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src/Cat.agda
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12
src/Cat.agda
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@ -0,0 +1,12 @@
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module Cat where
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import Cat.Categories.Sets
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import Cat.Categories.Cat
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import Cat.Categories.Rel
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import Cat.Category.Pathy
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import Cat.Category.Bij
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import Cat.Category.Free
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import Cat.Category.Properties
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import Cat.Category
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import Cat.Cubical
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import Cat.Functor
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@ -1,55 +1,174 @@
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{-# OPTIONS --cubical #-}
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{-# OPTIONS --cubical --allow-unsolved-metas #-}
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module Category.Categories.Cat where
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module Cat.Categories.Cat where
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open import Agda.Primitive
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open import Cubical
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open import Function
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open import Data.Product renaming (proj₁ to fst ; proj₂ to snd)
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open import Category
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open import Cat.Category
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open import Cat.Functor
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-- Tip from Andrea:
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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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eqpair : ∀ {ℓa ℓb} {A : Set ℓa} {B : Set ℓb} {a a' : A} {b b' : B}
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→ a ≡ a' → b ≡ b' → (a , b) ≡ (a' , b')
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eqpair eqa eqb i = eqa i , eqb 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* : f .func* ≡ g .func*)
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→ (eq→ : PathP (λ i → ∀ {x y} → A .Arrow x y → B .Arrow (eq* i x) (eq* i y))
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(f .func→) (g .func→))
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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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module _ {ℓ ℓ' : Level} where
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private
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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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assc : h ⊛ (g ⊛ f) ≡ (h ⊛ g) ⊛ f
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assc = {!!}
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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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eq* : func* (h ∘f (g ∘f f)) ≡ func* ((h ∘f g) ∘f f)
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eq* = refl
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eq→ : PathP
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(λ i → {x y : A .Object} → A .Arrow x y → D .Arrow (eq* i x) (eq* i y))
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(func→ (h ∘f (g ∘f f))) (func→ ((h ∘f g) ∘f f))
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eq→ = refl
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id-l = (h ∘f (g ∘f f)) .ident -- = func→ (h ∘f (g ∘f f)) (𝟙 A) ≡ 𝟙 D
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id-r = ((h ∘f g) ∘f f) .ident -- = func→ ((h ∘f g) ∘f f) (𝟙 A) ≡ 𝟙 D
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postulate eqI : PathP
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(λ i → ∀ {c : A .Object} → eq→ i (A .𝟙 {c}) ≡ D .𝟙 {eq* i c})
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(ident ((h ∘f (g ∘f f))))
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(ident ((h ∘f g) ∘f f))
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postulate 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) ≡ D ._⊕_ (eq→ i a') (eq→ i a))
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(distrib (h ∘f (g ∘f f))) (distrib ((h ∘f g) ∘f f))
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-- eqD = {!!}
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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
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(functor func* func→ idnt distrib)
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(functor func*₁ func→₁ idnt₁ distrib₁)
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eq-func* = {!!}
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assc : h ∘f (g ∘f f) ≡ (h ∘f g) ∘f f
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assc = lift-eq-functors eq* eq→ eqI eqD
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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 : (Functor.func* f) ∘ (Functor.func* idHere) ≡ Functor.func* f
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module _ {A B : Category ℓ ℓ'} {f : Functor A B} where
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lem : (func* f) ∘ (func* (identity {C = A})) ≡ func* f
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lem = refl
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ident-r : f ⊛ identity ≡ f
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ident-r = lift-eq (f ⊛ identity) f refl
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ident-l : identity ⊛ f ≡ f
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ident-l = {!!}
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-- lemmm : func→ {C = A} {D = B} (f ∘f identity) ≡ func→ f
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lemmm : PathP
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(λ i →
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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 ∘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 ∘f identity)) (ident f)
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-- lemz = {!!}
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postulate ident-r : f ∘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 ≡ 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 =
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Cat : Category (lsuc (ℓ ⊔ ℓ')) (ℓ ⊔ ℓ')
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Cat =
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record
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{ Object = Category {ℓ} {ℓ'}
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{ Object = Category ℓ ℓ'
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; Arrow = Functor
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; 𝟙 = identity
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; _⊕_ = functor-comp
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; assoc = {!!}
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; _⊕_ = _∘f_
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-- What gives here? Why can I not name the variables directly?
