ≡ Functor.func* g)
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-- TODO: Must transport here using the equality from above.
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-- Reason:
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-- func→ : Arrow A dom cod → Arrow B (func* dom) (func* cod)
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-- func→₁ : Arrow A dom cod → Arrow B (func*₁ dom) (func*₁ cod)
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-- In other words, func→ and func→₁ does not have the same type.
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-- → Functor.func→ f ≡ Functor.func→ g
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-- → Functor.ident f ≡ Functor.ident g
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-- → Functor.distrib f ≡ Functor.distrib g
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→ f ≡ g
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lift-eq f g eq* x = {!!}
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module _ {A B : Category {ℓ} {ℓ'}} {f : Functor A B} where
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module _ {A B : Category {ℓ} {ℓ'}} {f : Functor A B} where
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idHere = identity {ℓ} {ℓ'} {A}
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lem : (func* f) ∘ (func* (identity {C = A})) ≡ func* f
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lem : (Functor.func* f) ∘ (Functor.func* idHere) ≡ Functor.func* f
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lem = refl
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lem = refl
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ident-r : f ⊛ identity ≡ f
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-- lemmm : func→ {C = A} {D = B} (f ⊛ identity) ≡ func→ f
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ident-r = lift-eq (f ⊛ identity) f refl
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lemmm : PathP
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ident-l : identity ⊛ f ≡ f
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(λ i →
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ident-l = {!!}
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{x y : Object A} → Arrow A x y → Arrow B (func* f x) (func* f y))
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(func→ (f ⊛ identity)) (func→ f)
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lemmm = refl
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postulate lemz : PathP (λ i → {c : A .Object} → PathP (λ _ → Arrow B (func* f c) (func* f c)) (func→ f (A .𝟙)) (B .𝟙))
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(ident (f ⊛ identity)) (ident f)
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-- lemz = {!!}
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postulate ident-r : f ⊛ identity ≡ f
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-- ident-r = lift-eq-functors lem lemmm {!lemz!} {!!}
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postulate ident-l : identity ⊛ f ≡ f
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-- ident-l = lift-eq-functors lem lemmm {!refl!} {!!}
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CatCat : Category {lsuc (ℓ ⊔ ℓ')} {ℓ ⊔ ℓ'}
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CatCat : Category {lsuc (ℓ ⊔ ℓ')} {ℓ ⊔ ℓ'}
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CatCat =
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CatCat =
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@ -48,6 +69,54 @@ module _ {ℓ ℓ' : Level} where
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; Arrow = Functor
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; Arrow = Functor
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; 𝟙 = identity
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; 𝟙 = identity
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; _⊕_ = functor-comp
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; _⊕_ = functor-comp
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; assoc = {!!}
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-- What gives here? Why can I not name the variables directly?
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; assoc = λ {_ _ _ _ f g h} → assc {f = f} {g = g} {h = h}
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; ident = ident-r , ident-l
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; ident = ident-r , ident-l
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}
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}
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module _ {ℓ : Level} (C D : Category {ℓ} {ℓ}) where
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private
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proj₁ : Arrow CatCat (catProduct C D) C
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proj₁ = record { func* = fst ; func→ = fst ; ident = refl ; distrib = refl }
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proj₂ : Arrow CatCat (catProduct C D) D
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proj₂ = record { func* = snd ; func→ = snd ; ident = refl ; distrib = refl }
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module _ {X : Object (CatCat {ℓ} {ℓ})} (x₁ : Arrow CatCat X C) (x₂ : Arrow CatCat X D) where
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open Functor
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-- ident' : {c : Object X} → ((func→ x₁) {dom = c} (𝟙 X) , (func→ x₂) {dom = c} (𝟙 X)) ≡ 𝟙 (catProduct C D)
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-- ident' {c = c} = lift-eq (ident x₁) (ident x₂)
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x : Functor X (catProduct C D)
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x = record
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{ func* = λ x → (func* x₁) x , (func* x₂) x
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; func→ = λ x → func→ x₁ x , func→ x₂ x
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; ident = lift-eq (ident x₁) (ident x₂)
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; distrib = lift-eq (distrib x₁) (distrib x₂)
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}
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-- Need to "lift equality of functors"
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-- If I want to do this like I do it for pairs it's gonna be a pain.
