cat/src/Cat/Categories/Cat.agda

-- There is no category of categories in our interpretation
{-# OPTIONS --cubical --allow-unsolved-metas #-}

module Cat.Categories.Cat where

open import Agda.Primitive
open import Cubical
open import Function
open import Data.Product renaming (proj₁ to fst ; proj₂ to snd)

open import Cat.Category
open import Cat.Category.Functor
open import Cat.Category.Product
open import Cat.Category.Exponential hiding (_×_ ; product)
open import Cat.Category.NaturalTransformation

open import Cat.Equality
open Equality.Data.Product

open Functor using (func→ ; func*)
open Category using (Object ; 𝟙)

-- The category of categories
module _ (ℓ ℓ' : Level) where
  private
    module _ {𝔸 𝔹 ℂ 𝔻 : Category ℓ ℓ'} {F : Functor 𝔸 𝔹} {G : Functor 𝔹 ℂ} {H : Functor ℂ 𝔻} where
      assc : F[ H ∘ F[ G ∘ F ] ] ≡ F[ F[ H ∘ G ] ∘ F ]
      assc = Functor≡ refl

    module _ {ℂ 𝔻 : Category ℓ ℓ'} {F : Functor ℂ 𝔻} where
      ident-r : F[ F ∘ identity ] ≡ F
      ident-r = Functor≡ refl

      ident-l : F[ identity ∘ F ] ≡ F
      ident-l = Functor≡ refl

  RawCat : RawCategory (lsuc (ℓ ⊔ ℓ')) (ℓ ⊔ ℓ')
  RawCat =
    record
      { Object = Category ℓ ℓ'
      ; Arrow = Functor
      ; 𝟙 = identity
      ; _∘_ = F[_∘_]
      }
  private
    open RawCategory RawCat
    isAssociative : IsAssociative
    isAssociative {f = F} {G} {H} = assc {F = F} {G = G} {H = H}
    ident : IsIdentity identity
    ident = ident-r , ident-l
    -- NB! `ArrowsAreSets RawCat` is *not* provable. The type of functors,
    -- however, form a groupoid! Therefore there is no (1-)category of
    -- categories. There does, however, exist a 2-category of 1-categories.

  -- Because of the note above there is not category of categories.
  Cat : (unprovable : IsCategory RawCat) → Category (lsuc (ℓ ⊔ ℓ')) (ℓ ⊔ ℓ')
  Category.raw        (Cat _) = RawCat
  Category.isCategory (Cat unprovable) = unprovable
  -- Category.raw Cat _ = RawCat
  -- Category.isCategory Cat unprovable = unprovable

-- The following to some extend depends on the category of categories being a
-- category. In some places it may not actually be needed, however.
module CatProduct {ℓ ℓ' : Level} (ℂ 𝔻 : Category ℓ ℓ') where
  private
    :Object: = Object ℂ × Object 𝔻
    :Arrow:  : :Object: → :Object: → Set ℓ'
    :Arrow: (c , d) (c' , d') = ℂ [ c , c' ] × 𝔻 [ d , d' ]
    :𝟙: : {o : :Object:} → :Arrow: o o
    :𝟙: = 𝟙 ℂ , 𝟙 𝔻
    _:⊕:_ :
      {a b c : :Object:} →
      :Arrow: b c →
      :Arrow: a b →
      :Arrow: a c
    _:⊕:_ = λ { (bc∈C , bc∈D) (ab∈C , ab∈D) → ℂ [ bc∈C ∘ ab∈C ] , 𝔻 [ bc∈D ∘ ab∈D ]}

    :rawProduct: : RawCategory ℓ ℓ'
    RawCategory.Object :rawProduct: = :Object:
    RawCategory.Arrow :rawProduct: = :Arrow:
    RawCategory.𝟙 :rawProduct: = :𝟙:
    RawCategory._∘_ :rawProduct: = _:⊕:_
    open RawCategory :rawProduct:

    module ℂ = Category ℂ
    module 𝔻 = Category 𝔻
    open import Cubical.Sigma
    arrowsAreSets : ArrowsAreSets -- {A B : RawCategory.Object :rawProduct:} → isSet (Arrow A B)
    arrowsAreSets = setSig {sA = ℂ.arrowsAreSets} {sB = λ x → 𝔻.arrowsAreSets}
    isIdentity : IsIdentity :𝟙:
    isIdentity
      = Σ≡ (fst ℂ.isIdentity) (fst 𝔻.isIdentity)
      , Σ≡ (snd ℂ.isIdentity) (snd 𝔻.isIdentity)
    postulate univalent : Univalence.Univalent :rawProduct: isIdentity
    instance
      :isCategory: : IsCategory :rawProduct:
      IsCategory.isAssociative :isCategory: = Σ≡ ℂ.isAssociative 𝔻.isAssociative
      IsCategory.isIdentity :isCategory: = isIdentity
      IsCategory.arrowsAreSets :isCategory: = arrowsAreSets
      IsCategory.univalent :isCategory: = univalent

