Have yoneda without having a category of categories
I did break some things in Cat.Categories.Cat but since this is unprovable anyways it's not that big a deal.
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1bf565b87a
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@ -11,7 +11,7 @@ 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.Functor
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open import Cat.Category.Product
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open import Cat.Category.Exponential
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open import Cat.Category.Exponential hiding (_×_ ; product)
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open import Cat.Category.NaturalTransformation
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open import Cat.Equality
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@ -174,22 +174,19 @@ module _ {ℓ ℓ' : Level} (unprovable : IsCategory (RawCat ℓ ℓ')) where
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hasProducts = record { product = product }
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-- Basically proves that `Cat ℓ ℓ` is cartesian closed.
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module _ (ℓ : Level) (unprovable : IsCategory (RawCat ℓ ℓ)) where
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private
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module CatExponential {ℓ : Level} (ℂ 𝔻 : Category ℓ ℓ) where
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open Data.Product
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open import Cat.Categories.Fun
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Catℓ : Category (lsuc (ℓ ⊔ ℓ)) (ℓ ⊔ ℓ)
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Catℓ = Cat ℓ ℓ unprovable
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module _ (ℂ 𝔻 : Category ℓ ℓ) where
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Categoryℓ = Category ℓ ℓ
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open Fun ℂ 𝔻 renaming (identity to idN)
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private
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:obj: : Object Catℓ
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:obj: = Fun
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:func*: : Functor ℂ 𝔻 × Object ℂ → Object 𝔻
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:func*: (F , A) = func* F A
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prodObj : Categoryℓ
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prodObj = Fun
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module _ {dom cod : Functor ℂ 𝔻 × Object ℂ} where
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private
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F : Functor ℂ 𝔻
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@ -226,7 +223,7 @@ module _ (ℓ : Level) (unprovable : IsCategory (RawCat ℓ ℓ)) where
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result : 𝔻 [ func* F A , func* G B ]
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result = l
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_×p_ = product unprovable
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open CatProduct renaming (obj to _×p_) using ()
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module _ {c : Functor ℂ 𝔻 × Object ℂ} where
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private
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@ -244,7 +241,7 @@ module _ (ℓ : Level) (unprovable : IsCategory (RawCat ℓ ℓ)) where
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:ident: : :func→: {c} {c} (NT.identity F , 𝟙 ℂ {A = proj₂ c}) ≡ 𝟙 𝔻
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:ident: = begin
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:func→: {c} {c} (𝟙 (Product.obj (:obj: ×p ℂ)) {c}) ≡⟨⟩
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:func→: {c} {c} (𝟙 (prodObj ×p ℂ) {c}) ≡⟨⟩
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:func→: {c} {c} (idN F , 𝟙 ℂ) ≡⟨⟩
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𝔻 [ identityTrans F C ∘ func→ F (𝟙 ℂ)] ≡⟨⟩
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𝔻 [ 𝟙 𝔻 ∘ func→ F (𝟙 ℂ)] ≡⟨ proj₂ 𝔻.isIdentity ⟩
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@ -262,7 +259,7 @@ module _ (ℓ : Level) (unprovable : IsCategory (RawCat ℓ ℓ)) where
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H = H×C .proj₁
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C = H×C .proj₂
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-- Not entirely clear what this is at this point:
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_P⊕_ = Category._∘_ (Product.obj (:obj: ×p ℂ)) {F×A} {G×B} {H×C}
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_P⊕_ = Category._∘_ (prodObj ×p ℂ) {F×A} {G×B} {H×C}