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isUniqL : (CatCat ⊕ proj₁) x ≡ x₁
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isUniqL = lift-eq-functors refl refl {!!} {!!}
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isUniqR : (CatCat ⊕ proj₂) x ≡ x₂
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isUniqR = lift-eq-functors refl refl {!!} {!!}
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isUniq : (CatCat ⊕ proj₁) x ≡ x₁ × (CatCat ⊕ proj₂) x ≡ x₂
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isUniq = isUniqL , isUniqR
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uniq : ∃![ x ] ((CatCat ⊕ proj₁) x ≡ x₁ × (CatCat ⊕ proj₂) x ≡ x₂)
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uniq = x , isUniq
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instance
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isProduct : IsProduct CatCat proj₁ proj₂
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isProduct = uniq
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product : Product {ℂ = CatCat} C D
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product = record
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{ obj = catProduct C D
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; proj₁ = proj₁
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; proj₂ = proj₂
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}
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@ -25,9 +25,11 @@ Sets {ℓ} = record
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; ident = funExt (λ x → refl) , funExt (λ x → refl)
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; ident = funExt (λ x → refl) , funExt (λ x → refl)
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}
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}
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-- Covariant Presheaf
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Representable : {ℓ ℓ' : Level} → (ℂ : Category {ℓ} {ℓ'}) → Set (ℓ ⊔ lsuc ℓ')
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Representable : {ℓ ℓ' : Level} → (ℂ : Category {ℓ} {ℓ'}) → Set (ℓ ⊔ lsuc ℓ')
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Representable {ℓ' = ℓ'} ℂ = Functor ℂ (Sets {ℓ'})
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Representable {ℓ' = ℓ'} ℂ = Functor ℂ (Sets {ℓ'})
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-- The "co-yoneda" embedding.
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representable : {ℓ ℓ' : Level} → {ℂ : Category {ℓ} {ℓ'}} → Category.Object ℂ → Representable ℂ
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representable : {ℓ ℓ' : Level} → {ℂ : Category {ℓ} {ℓ'}} → Category.Object ℂ → Representable ℂ
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representable {ℂ = ℂ} A = record
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representable {ℂ = ℂ} A = record
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{ func* = λ B → ℂ.Arrow A B
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{ func* = λ B → ℂ.Arrow A B
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@ -38,9 +40,11 @@ representable {ℂ = ℂ} A = record
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where
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where
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open module ℂ = Category ℂ
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open module ℂ = Category ℂ
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-- Contravariant Presheaf
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Presheaf : {ℓ ℓ' : Level} → (ℂ : Category {ℓ} {ℓ'}) → Set (ℓ ⊔ lsuc ℓ')
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Presheaf : {ℓ ℓ' : Level} → (ℂ : Category {ℓ} {ℓ'}) → Set (ℓ ⊔ lsuc ℓ')
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Presheaf {ℓ' = ℓ'} ℂ = Functor (Opposite ℂ) (Sets {ℓ'})
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Presheaf {ℓ' = ℓ'} ℂ = Functor (Opposite ℂ) (Sets {ℓ'})
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-- Alternate name: `yoneda`
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presheaf : {ℓ ℓ' : Level} → {ℂ : Category {ℓ} {ℓ'}} → Category.Object (Opposite ℂ) → Presheaf ℂ
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presheaf : {ℓ ℓ' : Level} → {ℂ : Category {ℓ} {ℓ'}} → Category.Object (Opposite ℂ) → Presheaf ℂ
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presheaf {ℂ = ℂ} B = record
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presheaf {ℂ = ℂ} B = record
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{ func* = λ A → ℂ.Arrow A B
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{ func* = λ A → ℂ.Arrow A B
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@ -4,15 +4,29 @@ module Cat.Category where
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open import Agda.Primitive
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open import Agda.Primitive
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open import Data.Unit.Base
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open import Data.Unit.Base
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open import Data.Product renaming (proj₁ to fst ; proj₂ to snd)
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open import Data.Product renaming
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( proj₁ to fst
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; proj₂ to snd
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; ∃! to ∃!≈
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)
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open import Data.Empty
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open import Data.Empty
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open import Function
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open import Function
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open import Cubical
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open import Cubical
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∃! : ∀ {a b} {A : Set a}
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→ (A → Set b) → Set (a ⊔ b)
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∃! = ∃!≈ _≡_
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∃!-syntax : ∀ {a b} {A : Set a} → (A → Set b) → Set (a ⊔ b)
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∃!-syntax = ∃
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syntax ∃!-syntax (λ x → B) = ∃![ x ] B
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postulate undefined : {ℓ : Level} → {A : Set ℓ} → A
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postulate undefined : {ℓ : Level} → {A : Set ℓ} → A
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record Category {ℓ ℓ'} : Set (lsuc (ℓ' ⊔ ℓ)) where
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record Category {ℓ ℓ'} : Set (lsuc (ℓ' ⊔ ℓ)) where
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constructor category
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-- adding no-eta-equality can speed up type-checking.