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; isCategory = record
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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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}
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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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:Object: = C .Object × D .Object
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:Arrow: : :Object: → :Object: → Set ℓ
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:Arrow: (c , d) (c' , d') = Arrow C c c' × Arrow D d d'
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:𝟙: : {o : :Object:} → :Arrow: o o
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:𝟙: = C .𝟙 , D .𝟙
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_:⊕:_ :
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{a b c : :Object:} →
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:Arrow: b c →
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:Arrow: a b →
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:Arrow: a c
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_:⊕:_ = λ { (bc∈C , bc∈D) (ab∈C , ab∈D) → (C ._⊕_) bc∈C ab∈C , D ._⊕_ bc∈D ab∈D}
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instance
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:isCategory: : IsCategory :Object: :Arrow: :𝟙: _:⊕:_
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:isCategory: = record
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{ assoc = eqpair C.assoc D.assoc
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; ident
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= eqpair (fst C.ident) (fst D.ident)
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, eqpair (snd C.ident) (snd D.ident)
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}
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where
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open module C = IsCategory (C .isCategory)
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open module D = IsCategory (D .isCategory)
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:product: : Category ℓ ℓ
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:product: = record
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{ Object = :Object:
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; Arrow = :Arrow:
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; 𝟙 = :𝟙:
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; _⊕_ = _:⊕:_
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}
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proj₁ : Arrow Cat :product: C
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proj₁ = record { func* = fst ; func→ = fst ; ident = refl ; distrib = refl }
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proj₂ : Arrow Cat :product: D
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proj₂ = record { func* = snd ; func→ = snd ; ident = refl ; distrib = refl }
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module _ {X : Object (Cat {ℓ} {ℓ})} (x₁ : Arrow Cat X C) (x₂ : Arrow Cat 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 :product:
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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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postulate isUniqL : (Cat ⊕ proj₁) x ≡ x₁
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-- isUniqL = lift-eq-functors refl refl {!!} {!!}
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postulate isUniqR : (Cat ⊕ proj₂) x ≡ x₂
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-- isUniqR = lift-eq-functors refl refl {!!} {!!}
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isUniq : (Cat ⊕ proj₁) x ≡ x₁ × (Cat ⊕ proj₂) x ≡ x₂
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isUniq = isUniqL , isUniqR
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uniq : ∃![ x ] ((Cat ⊕ proj₁) x ≡ x₁ × (Cat ⊕ proj₂) x ≡ x₂)
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uniq = x , isUniq
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instance
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isProduct : IsProduct Cat proj₁ proj₂
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isProduct = uniq
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product : Product {ℂ = Cat} C D
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product = record
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{ obj = :product:
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; proj₁ = proj₁
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; proj₂ = proj₂
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}
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@ -154,12 +154,11 @@ module _ {A B C D : Set} {S : Subset (A × B)} {R : Subset (B × C)} {Q : Subset
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≡ (Σ[ b ∈ B ] (a , b) ∈ S × (Σ[ c ∈ C ] (b , c) ∈ R × (c , d) ∈ Q))
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is-assoc = equivToPath equi
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Rel : Category
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Rel : Category (lsuc lzero) (lsuc lzero)
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Rel = record
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{ Object = Set
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; Arrow = λ S R → Subset (S × R)
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; 𝟙 = λ {S} → Diag S
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; _⊕_ = λ {A B C} S R → λ {( a , c ) → Σ[ b ∈ B ] ( (a , b) ∈ R × (b , c) ∈ S )}
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; assoc = funExt is-assoc
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; ident = funExt ident-l , funExt ident-r
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; isCategory = record { assoc = funExt is-assoc ; ident = funExt ident-l , funExt ident-r }
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}
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@ -9,44 +9,45 @@ 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.Functor
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open Category
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-- Sets are built-in to Agda. The set of all small sets is called Set.
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Fun : {ℓ : Level} → ( T U : Set ℓ ) → Set ℓ
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Fun T U = T → U
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Sets : {ℓ : Level} → Category {lsuc ℓ} {ℓ}
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Sets : {ℓ : Level} → Category (lsuc ℓ) ℓ
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Sets {ℓ} = record
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{ Object = Set ℓ
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; Arrow = λ T U → Fun {ℓ} T U
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; 𝟙 = λ x → x
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; _⊕_ = λ g f x → g ( f x )
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; assoc = refl
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; ident = funExt (λ x → refl) , funExt (λ x → refl)
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; Arrow = λ T U → T → U
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; 𝟙 = id
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; _⊕_ = _∘′_
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; isCategory = record { assoc = refl ; ident = funExt (λ _ → refl) , funExt (λ _ → refl) }
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}
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where
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open import Function
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Representable : {ℓ ℓ' : Level} → (ℂ : Category {ℓ} {ℓ'}) → Set (ℓ ⊔ lsuc ℓ')
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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'} ℂ = Functor ℂ (Sets {ℓ'})
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representable : {ℓ ℓ' : Level} → {ℂ : Category {ℓ} {ℓ'}} → Category.Object ℂ → Representable ℂ
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-- The "co-yoneda" embedding.
