324 lines
12 KiB
Agda
324 lines
12 KiB
Agda
-- There is no category of categories in our interpretation
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{-# OPTIONS --cubical --allow-unsolved-metas #-}
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module Cat.Categories.Cat where
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open import Cat.Prelude renaming (fst to fst ; snd 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 hiding (_×_ ; product)
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import Cat.Category.NaturalTransformation
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open import Cat.Categories.Fun
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-- The category of categories
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module _ (ℓ ℓ' : Level) where
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RawCat : RawCategory (lsuc (ℓ ⊔ ℓ')) (ℓ ⊔ ℓ')
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RawCategory.Object RawCat = Category ℓ ℓ'
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RawCategory.Arrow RawCat = Functor
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RawCategory.identity RawCat = Functors.identity
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RawCategory._<<<_ RawCat = F[_∘_]
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-- NB! `ArrowsAreSets RawCat` is *not* provable. The type of functors,
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-- however, form a groupoid! Therefore there is no (1-)category of
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-- categories. There does, however, exist a 2-category of 1-categories.
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--
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-- Because of this there is no category of categories.
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Cat : (unprovable : IsCategory RawCat) → Category (lsuc (ℓ ⊔ ℓ')) (ℓ ⊔ ℓ')
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Category.raw (Cat _) = RawCat
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Category.isCategory (Cat unprovable) = unprovable
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-- | In the following we will pretend there is a category of categories when
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-- e.g. talking about it being cartesian closed. It still makes sense to
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-- construct these things even though that category does not exist.
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--
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-- If the notion of a category is later generalized to work on different
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-- homotopy levels, then the proof that the category of categories is cartesian
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-- closed will follow immediately from these constructions.
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-- | the category of categories have products.
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module CatProduct {ℓ ℓ' : Level} (ℂ 𝔻 : Category ℓ ℓ') where
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private
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module ℂ = Category ℂ
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module 𝔻 = Category 𝔻
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module _ where
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private
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Obj = ℂ.Object × 𝔻.Object
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Arr : Obj → Obj → Set ℓ'
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Arr (c , d) (c' , d') = ℂ [ c , c' ] × 𝔻 [ d , d' ]
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identity : {o : Obj} → Arr o o
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identity = ℂ.identity , 𝔻.identity
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_<<<_ :
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{a b c : Obj} →
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Arr b c →
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Arr a b →
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Arr 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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rawProduct : RawCategory ℓ ℓ'
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RawCategory.Object rawProduct = Obj
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RawCategory.Arrow rawProduct = Arr
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RawCategory.identity rawProduct = identity
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RawCategory._<<<_ rawProduct = _<<<_
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open RawCategory rawProduct
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arrowsAreSets : ArrowsAreSets
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arrowsAreSets = setSig {sA = ℂ.arrowsAreSets} {sB = λ x → 𝔻.arrowsAreSets}
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isIdentity : IsIdentity identity
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isIdentity
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= Σ≡ (fst ℂ.isIdentity) (fst 𝔻.isIdentity)
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, Σ≡ (snd ℂ.isIdentity) (snd 𝔻.isIdentity)
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isPreCategory : IsPreCategory rawProduct
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IsPreCategory.isAssociative isPreCategory = Σ≡ ℂ.isAssociative 𝔻.isAssociative
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IsPreCategory.isIdentity isPreCategory = isIdentity
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IsPreCategory.arrowsAreSets isPreCategory = arrowsAreSets
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postulate univalent : Univalence.Univalent isIdentity
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isCategory : IsCategory rawProduct
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IsCategory.isPreCategory isCategory = isPreCategory
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IsCategory.univalent isCategory = univalent
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object : Category ℓ ℓ'
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Category.raw object = rawProduct
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Category.isCategory object = isCategory
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fstF : Functor object ℂ
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fstF = record
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{ raw = record
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{ omap = fst ; fmap = fst }
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; isFunctor = record
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{ isIdentity = refl ; isDistributive = refl }
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}
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sndF : Functor object 𝔻
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sndF = record
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{ raw = record
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{ omap = snd ; fmap = snd }
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; isFunctor = record
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{ isIdentity = refl ; isDistributive = refl }
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}
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module _ {X : Category ℓ ℓ'} (x₁ : Functor X ℂ) (x₂ : Functor X 𝔻) where
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private
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x : Functor X object
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x = record