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no-eta-equality
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field
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field
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Object : Set ℓ
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Object : Set ℓ
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Arrow : Object → Object → Set ℓ'
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Arrow : Object → Object → Set ℓ'
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@ -36,7 +50,7 @@ module _ {ℓ ℓ' : Level} {ℂ : Category {ℓ} {ℓ'}} { A B : Object ℂ } w
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_+_ = ℂ._⊕_
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_+_ = ℂ._⊕_
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Isomorphism : (f : ℂ.Arrow A B) → Set ℓ'
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Isomorphism : (f : ℂ.Arrow A B) → Set ℓ'
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Isomorphism f = Σ[ g ∈ ℂ.Arrow B A ] g + f ≡ ℂ.𝟙 × f + g ≡ ℂ.𝟙
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Isomorphism f = Σ[ g ∈ ℂ.Arrow B A ] g ℂ.⊕ f ≡ ℂ.𝟙 × f + g ≡ ℂ.𝟙
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Epimorphism : {X : ℂ.Object } → (f : ℂ.Arrow A B) → Set ℓ'
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Epimorphism : {X : ℂ.Object } → (f : ℂ.Arrow A B) → Set ℓ'
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Epimorphism {X} f = ( g₀ g₁ : ℂ.Arrow B X ) → g₀ + f ≡ g₁ + f → g₀ ≡ g₁
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Epimorphism {X} f = ( g₀ g₁ : ℂ.Arrow B X ) → g₀ + f ≡ g₁ + f → g₀ ≡ g₁
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@ -92,14 +106,33 @@ _≅_ {ℂ = ℂ} A B = Σ[ f ∈ ℂ.Arrow A B ] (Isomorphism {ℂ = ℂ} f)
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where
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where
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open module ℂ = Category ℂ
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open module ℂ = Category ℂ
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Product : {ℓ : Level} → ( C D : Category {ℓ} {ℓ} ) → Category {ℓ} {ℓ}
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IsProduct : ∀ {ℓ ℓ'} (ℂ : Category {ℓ} {ℓ'}) {A B obj : Object ℂ} (π₁ : Arrow ℂ obj A) (π₂ : Arrow ℂ obj B) → Set (ℓ ⊔ ℓ')
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Product C D =
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IsProduct ℂ {A = A} {B = B} π₁ π₂
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= ∀ {X : ℂ.Object} (x₁ : ℂ.Arrow X A) (x₂ : ℂ.Arrow X B)
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→ ∃![ x ] (π₁ ℂ.⊕ x ≡ x₁ × π₂ ℂ.⊕ x ≡ x₂)
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where
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open module ℂ = Category ℂ
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-- Consider this style for efficiency:
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-- record R : Set where
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-- field
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-- isP : IsProduct {!!} {!!} {!!}
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record Product {ℓ ℓ' : Level} {ℂ : Category {ℓ} {ℓ'}} (A B : Category.Object ℂ) : Set (ℓ ⊔ ℓ') where
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no-eta-equality
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field
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obj : Category.Object ℂ
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proj₁ : Category.Arrow ℂ obj A
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proj₂ : Category.Arrow ℂ obj B
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{{isProduct}} : IsProduct ℂ proj₁ proj₂
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mutual
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catProduct : {ℓ : Level} → ( C D : Category {ℓ} {ℓ} ) → Category {ℓ} {ℓ}
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catProduct C D =
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record
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record
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{ Object = C.Object × D.Object
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{ Object = C.Object × D.Object
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; Arrow = λ { (c , d) (c' , d') →
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-- Why does "outlining with `arrowProduct` not work?
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let carr = C.Arrow c c'
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; Arrow = λ {(c , d) (c' , d') → Arrow C c c' × Arrow D d d'}
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darr = D.Arrow d d'
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in carr × darr}
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; 𝟙 = C.𝟙 , D.𝟙
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; 𝟙 = C.𝟙 , D.𝟙
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; _⊕_ = λ { (bc∈C , bc∈D) (ab∈C , ab∈D) → bc∈C C.⊕ ab∈C , bc∈D D.⊕ ab∈D}