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module _
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-- NaturalTransformation F G × ℂ .Arrow A B
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{θ×f : NaturalTransformation F G × ℂ [ A , B ]}
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@ -314,8 +311,9 @@ module _ (ℓ : Level) (unprovable : IsCategory (RawCat ℓ ℓ)) where
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open Category 𝔻
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module H = Functor H
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:eval: : Functor ((:obj: ×p ℂ) .Product.obj) 𝔻
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:eval: = record
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eval : Functor (CatProduct.obj prodObj ℂ) 𝔻
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-- :eval: : Functor (prodObj ×p ℂ) 𝔻
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eval = record
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{ raw = record
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{ func* = :func*:
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; func→ = λ {dom} {cod} → :func→: {dom} {cod}
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@ -326,12 +324,16 @@ module _ (ℓ : Level) (unprovable : IsCategory (RawCat ℓ ℓ)) where
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}
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}
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module _ (𝔸 : Category ℓ ℓ) (F : Functor ((𝔸 ×p ℂ) .Product.obj) 𝔻) where
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open HasProducts (hasProducts {ℓ} {ℓ} unprovable) renaming (_|×|_ to parallelProduct)
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module _ (𝔸 : Category ℓ ℓ) (F : Functor (𝔸 ×p ℂ) 𝔻) where
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-- open HasProducts (hasProducts {ℓ} {ℓ} unprovable) renaming (_|×|_ to parallelProduct)
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postulate
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transpose : Functor 𝔸 :obj:
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eq : Catℓ [ :eval: ∘ (parallelProduct transpose (𝟙 Catℓ {A = ℂ})) ] ≡ F
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parallelProduct
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: Functor 𝔸 prodObj → Functor ℂ ℂ
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→ Functor (𝔸 ×p ℂ) (prodObj ×p ℂ)
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transpose : Functor 𝔸 prodObj
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eq : F[ eval ∘ (parallelProduct transpose (identity {C = ℂ})) ] ≡ F
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-- eq : F[ :eval: ∘ {!!} ] ≡ F
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-- eq : Catℓ [ :eval: ∘ (HasProducts._|×|_ hasProducts transpose (𝟙 Catℓ {o = ℂ})) ] ≡ F
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-- eq' : (Catℓ [ :eval: ∘
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-- (record { product = product } HasProducts.|×| transpose)
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@ -344,20 +346,39 @@ module _ (ℓ : Level) (unprovable : IsCategory (RawCat ℓ ℓ)) where
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-- :eval: ∘ (parallelProduct F~ (𝟙 Catℓ {o = ℂ}))] ≡ F) catTranspose =
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-- transpose , eq
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postulate :isExponential: : IsExponential Catℓ ℂ 𝔻 :obj: :eval:
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-- :isExponential: : IsExponential Catℓ ℂ 𝔻 :obj: :eval:
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-- :isExponential: = {!catTranspose!}
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-- where
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-- open HasProducts (hasProducts {ℓ} {ℓ} unprovable) using (_|×|_)
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-- :isExponential: = λ 𝔸 F → transpose 𝔸 F , eq' 𝔸 F
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module _ (ℓ : Level) (unprovable : IsCategory (RawCat ℓ ℓ)) where
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private
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Catℓ : Category (lsuc (ℓ ⊔ ℓ)) (ℓ ⊔ ℓ)
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Catℓ = Cat ℓ ℓ unprovable
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module _ (ℂ 𝔻 : Category ℓ ℓ) where
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open CatExponential ℂ 𝔻 using (prodObj ; eval)
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-- Putting in the type annotation causes Agda to loop indefinitely.
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-- eval' : Functor (CatProduct.obj prodObj ℂ) 𝔻
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-- Likewise, using it below also results in this.