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{ raw = record
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{ omap = λ x → x₁.omap x , x₂.omap x
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; fmap = λ x → x₁.fmap x , x₂.fmap x
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}
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; isFunctor = record
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{ isIdentity = Σ≡ x₁.isIdentity x₂.isIdentity
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; isDistributive = Σ≡ x₁.isDistributive x₂.isDistributive
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}
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}
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where
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open module x₁ = Functor x₁
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open module x₂ = Functor x₂
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isUniqL : F[ fstF ∘ x ] ≡ x₁
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isUniqL = Functor≡ refl
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isUniqR : F[ sndF ∘ x ] ≡ x₂
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isUniqR = Functor≡ refl
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isUniq : F[ fstF ∘ x ] ≡ x₁ × F[ sndF ∘ x ] ≡ x₂
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isUniq = isUniqL , isUniqR
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isProduct : ∃![ x ] (F[ fstF ∘ x ] ≡ x₁ × F[ sndF ∘ x ] ≡ x₂)
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isProduct = x , isUniq , uq
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where
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module _ {y : Functor X object} (eq : F[ fstF ∘ y ] ≡ x₁ × F[ sndF ∘ y ] ≡ x₂) where
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omapEq : Functor.omap x ≡ Functor.omap y
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omapEq = {!!}
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-- fmapEq : (λ i → {!{A B : ?} → Arrow A B → 𝔻 [ ? A , ? B ]!}) [ Functor.fmap x ≡ Functor.fmap y ]
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-- fmapEq = {!!}
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rawEq : Functor.raw x ≡ Functor.raw y
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rawEq = {!!}
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uq : x ≡ y
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uq = Functor≡ rawEq
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module _ {ℓ ℓ' : Level} (unprovable : IsCategory (RawCat ℓ ℓ')) where
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private
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Catℓ = Cat ℓ ℓ' unprovable
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module _ (ℂ 𝔻 : Category ℓ ℓ') where
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private
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module P = CatProduct ℂ 𝔻
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rawProduct : RawProduct Catℓ ℂ 𝔻
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RawProduct.object rawProduct = P.object
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RawProduct.fst rawProduct = P.fstF
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RawProduct.snd rawProduct = P.sndF
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isProduct : IsProduct Catℓ _ _ rawProduct
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IsProduct.ump isProduct = P.isProduct
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product : Product Catℓ ℂ 𝔻
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Product.raw product = rawProduct
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Product.isProduct product = isProduct
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instance
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hasProducts : HasProducts Catℓ
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hasProducts = record { product = product }
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-- | The category of categories have expoentntials - and because it has products
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-- it is therefory also cartesian closed.
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module CatExponential {ℓ : Level} (ℂ 𝔻 : Category ℓ ℓ) where
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open Cat.Category.NaturalTransformation ℂ 𝔻
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renaming (identity to identityNT)
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using ()
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private
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module ℂ = Category ℂ
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module 𝔻 = Category 𝔻
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Categoryℓ = Category ℓ ℓ
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open Fun ℂ 𝔻 renaming (identity to idN)
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omap : Functor ℂ 𝔻 × ℂ.Object → 𝔻.Object
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omap (F , A) = Functor.omap F A
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-- The exponential object
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object : Categoryℓ
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object = Fun
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module _ {dom cod : Functor ℂ 𝔻 × ℂ.Object} where
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open Σ dom renaming (fst to F ; snd to A)
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open Σ cod renaming (fst to G ; snd to B)
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private
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module F = Functor F
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module G = Functor G
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fmap : (pobj : NaturalTransformation F G × ℂ [ A , B ])
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→ 𝔻 [ F.omap A , G.omap B ]
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fmap ((θ , θNat) , f) = 𝔻 [ θ B ∘ F.fmap f ]
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-- Alternatively:
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--
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-- fmap ((θ , θNat) , f) = 𝔻 [ G.fmap f ∘ θ A ]
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--
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-- Since they are equal by naturality of θ.
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open CatProduct renaming (object to _⊗_) using ()
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module _ {c : Functor ℂ 𝔻 × ℂ.Object} where
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open Σ c renaming (fst to F ; snd to C)
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ident : fmap {c} {c} (identityNT F , ℂ.identity {A = snd c}) ≡ 𝔻.identity
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ident = begin
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fmap {c} {c} (Category.identity (object ⊗ ℂ) {c}) ≡⟨⟩
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fmap {c} {c} (idN F , ℂ.identity) ≡⟨⟩
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𝔻 [ identityTrans F C ∘ F.fmap ℂ.identity ] ≡⟨⟩
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𝔻 [ 𝔻.identity ∘ F.fmap ℂ.identity ] ≡⟨ 𝔻.leftIdentity ⟩
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F.fmap ℂ.identity ≡⟨ F.isIdentity ⟩
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𝔻.identity ∎
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where
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module F = Functor F
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module _ {F×A G×B H×C : Functor ℂ 𝔻 × ℂ.Object} where