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; _⊕_ = λ { (bc∈C , bc∈D) (ab∈C , ab∈D) → bc∈C C.⊕ ab∈C , bc∈D D.⊕ ab∈D}
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; assoc = eqpair C.assoc D.assoc
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; assoc = eqpair C.assoc D.assoc
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eqpair : {ℓ : Level} → { A : Set ℓ } → { B : Set ℓ } → { a a' : A } → { b b' : B } → a ≡ a' → b ≡ b' → (a , b) ≡ (a' , b')
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eqpair : {ℓ : Level} → { A : Set ℓ } → { B : Set ℓ } → { a a' : A } → { b b' : B } → a ≡ a' → b ≡ b' → (a , b) ≡ (a' , b')
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eqpair {a = a} {b = b} eqa eqb = subst eqa (subst eqb (refl {x = (a , b)}))
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eqpair {a = a} {b = b} eqa eqb = subst eqa (subst eqb (refl {x = (a , b)}))
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-- arrowProduct : ∀ {ℓ} {C D : Category {ℓ} {ℓ}} → (Object C) × (Object D) → (Object C) × (Object D) → Set ℓ
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-- arrowProduct = {!!}
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-- Arrows in the product-category
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arrowProduct : ∀ {ℓ} {C D : Category {ℓ} {ℓ}} (c d : Object (catProduct C D)) → Set ℓ
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arrowProduct {C = C} {D = D} (c , d) (c' , d') = Arrow C c c' × Arrow D d d'
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Opposite : ∀ {ℓ ℓ'} → Category {ℓ} {ℓ'} → Category {ℓ} {ℓ'}
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Opposite : ∀ {ℓ ℓ'} → Category {ℓ} {ℓ'} → Category {ℓ} {ℓ'}
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Opposite ℂ =
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Opposite ℂ =
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record
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record
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@ -128,15 +169,21 @@ Opposite ℂ =
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where
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where
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open module ℂ = Category ℂ
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open module ℂ = Category ℂ
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-- A consequence of no-eta-equality; `Opposite-is-involution` is no longer
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-- definitional - i.e.; you must match on the fields:
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--
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-- Opposite-is-involution : ∀ {ℓ ℓ'} → {C : Category {ℓ} {ℓ'}} → Opposite (Opposite C) ≡ C
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-- Object (Opposite-is-involution {C = C} i) = Object C
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-- Arrow (Opposite-is-involution i) = {!!}
|
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|
-- 𝟙 (Opposite-is-involution i) = {!!}
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-- _⊕_ (Opposite-is-involution i) = {!!}
|
||||||
|
-- assoc (Opposite-is-involution i) = {!!}
|
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|
-- ident (Opposite-is-involution i) = {!!}
|
||||||
|
|
||||||
Hom : {ℓ ℓ' : Level} → (ℂ : Category {ℓ} {ℓ'}) → (A B : Object ℂ) → Set ℓ'
|
Hom : {ℓ ℓ' : Level} → (ℂ : Category {ℓ} {ℓ'}) → (A B : Object ℂ) → Set ℓ'
|
||||||
Hom ℂ A B = Arrow ℂ A B
|
Hom ℂ A B = Arrow ℂ A B
|
||||||
|
|
||||||
module _ {ℓ ℓ' : Level} {ℂ : Category {ℓ} {ℓ'}} where
|
module _ {ℓ ℓ' : Level} {ℂ : Category {ℓ} {ℓ'}} where
|
||||||
private
|
HomFromArrow : (A : ℂ .Object) → {B B' : ℂ .Object} → (g : ℂ .Arrow B B')
|
||||||
Obj = Object ℂ
|
|
||||||
Arr = Arrow ℂ
|
|
||||||
_+_ = _⊕_ ℂ
|
|
||||||
|
|
||||||
HomFromArrow : (A : Obj) → {B B' : Obj} → (g : Arr B B')
|
|
||||||
→ Hom ℂ A B → Hom ℂ A B'
|
→ Hom ℂ A B → Hom ℂ A B'
|
||||||
HomFromArrow _A g = λ f → g + f
|
HomFromArrow _A = _⊕_ ℂ
|
||||||
|
|
|
@ -23,7 +23,7 @@ module _ {A : Set} {a : A} {P : A → Set} where
|
||||||
w x = {!!}
|
w x = {!!}
|
||||||
|
|
||||||
vw-bij : (a : P a) → (w ∘ v) a ≡ a
|
vw-bij : (a : P a) → (w ∘ v) a ≡ a
|
||||||
vw-bij a = ?
|
vw-bij a = {!!}
|
||||||
-- tubij a with (t ∘ u) a
|
-- tubij a with (t ∘ u) a
|
||||||
-- ... | q = {!!}
|
-- ... | q = {!!}
|
||||||
|
|
||||||
|
|
|
@ -10,8 +10,13 @@ open import Data.Empty
|
||||||
|
|
||||||
open import Cat.Category
|
open import Cat.Category
|
||||||
|
|
||||||
|
-- See chapter 1 for a discussion on how presheaf categories are CwF's.
|
||||||
|
|
||||||
|
-- See section 6.8 in Huber's thesis for details on how to implement the
|
||||||
|
-- categorical version of CTT
|
||||||
|
|
||||||
module _ {ℓ ℓ' : Level} (Ns : Set ℓ) where
|
module _ {ℓ ℓ' : Level} (Ns : Set ℓ) where
|
||||||
-- Σ is the "namespace"
|
-- Ns is the "namespace"
|
||||||
ℓo = (lsuc lzero ⊔ ℓ)
|
ℓo = (lsuc lzero ⊔ ℓ)
|
||||||
|
|
||||||
FiniteDecidableSubset : Set ℓ
|
FiniteDecidableSubset : Set ℓ
|
||||||
|
|
Loading…
Reference in a new issue