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representable : ∀ {ℓ ℓ'} {ℂ : Category ℓ ℓ'} → Category.Object ℂ → Representable ℂ
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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 f g → f ℂ.⊕ g
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; ident = funExt λ _ → snd ℂ.ident
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; distrib = funExt λ x → sym ℂ.assoc
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{ func* = λ B → ℂ .Arrow A B
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; func→ = ℂ ._⊕_
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; ident = funExt λ _ → snd ident
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; distrib = funExt λ x → sym assoc
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}
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where
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open module ℂ = Category ℂ
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open IsCategory (ℂ .isCategory)
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Presheaf : {ℓ ℓ' : Level} → (ℂ : Category {ℓ} {ℓ'}) → Set (ℓ ⊔ lsuc ℓ')
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-- Contravariant Presheaf
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Presheaf : ∀ {ℓ ℓ'} (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' : Level} → {ℂ : Category {ℓ} {ℓ'}} → Category.Object (Opposite ℂ) → Presheaf ℂ
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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 {ℂ = ℂ} B = record
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{ func* = λ A → ℂ.Arrow A B
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; func→ = λ f g → g ℂ.⊕ f
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; ident = funExt λ x → fst ℂ.ident
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; distrib = funExt λ x → ℂ.assoc
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{ func* = λ A → ℂ .Arrow A B
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; func→ = λ f g → ℂ ._⊕_ g f
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; ident = funExt λ x → fst ident
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; distrib = funExt λ x → assoc
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}
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where
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open module ℂ = Category ℂ
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open IsCategory (ℂ .isCategory)
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|
|
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|
|
@ -4,139 +4,119 @@ module Cat.Category where
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|
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open import Agda.Primitive
|
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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 Function
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open import Cubical
|
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|
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postulate undefined : {ℓ : Level} → {A : Set ℓ} → A
|
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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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|
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record Category {ℓ ℓ'} : Set (lsuc (ℓ' ⊔ ℓ)) where
|
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constructor category
|
||||
∃!-syntax : ∀ {a b} {A : Set a} → (A → Set b) → Set (a ⊔ b)
|
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∃!-syntax = ∃
|
||||
|
||||