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eval' : _
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eval' = eval
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-- private
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-- -- module _ (ℂ 𝔻 : Category ℓ ℓ) where
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-- postulate :isExponential: : IsExponential Catℓ ℂ 𝔻 prodObj :eval:
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-- -- :isExponential: : IsExponential Catℓ ℂ 𝔻 :obj: :eval:
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-- -- :isExponential: = {!catTranspose!}
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-- -- where
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-- -- open HasProducts (hasProducts {ℓ} {ℓ} unprovable) using (_|×|_)
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-- -- :isExponential: = λ 𝔸 F → transpose 𝔸 F , eq' 𝔸 F
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-- :exponent: : Exponential (Cat ℓ ℓ) A B
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:exponent: : Exponential Catℓ ℂ 𝔻
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:exponent: = record
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{ obj = :obj:
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; eval = :eval:
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; isExponential = :isExponential:
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-- -- :exponent: : Exponential (Cat ℓ ℓ) A B
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exponent : Exponential Catℓ ℂ 𝔻
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exponent = record
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{ obj = prodObj
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; eval = {!evalll'!}
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; isExponential = {!:isExponential:!}
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}
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where
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open HasProducts (hasProducts unprovable) renaming (_×_ to _×p_)
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open import Cat.Categories.Fun
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open Fun
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-- _×p_ = CatProduct.obj -- prodObj ℂ
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-- eval' : Functor CatP.obj 𝔻
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hasExponentials : HasExponentials Catℓ
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hasExponentials = record { exponent = :exponent: }
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hasExponentials = record { exponent = exponent }
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@ -1,40 +1,44 @@
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module Cat.Category.Exponential where
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open import Agda.Primitive
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open import Data.Product
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open import Data.Product hiding (_×_)
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open import Cubical
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open import Cat.Category
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open import Cat.Category.Product
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open Category
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module _ {ℓ ℓ'} (ℂ : Category ℓ ℓ') {{hasProducts : HasProducts ℂ}} where
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open HasProducts hasProducts
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open Product hiding (obj)
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private
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_×p_ : (A B : Object ℂ) → Object ℂ
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_×p_ A B = Product.obj (product A B)
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open Category ℂ
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open HasProducts hasProducts public
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module _ (B C : Object ℂ) where
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IsExponential : (Cᴮ : Object ℂ) → ℂ [ Cᴮ ×p B , C ] → Set (ℓ ⊔ ℓ')
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IsExponential Cᴮ eval = ∀ (A : Object ℂ) (f : ℂ [ A ×p B , C ])
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module _ (B C : Object) where
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record IsExponential'
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(Cᴮ : Object)
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(eval : ℂ [ Cᴮ × B , C ]) : Set (ℓ ⊔ ℓ') where
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field
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uniq
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: ∀ (A : Object) (f : ℂ [ A × B , C ])
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→ ∃![ f~ ] (ℂ [ eval ∘ f~ |×| Category.𝟙 ℂ ] ≡ f)
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IsExponential : (Cᴮ : Object) → ℂ [ Cᴮ × B , C ] → Set (ℓ ⊔ ℓ')
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IsExponential Cᴮ eval = ∀ (A : Object) (f : ℂ [ A × B , C ])
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→ ∃![ f~ ] (ℂ [ eval ∘ f~ |×| Category.𝟙 ℂ ] ≡ f)
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record Exponential : Set (ℓ ⊔ ℓ') where
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field
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-- obj ≡ Cᴮ
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obj : Object ℂ
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eval : ℂ [ obj ×p B , C ]
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obj : Object
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eval : ℂ [ obj × B , C ]
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{{isExponential}} : IsExponential obj eval
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-- If I make this an instance-argument then the instance resolution
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-- algorithm goes into an infinite loop. Why?
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exponentialsHaveProducts : HasProducts ℂ
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exponentialsHaveProducts = hasProducts
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transpose : (A : Object ℂ) → ℂ [ A ×p B , C ] → ℂ [ A , obj ]
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transpose : (A : Object) → ℂ [ A × B , C ] → ℂ [ A , obj ]
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transpose A f = proj₁ (isExponential A f)
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record HasExponentials {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') {{_ : HasProducts ℂ}} : Set (ℓ ⊔ ℓ') where
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open Category ℂ
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open Exponential public
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field
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exponent : (A B : Object ℂ) → Exponential ℂ A B
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exponent : (A B : Object) → Exponential ℂ A B
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_⇑_ : (A B : Object) → Object
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A ⇑ B = (exponent A B) .obj
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@ -27,9 +27,10 @@ module _ (ℓa ℓb : Level) where
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open Category category public
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field
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{{hasProducts}} : HasProducts category
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mempty : Object
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empty : Object
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-- aka. tensor product, monoidal product.