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open Σ F×A renaming (fst to F ; snd to A)
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open Σ G×B renaming (fst to G ; snd to B)
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open Σ H×C renaming (fst to H ; snd to C)
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private
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module F = Functor F
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module G = Functor G
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module H = Functor H
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module _
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{θ×f : NaturalTransformation F G × ℂ [ A , B ]}
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{η×g : NaturalTransformation G H × ℂ [ B , C ]} where
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open Σ θ×f renaming (fst to θNT ; snd to f)
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open Σ θNT renaming (fst to θ ; snd to θNat)
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open Σ η×g renaming (fst to ηNT ; snd to g)
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open Σ ηNT renaming (fst to η ; snd to ηNat)
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private
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ηθNT : NaturalTransformation F H
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ηθNT = NT[_∘_] {F} {G} {H} ηNT θNT
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open Σ ηθNT renaming (fst to ηθ ; snd to ηθNat)
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isDistributive :
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𝔻 [ 𝔻 [ η C ∘ θ C ] ∘ F.fmap ( ℂ [ g ∘ f ] ) ]
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≡ 𝔻 [ 𝔻 [ η C ∘ G.fmap g ] ∘ 𝔻 [ θ B ∘ F.fmap f ] ]
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isDistributive = begin
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𝔻 [ (ηθ C) ∘ F.fmap (ℂ [ g ∘ f ]) ]
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≡⟨ ηθNat (ℂ [ g ∘ f ]) ⟩
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𝔻 [ H.fmap (ℂ [ g ∘ f ]) ∘ (ηθ A) ]
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≡⟨ cong (λ φ → 𝔻 [ φ ∘ ηθ A ]) (H.isDistributive) ⟩
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𝔻 [ 𝔻 [ H.fmap g ∘ H.fmap f ] ∘ (ηθ A) ]
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≡⟨ sym 𝔻.isAssociative ⟩
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𝔻 [ H.fmap g ∘ 𝔻 [ H.fmap f ∘ ηθ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ H.fmap g ∘ φ ]) 𝔻.isAssociative ⟩
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𝔻 [ H.fmap g ∘ 𝔻 [ 𝔻 [ H.fmap f ∘ η A ] ∘ θ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ H.fmap g ∘ φ ]) (cong (λ φ → 𝔻 [ φ ∘ θ A ]) (sym (ηNat f))) ⟩
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𝔻 [ H.fmap g ∘ 𝔻 [ 𝔻 [ η B ∘ G.fmap f ] ∘ θ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ H.fmap g ∘ φ ]) (sym 𝔻.isAssociative) ⟩
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𝔻 [ H.fmap g ∘ 𝔻 [ η B ∘ 𝔻 [ G.fmap f ∘ θ A ] ] ]
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≡⟨ 𝔻.isAssociative ⟩
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𝔻 [ 𝔻 [ H.fmap g ∘ η B ] ∘ 𝔻 [ G.fmap f ∘ θ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ φ ∘ 𝔻 [ G.fmap f ∘ θ A ] ]) (sym (ηNat g)) ⟩
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𝔻 [ 𝔻 [ η C ∘ G.fmap g ] ∘ 𝔻 [ G.fmap f ∘ θ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ 𝔻 [ η C ∘ G.fmap g ] ∘ φ ]) (sym (θNat f)) ⟩
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𝔻 [ 𝔻 [ η C ∘ G.fmap g ] ∘ 𝔻 [ θ B ∘ F.fmap f ] ] ∎
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eval : Functor (CatProduct.object object ℂ) 𝔻
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eval = record
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{ raw = record
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{ omap = omap
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; fmap = λ {dom} {cod} → fmap {dom} {cod}
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}
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; isFunctor = record
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{ isIdentity = λ {o} → ident {o}
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; isDistributive = λ {f u n k y} → isDistributive {f} {u} {n} {k} {y}
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}
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}
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module _ (𝔸 : Category ℓ ℓ) (F : Functor (𝔸 ⊗ ℂ) 𝔻) where
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postulate
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parallelProduct
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: Functor 𝔸 object → Functor ℂ ℂ
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→ Functor (𝔸 ⊗ ℂ) (object ⊗ ℂ)
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transpose : Functor 𝔸 object
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eq : F[ eval ∘ (parallelProduct transpose (Functors.identity {ℂ = ℂ})) ] ≡ F
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-- eq : F[ :eval: ∘ {!!} ] ≡ F
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-- eq : Catℓ [ :eval: ∘ (HasProducts._|×|_ hasProducts transpose (identity Catℓ {o = ℂ})) ] ≡ F
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-- eq' : (Catℓ [ :eval: ∘
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-- (record { product = product } HasProducts.|×| transpose)
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-- (identity Catℓ)
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-- ])
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-- ≡ F
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-- For some reason after `e8215b2c051062c6301abc9b3f6ec67106259758`
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-- `catTranspose` makes Agda hang. catTranspose : ∃![ F~ ] (Catℓ [
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-- :eval: ∘ (parallelProduct F~ (identity Catℓ {o = ℂ}))] ≡ F) catTranspose =
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-- transpose , eq
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-- We don't care about filling out the holes below since they are anyways hidden
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-- behind an unprovable statement.
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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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module CatExp = CatExponential ℂ 𝔻
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_⊗_ = CatProduct.object
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-- Filling the hole causes Agda to loop indefinitely.
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eval : Functor (CatExp.object ⊗ ℂ) 𝔻
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eval = {!CatExp.eval!}
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isExponential : IsExponential Catℓ ℂ 𝔻 CatExp.object eval
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isExponential = {!CatExp.isExponential!}
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exponent : Exponential Catℓ ℂ 𝔻
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exponent = record
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{ obj = CatExp.object
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; eval = eval
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; isExponential = isExponential
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}
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hasExponentials : HasExponentials Catℓ
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hasExponentials = record { exponent = exponent }
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