syntax ∃!-syntax (λ x → B) = ∃![ x ] B
|
||||
|
||||
record IsCategory {ℓ ℓ' : Level}
|
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(Object : Set ℓ)
|
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(Arrow : Object → Object → Set ℓ')
|
||||
(𝟙 : {o : Object} → Arrow o o)
|
||||
(_⊕_ : { a b c : Object } → Arrow b c → Arrow a b → Arrow a c)
|
||||
: Set (lsuc (ℓ' ⊔ ℓ)) where
|
||||
field
|
||||
assoc : {A B C D : Object} { f : Arrow A B } { g : Arrow B C } { h : Arrow C D }
|
||||
→ h ⊕ (g ⊕ f) ≡ (h ⊕ g) ⊕ f
|
||||
ident : {A B : Object} {f : Arrow A B}
|
||||
→ f ⊕ 𝟙 ≡ f × 𝟙 ⊕ f ≡ f
|
||||
|
||||
-- open IsCategory public
|
||||
|
||||
record Category (ℓ ℓ' : Level) : Set (lsuc (ℓ' ⊔ ℓ)) where
|
||||
-- adding no-eta-equality can speed up type-checking.
|
||||
no-eta-equality
|
||||
field
|
||||
Object : Set ℓ
|
||||
Arrow : Object → Object → Set ℓ'
|
||||
𝟙 : {o : Object} → Arrow o o
|
||||
_⊕_ : { a b c : Object } → Arrow b c → Arrow a b → Arrow a c
|
||||
assoc : { A B C D : Object } { f : Arrow A B } { g : Arrow B C } { h : Arrow C D }
|
||||
→ h ⊕ (g ⊕ f) ≡ (h ⊕ g) ⊕ f
|
||||
ident : { A B : Object } { f : Arrow A B }
|
||||
→ f ⊕ 𝟙 ≡ f × 𝟙 ⊕ f ≡ f
|
||||
{{isCategory}} : IsCategory Object Arrow 𝟙 _⊕_
|
||||
infixl 45 _⊕_
|
||||
domain : { a b : Object } → Arrow a b → Object
|
||||
domain {a = a} _ = a
|
||||
codomain : { a b : Object } → Arrow a b → Object
|
||||
codomain {b = b} _ = b
|
||||
|
||||
open Category public
|
||||
open Category
|
||||
|
||||
module _ {ℓ ℓ' : Level} {ℂ : Category {ℓ} {ℓ'}} { A B : Object ℂ } where
|
||||
private
|
||||
open module ℂ = Category ℂ
|
||||
_+_ = ℂ._⊕_
|
||||
module _ {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} where
|
||||
module _ { A B : ℂ .Object } where
|
||||
Isomorphism : (f : ℂ .Arrow A B) → Set ℓ'
|
||||
Isomorphism f = Σ[ g ∈ ℂ .Arrow B A ] ℂ ._⊕_ g f ≡ ℂ .𝟙 × ℂ ._⊕_ f g ≡ ℂ .𝟙
|
||||
|
||||
Isomorphism : (f : ℂ.Arrow A B) → Set ℓ'
|
||||
Isomorphism f = Σ[ g ∈ ℂ.Arrow B A ] g + f ≡ ℂ.𝟙 × f + g ≡ ℂ.𝟙
|
||||
Epimorphism : {X : ℂ .Object } → (f : ℂ .Arrow A B) → Set ℓ'
|
||||
Epimorphism {X} f = ( g₀ g₁ : ℂ .Arrow B X ) → ℂ ._⊕_ g₀ f ≡ ℂ ._⊕_ g₁ f → g₀ ≡ g₁
|
||||
|
||||
Epimorphism : {X : ℂ.Object } → (f : ℂ.Arrow A B) → Set ℓ'
|
||||
Epimorphism {X} f = ( g₀ g₁ : ℂ.Arrow B X ) → g₀ + f ≡ g₁ + f → g₀ ≡ g₁
|
||||
Monomorphism : {X : ℂ .Object} → (f : ℂ .Arrow A B) → Set ℓ'
|
||||
Monomorphism {X} f = ( g₀ g₁ : ℂ .Arrow X A ) → ℂ ._⊕_ f g₀ ≡ ℂ ._⊕_ f g₁ → g₀ ≡ g₁
|
||||
|
||||
Monomorphism : {X : ℂ.Object} → (f : ℂ.Arrow A B) → Set ℓ'
|
||||
Monomorphism {X} f = ( g₀ g₁ : ℂ.Arrow X A ) → f + g₀ ≡ f + g₁ → g₀ ≡ g₁
|
||||
-- Isomorphism of objects
|
||||
_≅_ : (A B : Object ℂ) → Set ℓ'
|
||||
_≅_ A B = Σ[ f ∈ ℂ .Arrow A B ] (Isomorphism f)
|
||||
|
||||
iso-is-epi : ∀ {X} (f : ℂ.Arrow A B) → Isomorphism f → Epimorphism {X = X} f
|
||||
-- Idea: Pre-compose with f- on both sides of the equality of eq to get
|
||||
-- g₀ + f + f- ≡ g₁ + f + f-
|
||||
-- which by left-inv reduces to the goal.
|
||||
iso-is-epi f (f- , left-inv , right-inv) g₀ g₁ eq =
|
||||
trans (sym (fst ℂ.ident))
|
||||
( trans (cong (_+_ g₀) (sym right-inv))
|
||||
( trans ℂ.assoc
|
||||
( trans (cong (λ x → x + f-) eq)
|
||||
( trans (sym ℂ.assoc)
|
||||
( trans (cong (_+_ g₁) right-inv) (fst ℂ.ident))
|
||||
)
|
||||
)
|
||||
)
|
||||
)
|
||||
module _ {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') {A B obj : Object ℂ} where
|
||||
IsProduct : (π₁ : Arrow ℂ obj A) (π₂ : Arrow ℂ obj B) → Set (ℓ ⊔ ℓ')
|
||||
IsProduct π₁ π₂
|
||||
= ∀ {X : ℂ .Object} (x₁ : ℂ .Arrow X A) (x₂ : ℂ .Arrow X B)
|
||||
→ ∃![ x ] (ℂ ._⊕_ π₁ x ≡ x₁ × ℂ ._⊕_ π₂ x ≡ x₂)
|