  obj : Category ℓ ℓ'
  Category.raw obj = :rawProduct:

  proj₁ : Functor obj ℂ
  proj₁ = record
    { raw = record { func* = fst ; func→ = fst }
    ; isFunctor = record { isIdentity = refl ; isDistributive = refl }
    }

  proj₂ : Functor obj 𝔻
  proj₂ = record
    { raw = record { func* = snd ; func→ = snd }
    ; isFunctor = record { isIdentity = refl ; isDistributive = refl }
    }

  module _ {X : Category ℓ ℓ'} (x₁ : Functor X ℂ) (x₂ : Functor X 𝔻) where
    private
      x : Functor X obj
      x = record
        { raw = record
          { func* = λ x → x₁.func* x , x₂.func* x
          ; func→ = λ x → x₁.func→ x , x₂.func→ x
          }
        ; isFunctor = record
          { isIdentity   = Σ≡ x₁.isIdentity x₂.isIdentity
          ; isDistributive = Σ≡ x₁.isDistributive x₂.isDistributive
          }
        }
        where
          open module x₁ = Functor x₁
          open module x₂ = Functor x₂

      isUniqL : F[ proj₁ ∘ x ] ≡ x₁
      isUniqL = Functor≡ refl

      isUniqR : F[ proj₂ ∘ x ] ≡ x₂
      isUniqR = Functor≡ refl

      isUniq : F[ proj₁ ∘ x ] ≡ x₁ × F[ proj₂ ∘ x ] ≡ x₂
      isUniq = isUniqL , isUniqR

    isProduct : ∃![ x ] (F[ proj₁ ∘ x ] ≡ x₁ × F[ proj₂ ∘ x ] ≡ x₂)
    isProduct = x , isUniq

module _ {ℓ ℓ' : Level} (unprovable : IsCategory (RawCat ℓ ℓ')) where
  private
    Catℓ = Cat ℓ ℓ' unprovable

  module _ (ℂ 𝔻 : Category ℓ ℓ') where
    private
      module P = CatProduct ℂ 𝔻

      rawProduct : RawProduct {ℂ = Catℓ} ℂ 𝔻
      RawProduct.obj   rawProduct = P.obj
      RawProduct.proj₁ rawProduct = P.proj₁
      RawProduct.proj₂ rawProduct = P.proj₂

      isProduct : IsProduct Catℓ rawProduct
      IsProduct.isProduct isProduct = P.isProduct

    product : Product {ℂ = Catℓ} ℂ 𝔻
    Product.raw       product = rawProduct
    Product.isProduct product = isProduct

  instance
    hasProducts : HasProducts Catℓ
    hasProducts = record { product = product }

-- Basically proves that `Cat ℓ ℓ` is cartesian closed.
module CatExponential {ℓ : Level} (ℂ 𝔻 : Category ℓ ℓ) where
  open Data.Product
  open import Cat.Categories.Fun

  Categoryℓ = Category ℓ ℓ
  open Fun ℂ 𝔻 renaming (identity to idN)
  private
    :func*: : Functor ℂ 𝔻 × Object ℂ → Object 𝔻
    :func*: (F , A) = func* F A

  prodObj : Categoryℓ
  prodObj = Fun

  module _ {dom cod : Functor ℂ 𝔻 × Object ℂ} where
    private
      F : Functor ℂ 𝔻
      F = proj₁ dom
      A : Object ℂ
      A = proj₂ dom

      G : Functor ℂ 𝔻
      G = proj₁ cod
      B : Object ℂ
      B = proj₂ cod

    :func→: : (pobj : NaturalTransformation F G × ℂ [ A , B ])
      → 𝔻 [ func* F A , func* G B ]
    :func→: ((θ , θNat) , f) = result
      where
        θA : 𝔻 [ func* F A , func* G A ]
        θA = θ A
        θB : 𝔻 [ func* F B , func* G B ]
        θB = θ B
        F→f : 𝔻 [ func* F A , func* F B ]
        F→f = func→ F f
        G→f : 𝔻 [ func* G A , func* G B ]
        G→f = func→ G f
        l : 𝔻 [ func* F A , func* G B ]
        l = 𝔻 [ θB ∘ F→f ]
        r : 𝔻 [ func* F A , func* G B ]
        r = 𝔻 [ G→f ∘ θA ]
        -- There are two choices at this point,
        -- but I suppose the whole point is that
        -- by `θNat f` we have `l ≡ r`
        --     lem : 𝔻 [ θ B ∘ F .func→ f ] ≡ 𝔻 [ G .func→ f ∘ θ A ]
        --     lem = θNat f
        result : 𝔻 [ func* F A , func* G B ]
        result = l

  open CatProduct renaming (obj to _×p_) using ()

  module _ {c : Functor ℂ 𝔻 × Object ℂ} where
    private
      F : Functor ℂ 𝔻
      F = proj₁ c
      C : Object ℂ
      C = proj₂ c