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mappend : Functor (category × category) category
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append : Functor (category × category) category
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open HasProducts hasProducts public
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record MonoidalCategory : Set ℓ where
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field
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@ -40,10 +41,10 @@ module _ {ℓa ℓb : Level} (ℂ : MonoidalCategory ℓa ℓb) where
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private
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ℓ = ℓa ⊔ ℓb
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module MC = MonoidalCategory ℂ
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open HasProducts MC.hasProducts
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open MonoidalCategory ℂ public
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record Monoid : Set ℓ where
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field
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carrier : MC.Object
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mempty : MC.Arrow (carrier × carrier) carrier
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mappend : MC.Arrow MC.mempty carrier
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carrier : Object
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mempty : Arrow empty carrier
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mappend : Arrow (carrier × carrier) carrier
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@ -31,6 +31,7 @@ record Product {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} (A B : Object ℂ) :
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proj₂ : ℂ [ obj , B ]
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{{isProduct}} : IsProduct ℂ proj₁ proj₂
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-- | Arrow product
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_P[_×_] : ∀ {X} → (π₁ : ℂ [ X , A ]) (π₂ : ℂ [ X , B ])
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→ ℂ [ X , obj ]
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_P[_×_] π₁ π₂ = proj₁ (isProduct π₁ π₂)
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@ -39,16 +40,21 @@ record HasProducts {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') : Set (ℓ ⊔
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field
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product : ∀ (A B : Object ℂ) → Product {ℂ = ℂ} A B
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open Product
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open Product hiding (obj)
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_×_ : (A B : Object ℂ) → Object ℂ
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A × B = Product.obj (product A B)
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-- The product mentioned in awodey in Def 6.1 is not the regular product of arrows.
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-- It's a "parallel" product
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_|×|_ : {A A' B B' : Object ℂ} → ℂ [ A , A' ] → ℂ [ B , B' ]
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→ ℂ [ A × B , A' × B' ]
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_|×|_ {A = A} {A' = A'} {B = B} {B' = B'} a b
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= product A' B'
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P[ ℂ [ a ∘ (product A B) .proj₁ ]
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× ℂ [ b ∘ (product A B) .proj₂ ]
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module _ (A B : Object ℂ) where
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open Product (product A B)
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_×_ : Object ℂ
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_×_ = obj
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-- | Parallel product of arrows
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--
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-- The product mentioned in awodey in Def 6.1 is not the regular product of
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-- arrows. It's a "parallel" product
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module _ {A A' B B' : Object ℂ} where
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open Product (product A B) hiding (_P[_×_]) renaming (proj₁ to fst ; proj₂ to snd)
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_|×|_ : ℂ [ A , A' ] → ℂ [ B , B' ] → ℂ [ A × B , A' × B' ]
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a |×| b = product A' B'
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P[ ℂ [ a ∘ fst ]
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× ℂ [ b ∘ snd ]
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]
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@ -15,7 +15,7 @@ open Equality.Data.Product
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-- category of categories (since it doesn't exist).
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open import Cat.Categories.Cat using (RawCat)
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module _ {ℓ : Level} {ℂ : Category ℓ ℓ} (unprovable : IsCategory (RawCat ℓ ℓ)) where
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module _ {ℓ : Level} {ℂ : Category ℓ ℓ} where
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private
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open import Cat.Categories.Fun
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open import Cat.Categories.Sets
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@ -24,15 +24,17 @@ module _ {ℓ : Level} {ℂ : Category ℓ ℓ} (unprovable : IsCategory (RawCat
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open Functor
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𝓢 = Sets ℓ
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open Fun (opposite ℂ) 𝓢
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Catℓ : Category _ _
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Catℓ = Cat.Cat ℓ ℓ unprovable
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prshf = presheaf ℂ
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module ℂ = Category ℂ
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_⇑_ : (A B : Category.Object Catℓ) → Category.Object Catℓ
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A ⇑ B = (exponent A B) .obj
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where
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open HasExponentials (Cat.hasExponentials ℓ unprovable)
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-- There is no (small) category of categories. So we won't use _⇑_ from
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-- `HasExponential`
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--
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-- open HasExponentials (Cat.hasExponentials ℓ unprovable) using (_⇑_)
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--
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-- In stead we'll use an ad-hoc definition -- which is definitionally
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-- equivalent to that other one.
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_⇑_ = Cat.CatExponential.prodObj
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module _ {A B : ℂ.Object} (f : ℂ [ A , B ]) where
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:func→: : NaturalTransformation (prshf A) (prshf B)
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