||||
|
||||
iso-is-mono : ∀ {X} (f : ℂ.Arrow A B ) → Isomorphism f → Monomorphism {X = X} f
|
||||
-- For the next goal we do something similar: Post-compose with f- and use
|
||||
-- right-inv to get the goal.
|
||||
iso-is-mono f (f- , (left-inv , right-inv)) g₀ g₁ eq =
|
||||
trans (sym (snd ℂ.ident))
|
||||
( trans (cong (λ x → x + g₀) (sym left-inv))
|
||||
( trans (sym ℂ.assoc)
|
||||
( trans (cong (_+_ f-) eq)
|
||||
( trans ℂ.assoc
|
||||
( trans (cong (λ x → x + g₁) left-inv) (snd ℂ.ident)
|
||||
)
|
||||
)
|
||||
)
|
||||
)
|
||||
)
|
||||
-- Tip from Andrea; Consider this style for efficiency:
|
||||
-- record IsProduct {ℓ ℓ' : Level} (ℂ : Category {ℓ} {ℓ'})
|
||||
-- {A B obj : Object ℂ} (π₁ : Arrow ℂ obj A) (π₂ : Arrow ℂ obj B) : Set (ℓ ⊔ ℓ') where
|
||||
-- field
|
||||
-- isProduct : ∀ {X : ℂ .Object} (x₁ : ℂ .Arrow X A) (x₂ : ℂ .Arrow X B)
|
||||
-- → ∃![ x ] (ℂ ._⊕_ π₁ x ≡ x₁ × ℂ. _⊕_ π₂ x ≡ x₂)
|
||||
|
||||
iso-is-epi-mono : ∀ {X} (f : ℂ.Arrow A B ) → Isomorphism f → Epimorphism {X = X} f × Monomorphism {X = X} f
|
||||
iso-is-epi-mono f iso = iso-is-epi f iso , iso-is-mono f iso
|
||||
record Product {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} (A B : ℂ .Object) : Set (ℓ ⊔ ℓ') where
|
||||
no-eta-equality
|
||||
field
|
||||
obj : ℂ .Object
|
||||
proj₁ : ℂ .Arrow obj A
|
||||
proj₂ : ℂ .Arrow obj B
|
||||
{{isProduct}} : IsProduct ℂ proj₁ proj₂
|
||||
|
||||
{-
|
||||
epi-mono-is-not-iso : ∀ {ℓ ℓ'} → ¬ ((ℂ : Category {ℓ} {ℓ'}) {A B X : Object ℂ} (f : Arrow ℂ A B ) → Epimorphism {ℂ = ℂ} {X = X} f → Monomorphism {ℂ = ℂ} {X = X} f → Isomorphism {ℂ = ℂ} f)
|
||||
epi-mono-is-not-iso f =
|
||||
let k = f {!!} {!!} {!!} {!!}
|
||||
in {!!}
|
||||
-}
|
||||
|
||||
-- Isomorphism of objects
|
||||
_≅_ : { ℓ ℓ' : Level } → { ℂ : Category {ℓ} {ℓ'} } → ( A B : Object ℂ ) → Set ℓ'
|
||||
_≅_ {ℂ = ℂ} A B = Σ[ f ∈ ℂ.Arrow A B ] (Isomorphism {ℂ = ℂ} f)
|
||||
where
|
||||
open module ℂ = Category ℂ
|
||||
|
||||
Product : {ℓ : Level} → ( C D : Category {ℓ} {ℓ} ) → Category {ℓ} {ℓ}
|
||||
Product C D =
|
||||
module _ {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') where
|
||||
Opposite : Category ℓ ℓ'
|
||||
Opposite =
|
||||
record
|
||||
{ Object = C.Object × D.Object
|
||||
; Arrow = λ { (c , d) (c' , d') →
|
||||
let carr = C.Arrow c c'
|
||||
darr = D.Arrow d d'
|
||||
in carr × darr}
|
||||
; 𝟙 = C.𝟙 , D.𝟙
|
||||
; _⊕_ = λ { (bc∈C , bc∈D) (ab∈C , ab∈D) → bc∈C C.⊕ ab∈C , bc∈D D.⊕ ab∈D}
|
||||
; assoc = eqpair C.assoc D.assoc
|
||||
; ident =
|
||||
let (Cl , Cr) = C.ident
|
||||
(Dl , Dr) = D.ident
|
||||
in eqpair Cl Dl , eqpair Cr Dr
|
||||
{ Object = ℂ .Object
|
||||
; Arrow = flip (ℂ .Arrow)
|
||||
; 𝟙 = ℂ .𝟙
|
||||
; _⊕_ = flip (ℂ ._⊕_)
|
||||
; isCategory = record { assoc = sym assoc ; ident = swap ident }
|
||||
}
|
||||
where
|
||||
open module C = Category C
|
||||
open module D = Category D
|
||||
-- Two pairs are equal if their components are equal.
|
||||
eqpair : {ℓ : Level} → { A : Set ℓ } → { B : Set ℓ } → { a a' : A } → { b b' : B } → a ≡ a' → b ≡ b' → (a , b) ≡ (a' , b')
|
||||
eqpair {a = a} {b = b} eqa eqb = subst eqa (subst eqb (refl {x = (a , b)}))
|
||||
open IsCategory (ℂ .isCategory)
|
||||
|
||||
Opposite : ∀ {ℓ ℓ'} → Category {ℓ} {ℓ'} → Category {ℓ} {ℓ'}
|
||||
Opposite ℂ =
|
||||
record
|
||||
{ Object = ℂ.Object
|
||||
; Arrow = λ A B → ℂ.Arrow B A
|
||||
; 𝟙 = ℂ.𝟙
|
||||
; _⊕_ = λ g f → f ℂ.⊕ g
|
||||
; assoc = sym ℂ.assoc
|
||||
; ident = swap ℂ.ident
|
||||
}
|
||||
where
|
||||
open module ℂ = Category ℂ
|
||||
-- A consequence of no-eta-equality; `Opposite-is-involution` is no longer
|
||||
-- definitional - i.e.; you must match on the fields:
|
||||
--
|
||||
-- Opposite-is-involution : ∀ {ℓ ℓ'} → {C : Category {ℓ} {ℓ'}} → Opposite (Opposite C) ≡ C
|
||||
-- Object (Opposite-is-involution {C = C} i) = Object C
|
||||
-- Arrow (Opposite-is-involution i) = {!!}
|
||||
-- 𝟙 (Opposite-is-involution i) = {!!}
|
||||
-- _⊕_ (Opposite-is-involution i) = {!!}
|
||||
-- assoc (Opposite-is-involution i) = {!!}
|
||||
-- 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
|
||||
|
||||
module _ {ℓ ℓ' : Level} {ℂ : Category {ℓ} {ℓ'}} where
|
||||
private
|
||||
Obj = Object ℂ
|
||||
Arr = Arrow ℂ
|
||||
_+_ = _⊕_ ℂ
|
||||
|
||||
HomFromArrow : (A : Obj) → {B B' : Obj} → (g : Arr B B')
|
||||
module _ {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} where
|
||||
HomFromArrow : (A : ℂ .Object) → {B B' : ℂ .Object} → (g : ℂ .Arrow B B')
|
||||
→ Hom ℂ A B → Hom ℂ A B'
|
||||
HomFromArrow _A g = λ f → g + f
|
||||
HomFromArrow _A = _⊕_ ℂ
|
||||
|
|
|
|||
|
|
@ -1,6 +1,9 @@
|
|||
{-# OPTIONS --cubical #-}
|
||||
{-# OPTIONS --cubical --allow-unsolved-metas #-}
|
||||
|
||||
module Cat.Category.Bij where
|
||||
|
||||
open import Cubical.PathPrelude hiding ( Id )
|
||||
open import Cubical.FromStdLib
|
||||
|
||||
module _ {A : Set} {a : A} {P : A → Set} where
|
||||
Q : A → Set
|
||||
|
|
@ -20,7 +23,7 @@ module _ {A : Set} {a : A} {P : A → Set} where
|
|||
w x = {!!}
|
||||
|
||||
vw-bij : (a : P a) → (w ∘ v) a ≡ a
|
||||
vw-bij a = refl
|
||||
vw-bij a = {!!}
|
||||
-- tubij a with (t ∘ u) a
|
||||
-- ... | q = {!!}
|
||||
|
||||
|
|
|
|||
|
|
@ -1,21 +1,20 @@
|
|||
module Category.Free where
|
||||
module Cat.Category.Free where
|
||||
|
||||
open import Agda.Primitive
|
||||
open import Cubical.PathPrelude hiding (Path)
|
||||
open import Data.Product
|
||||
|
||||
open import Category as C
|
||||
open import Cat.Category as C
|
||||
|
||||
module _ {ℓ ℓ' : Level} (ℂ : Category {ℓ} {ℓ'}) where
|
||||
module _ {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') where
|
||||
private
|
||||
open module ℂ = Category ℂ
|
||||
Obj = ℂ.Object
|
||||
|
||||
Path : ( a b : Obj ) → Set
|
||||
Path a b = undefined
|
||||
|
||||
postulate emptyPath : (o : Obj) → Path o o
|
||||
|
||||
postulate concatenate : {a b c : Obj} → Path b c → Path a b → Path a c
|
||||
postulate
|
||||
Path : ( a b : Obj ) → Set ℓ'
|
||||
emptyPath : (o : Obj) → Path o o
|
||||
concatenate : {a b c : Obj} → Path b c → Path a b → Path a c
|
||||
|
||||
private
|
||||
module _ {A B C D : Obj} {r : Path A B} {q : Path B C} {p : Path C D} where
|
||||
|
|
@ -27,12 +26,11 @@ module _ {ℓ ℓ' : Level} (ℂ : Category {ℓ} {ℓ'}) where
|
|||
ident-r : concatenate {A} {A} {B} p (emptyPath A) ≡ p
|
||||
ident-l : concatenate {A} {B} {B} (emptyPath B) p ≡ p
|
||||
|
||||
Free : Category
|
||||
Free : Category ℓ ℓ'
|
||||
Free = record
|
||||
{ Object = Obj
|
||||
; Arrow = Path
|
||||
; 𝟙 = λ {o} → emptyPath o
|
||||
; _⊕_ = λ {a b c} → concatenate {a} {b} {c}
|
||||
; assoc = p-assoc
|
||||
; ident = ident-r , ident-l
|
||||
; isCategory = record { assoc = p-assoc ; ident = ident-r , ident-l }
|
||||
}
|
||||
|
|
|
|||
|
|
@ -1,7 +1,8 @@
|
|||
{-# OPTIONS --cubical #-}
|
||||
|
||||
module Category.Pathy where
|
||||
module Cat.Category.Pathy where
|
||||
|
||||
open import Level
|
||||
open import Cubical.PathPrelude
|
||||
|
||||
{-
|
||||
|
|
|
|||
|
|
@ -2,22 +2,64 @@
|
|||
|
||||
module Cat.Category.Properties where
|
||||
|
||||
open import Agda.Primitive
|
||||
open import Data.Product
|
||||
open import Cubical.PathPrelude
|
||||
|
||||
open import Cat.Category
|
||||
open import Cat.Functor
|
||||
open import Cat.Categories.Sets
|
||||
|
||||