    -- NaturalTransformation F G × ℂ .Arrow A B
    -- :ident: : :func→: {c} {c} (identityNat F , ℂ .𝟙) ≡ 𝔻 .𝟙
    -- :ident: = trans (proj₂ 𝔻.isIdentity) (F .isIdentity)
    --   where
    --     open module 𝔻 = IsCategory (𝔻 .isCategory)
    -- Unfortunately the equational version has some ambigous arguments.

    :ident: : :func→: {c} {c} (NT.identity F , 𝟙 ℂ {A = proj₂ c}) ≡ 𝟙 𝔻
    :ident: = begin
      :func→: {c} {c} (𝟙 (prodObj ×p ℂ) {c})    ≡⟨⟩
      :func→: {c} {c} (idN F , 𝟙 ℂ)             ≡⟨⟩
      𝔻 [ identityTrans F C ∘ func→ F (𝟙 ℂ)]    ≡⟨⟩
      𝔻 [ 𝟙 𝔻 ∘ func→ F (𝟙 ℂ)]                  ≡⟨ proj₂ 𝔻.isIdentity ⟩
      func→ F (𝟙 ℂ)                             ≡⟨ F.isIdentity ⟩
      𝟙 𝔻                                       ∎
      where
        open module 𝔻 = Category 𝔻
        open module F = Functor F

  module _ {F×A G×B H×C : Functor ℂ 𝔻 × Object ℂ} where
    F = F×A .proj₁
    A = F×A .proj₂
    G = G×B .proj₁
    B = G×B .proj₂
    H = H×C .proj₁
    C = H×C .proj₂
    -- Not entirely clear what this is at this point:
    _P⊕_ = Category._∘_ (prodObj ×p ℂ) {F×A} {G×B} {H×C}
    module _
      -- NaturalTransformation F G × ℂ .Arrow A B
      {θ×f : NaturalTransformation F G × ℂ [ A , B ]}
      {η×g : NaturalTransformation G H × ℂ [ B , C ]} where
      private
        θ : Transformation F G
        θ = proj₁ (proj₁ θ×f)
        θNat : Natural F G θ
        θNat = proj₂ (proj₁ θ×f)
        f : ℂ [ A , B ]
        f = proj₂ θ×f
        η : Transformation G H
        η = proj₁ (proj₁ η×g)
        ηNat : Natural G H η
        ηNat = proj₂ (proj₁ η×g)
        g : ℂ [ B , C ]
        g = proj₂ η×g

        ηθNT : NaturalTransformation F H
        ηθNT = Category._∘_ Fun {F} {G} {H} (η , ηNat) (θ , θNat)

        ηθ = proj₁ ηθNT
        ηθNat = proj₂ ηθNT

      :isDistributive: :
          𝔻 [ 𝔻 [ η C ∘ θ C ] ∘ func→ F ( ℂ [ g ∘ f ] ) ]
        ≡ 𝔻 [ 𝔻 [ η C ∘ func→ G g ] ∘ 𝔻 [ θ B ∘ func→ F f ] ]
      :isDistributive: = begin
        𝔻 [ (ηθ C) ∘ func→ F (ℂ [ g ∘ f ]) ]
          ≡⟨ ηθNat (ℂ [ g ∘ f ]) ⟩
        𝔻 [ func→ H (ℂ [ g ∘ f ]) ∘ (ηθ A) ]
          ≡⟨ cong (λ φ → 𝔻 [ φ ∘ ηθ A ]) (H.isDistributive) ⟩
        𝔻 [ 𝔻 [ func→ H g ∘ func→ H f ] ∘ (ηθ A) ]
          ≡⟨ sym isAssociative ⟩
        𝔻 [ func→ H g ∘ 𝔻 [ func→ H f ∘ ηθ A ] ]
          ≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) isAssociative ⟩
        𝔻 [ func→ H g ∘ 𝔻 [ 𝔻 [ func→ H f ∘ η A ] ∘ θ A ] ]
          ≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) (cong (λ φ → 𝔻 [ φ ∘ θ A ]) (sym (ηNat f))) ⟩
        𝔻 [ func→ H g ∘ 𝔻 [ 𝔻 [ η B ∘ func→ G f ] ∘ θ A ] ]
          ≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) (sym isAssociative) ⟩
        𝔻 [ func→ H g ∘ 𝔻 [ η B ∘ 𝔻 [ func→ G f ∘ θ A ] ] ]
          ≡⟨ isAssociative ⟩
        𝔻 [ 𝔻 [ func→ H g ∘ η B ] ∘ 𝔻 [ func→ G f ∘ θ A ] ]
          ≡⟨ cong (λ φ → 𝔻 [ φ ∘ 𝔻 [ func→ G f ∘ θ A ] ]) (sym (ηNat g)) ⟩
        𝔻 [ 𝔻 [ η C ∘ func→ G g ] ∘ 𝔻 [ func→ G f ∘ θ A ] ]
          ≡⟨ cong (λ φ → 𝔻 [ 𝔻 [ η C ∘ func→ G g ] ∘ φ ]) (sym (θNat f)) ⟩
        𝔻 [ 𝔻 [ η C ∘ func→ G g ] ∘ 𝔻 [ θ B ∘ func→ F f ] ] ∎
        where
          open Category 𝔻
          module H = Functor H