module _ {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} { A B : ℂ .Category.Object } {X : ℂ .Category.Object} (f : ℂ .Category.Arrow A B) where
|
||||
open Category ℂ
|
||||
open IsCategory (isCategory)
|
||||
|
||||
iso-is-epi : Isomorphism {ℂ = ℂ} f → Epimorphism {ℂ = ℂ} {X = X} f
|
||||
iso-is-epi (f- , left-inv , right-inv) g₀ g₁ eq =
|
||||
begin
|
||||
g₀ ≡⟨ sym (proj₁ ident) ⟩
|
||||
g₀ ⊕ 𝟙 ≡⟨ cong (_⊕_ g₀) (sym right-inv) ⟩
|
||||
g₀ ⊕ (f ⊕ f-) ≡⟨ assoc ⟩
|
||||
(g₀ ⊕ f) ⊕ f- ≡⟨ cong (λ φ → φ ⊕ f-) eq ⟩
|
||||
(g₁ ⊕ f) ⊕ f- ≡⟨ sym assoc ⟩
|
||||
g₁ ⊕ (f ⊕ f-) ≡⟨ cong (_⊕_ g₁) right-inv ⟩
|
||||
g₁ ⊕ 𝟙 ≡⟨ proj₁ ident ⟩
|
||||
g₁ ∎
|
||||
|
||||
iso-is-mono : Isomorphism {ℂ = ℂ} f → Monomorphism {ℂ = ℂ} {X = X} f
|
||||
iso-is-mono (f- , (left-inv , right-inv)) g₀ g₁ eq =
|
||||
begin
|
||||
g₀ ≡⟨ sym (proj₂ ident) ⟩
|
||||
𝟙 ⊕ g₀ ≡⟨ cong (λ φ → φ ⊕ g₀) (sym left-inv) ⟩
|
||||
(f- ⊕ f) ⊕ g₀ ≡⟨ sym assoc ⟩
|
||||
f- ⊕ (f ⊕ g₀) ≡⟨ cong (_⊕_ f-) eq ⟩
|
||||
f- ⊕ (f ⊕ g₁) ≡⟨ assoc ⟩
|
||||
(f- ⊕ f) ⊕ g₁ ≡⟨ cong (λ φ → φ ⊕ g₁) left-inv ⟩
|
||||
𝟙 ⊕ g₁ ≡⟨ proj₂ ident ⟩
|
||||
g₁ ∎
|
||||
|
||||
iso-is-epi-mono : Isomorphism {ℂ = ℂ} f → Epimorphism {ℂ = ℂ} {X = X} f × Monomorphism {ℂ = ℂ} {X = X} f
|
||||
iso-is-epi-mono iso = iso-is-epi iso , iso-is-mono iso
|
||||
|
||||
{-
|
||||
epi-mono-is-not-iso : ∀ {ℓ ℓ'} → ¬ ((ℂ : Category {ℓ} {ℓ'}) {A B X : Object ℂ} (f : Arrow ℂ A B ) → Epimorphism {ℂ = ℂ} {X = X} f → Monomorphism {ℂ = ℂ} {X = X} f → Isomorphism {ℂ = ℂ} f)
|
||||
epi-mono-is-not-iso f =
|
||||
let k = f {!!} {!!} {!!} {!!}
|
||||
in {!!}
|
||||
-}
|
||||
|
||||
|
||||
module _ {ℓa ℓa' ℓb ℓb'} where
|
||||
Exponential : Category {ℓa} {ℓa'} → Category {ℓb} {ℓb'} → Category {{!!}} {{!!}}
|
||||
Exponential : Category ℓa ℓa' → Category ℓb ℓb' → Category {!!} {!!}
|
||||
Exponential A B = record
|
||||
{ Object = {!!}
|
||||
; Arrow = {!!}
|
||||
; 𝟙 = {!!}
|
||||
; _⊕_ = {!!}
|
||||
; assoc = {!!}
|
||||
; ident = {!!}
|
||||
; isCategory = {!!}
|
||||
}
|
||||
|
||||
_⇑_ = Exponential
|
||||
|
||||
yoneda : ∀ {ℓ ℓ'} → {ℂ : Category {ℓ} {ℓ'}} → Functor ℂ (Sets ⇑ (Opposite ℂ))
|
||||
yoneda : ∀ {ℓ ℓ'} → {ℂ : Category ℓ ℓ'} → Functor ℂ (Sets ⇑ (Opposite ℂ))
|
||||
yoneda = {!!}
|
||||
|
|
|
|||
|
|
@ -1,3 +1,4 @@
|
|||
{-# OPTIONS --allow-unsolved-metas #-}
|
||||
module Cat.Cubical where
|
||||
|
||||
open import Agda.Primitive
|
||||
|
|
@ -9,8 +10,13 @@ open import Data.Empty
|
|||
|
||||
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
|
||||
-- Σ is the "namespace"
|
||||
-- Ns is the "namespace"
|
||||
ℓo = (lsuc lzero ⊔ ℓ)
|
||||
|
||||
FiniteDecidableSubset : Set ℓ
|
||||
|
|
@ -36,13 +42,12 @@ module _ {ℓ ℓ' : Level} (Ns : Set ℓ) where
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Mor = Σ themap rules
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-- The category of names and substitutions
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ℂ : Category -- {ℓo} {lsuc lzero ⊔ ℓo}
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ℂ : Category ℓ ℓ -- ℓo (lsuc lzero ⊔ ℓo)
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ℂ = record
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-- { Object = FiniteDecidableSubset
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{ Object = Ns → Bool
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; Arrow = Mor
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; 𝟙 = {!!}
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; _⊕_ = {!!}
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; assoc = {!!}
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; ident = {!!}
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; isCategory = ?
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||||
}
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||||
|
|
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|
|
@ -6,7 +6,7 @@ open import Function
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|
||||
open import Cat.Category
|
||||
|
||||
record Functor {ℓc ℓc' ℓd ℓd'} (C : Category {ℓc} {ℓc'}) (D : Category {ℓd} {ℓd'})
|
||||
record Functor {ℓc ℓc' ℓd ℓd'} (C : Category ℓc ℓc') (D : Category ℓd ℓd')
|
||||
: Set (ℓc ⊔ ℓc' ⊔ ℓd ⊔ ℓd') where
|
||||
private
|
||||
open module C = Category C
|
||||
|
|
@ -21,43 +21,41 @@ record Functor {ℓc ℓc' ℓd ℓd'} (C : Category {ℓc} {ℓc'}) (D : Catego
|
|||
distrib : { c c' c'' : C.Object} {a : C.Arrow c c'} {a' : C.Arrow c' c''}
|
||||
→ func→ (a' C.⊕ a) ≡ func→ a' D.⊕ func→ a
|
||||
|
||||
module _ {ℓ ℓ' : Level} {A B C : Category {ℓ} {ℓ'}} (F : Functor B C) (G : Functor A B) where
|
||||
module _ {ℓ ℓ' : Level} {A B C : Category ℓ ℓ'} (F : Functor B C) (G : Functor A B) where
|
||||
open Functor
|
||||
open Category
|
||||
private
|
||||
open module F = Functor F
|
||||
open module G = Functor G
|
||||
open module A = Category A
|
||||
open module B = Category B
|
||||
open module C = Category C
|
||||
F* = F .func*
|
||||
F→ = F .func→
|
||||
G* = G .func*
|
||||
G→ = G .func→
|
||||
_A⊕_ = A ._⊕_
|
||||
_B⊕_ = B ._⊕_
|
||||
_C⊕_ = C ._⊕_
|
||||
module _ {a0 a1 a2 : A .Object} {α0 : A .Arrow a0 a1} {α1 : A .Arrow a1 a2} where
|
||||
|
||||
F* = F.func*
|
||||
F→ = F.func→
|
||||
G* = G.func*
|
||||
G→ = G.func→
|
||||
module _ {a0 a1 a2 : A.Object} {α0 : A.Arrow a0 a1} {α1 : A.Arrow a1 a2} where
|
||||
|
||||
dist : (F→ ∘ G→) (α1 A.⊕ α0) ≡ (F→ ∘ G→) α1 C.⊕ (F→ ∘ G→) α0
|
||||
dist : (F→ ∘ G→) (α1 A⊕ α0) ≡ (F→ ∘ G→) α1 C⊕ (F→ ∘ G→) α0
|
||||
dist = begin
|
||||
(F→ ∘ G→) (α1 A.⊕ α0) ≡⟨ refl ⟩
|
||||
F→ (G→ (α1 A.⊕ α0)) ≡⟨ cong F→ G.distrib ⟩
|
||||
F→ ((G→ α1) B.⊕ (G→ α0)) ≡⟨ F.distrib ⟩
|
||||
(F→ ∘ G→) α1 C.⊕ (F→ ∘ G→) α0 ∎
|
||||
(F→ ∘ G→) (α1 A⊕ α0) ≡⟨ refl ⟩
|
||||
F→ (G→ (α1 A⊕ α0)) ≡⟨ cong F→ (G .distrib)⟩
|
||||
F→ ((G→ α1) B⊕ (G→ α0)) ≡⟨ F .distrib ⟩
|
||||
(F→ ∘ G→) α1 C⊕ (F→ ∘ G→) α0 ∎
|
||||
|
||||
functor-comp : Functor A C
|
||||
functor-comp =
|
||||
_∘f_ : Functor A C
|
||||
_∘f_ =
|
||||
record
|
||||
{ func* = F* ∘ G*
|
||||
; func→ = F→ ∘ G→
|
||||
; ident = begin
|
||||
(F→ ∘ G→) (A.𝟙) ≡⟨ refl ⟩
|
||||
F→ (G→ (A.𝟙)) ≡⟨ cong F→ G.ident ⟩
|
||||
F→ (B.𝟙) ≡⟨ F.ident ⟩
|
||||
C.𝟙 ∎
|
||||
(F→ ∘ G→) (A .𝟙) ≡⟨ refl ⟩
|
||||
F→ (G→ (A .𝟙)) ≡⟨ cong F→ (G .ident)⟩
|
||||
F→ (B .𝟙) ≡⟨ F .ident ⟩
|
||||
C .𝟙 ∎
|
||||
; distrib = dist
|
||||
}
|
||||
|
||||
-- The identity functor
|
||||
identity : {ℓ ℓ' : Level} → {C : Category {ℓ} {ℓ'}} → Functor C C
|
||||
-- Identity = record { F* = λ x → x ; F→ = λ x → x ; ident = refl ; distrib = refl }
|
||||
identity : ∀ {ℓ ℓ'} → {C : Category ℓ ℓ'} → Functor C C
|
||||
identity = record
|
||||
{ func* = λ x → x
|
||||
; func→ = λ x → x
|
||||
|
|
|
|||
Loading…
Reference in a new issue