  eval : Functor (CatProduct.obj prodObj ℂ) 𝔻
  -- :eval: : Functor (prodObj ×p ℂ) 𝔻
  eval = record
    { raw = record
      { func* = :func*:
      ; func→ = λ {dom} {cod} → :func→: {dom} {cod}
      }
    ; isFunctor = record
      { isIdentity = λ {o} → :ident: {o}
      ; isDistributive = λ {f u n k y} → :isDistributive: {f} {u} {n} {k} {y}
      }
    }

  module _ (𝔸 : Category ℓ ℓ) (F : Functor (𝔸 ×p ℂ) 𝔻) where
    -- open HasProducts (hasProducts {ℓ} {ℓ} unprovable) renaming (_|×|_ to parallelProduct)

    postulate
      parallelProduct
        : Functor 𝔸 prodObj → Functor ℂ ℂ
        → Functor (𝔸 ×p ℂ) (prodObj ×p ℂ)
      transpose : Functor 𝔸 prodObj
      eq : F[ eval ∘ (parallelProduct transpose (identity {C = ℂ})) ] ≡ F
      -- eq : F[ :eval: ∘ {!!} ] ≡ F
      -- eq : Catℓ [ :eval: ∘ (HasProducts._|×|_ hasProducts transpose (𝟙 Catℓ {o = ℂ})) ] ≡ F
      -- eq' : (Catℓ [ :eval: ∘
      --   (record { product = product } HasProducts.|×| transpose)
      --   (𝟙 Catℓ)
      --   ])
      --   ≡ F

    -- For some reason after `e8215b2c051062c6301abc9b3f6ec67106259758`
    -- `catTranspose` makes Agda hang. catTranspose : ∃![ F~ ] (Catℓ [
    -- :eval: ∘ (parallelProduct F~ (𝟙 Catℓ {o = ℂ}))] ≡ F) catTranspose =
    -- transpose , eq

module _ (ℓ : Level) (unprovable : IsCategory (RawCat ℓ ℓ)) where
  private
    Catℓ : Category (lsuc (ℓ ⊔ ℓ)) (ℓ ⊔ ℓ)
    Catℓ = Cat ℓ ℓ unprovable
  module _ (ℂ 𝔻 : Category ℓ ℓ) where
    open CatExponential ℂ 𝔻 using (prodObj ; eval)
    -- Putting in the type annotation causes Agda to loop indefinitely.
    -- eval' : Functor (CatProduct.obj prodObj ℂ) 𝔻
    -- Likewise, using it below also results in this.
    eval' : _
    eval' = eval
  --   private
  --     -- module _ (ℂ 𝔻 : Category ℓ ℓ) where
  --       postulate :isExponential: : IsExponential Catℓ ℂ 𝔻 prodObj :eval:
  --       -- :isExponential: : IsExponential Catℓ ℂ 𝔻 :obj: :eval:
  --       -- :isExponential: = {!catTranspose!}
  --       --   where
  --       --     open HasProducts (hasProducts {ℓ} {ℓ} unprovable) using (_|×|_)
  --       -- :isExponential: = λ 𝔸 F → transpose 𝔸 F , eq' 𝔸 F

  --       -- :exponent: : Exponential (Cat ℓ ℓ) A B
    exponent : Exponential Catℓ ℂ 𝔻
    exponent = record
      { obj = prodObj
      ; eval = {!evalll'!}
      ; isExponential = {!:isExponential:!}
      }
      where
        open HasProducts (hasProducts unprovable) renaming (_×_ to _×p_)
        open import Cat.Categories.Fun
        open Fun
        -- _×p_ = CatProduct.obj -- prodObj ℂ
        -- eval' : Functor CatP.obj 𝔻

  hasExponentials : HasExponentials Catℓ
  hasExponentials = record { exponent = exponent }