cat/src/Cat/Categories/Cat.agda

{-# 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.Functor

-- Tip from Andrea:
-- Use co-patterns - they help with showing more understandable types in goals.
lift-eq : ∀ {ℓ} {A B : Set ℓ} {a a' : A} {b b' : B} → a ≡ a' → b ≡ b' → (a , b) ≡ (a' , b')
fst (lift-eq a b i) = a i
snd (lift-eq a b i) = b i

eqpair : ∀ {ℓa ℓb} {A : Set ℓa} {B : Set ℓb} {a a' : A} {b b' : B}
  → a ≡ a' → b ≡ b' → (a , b) ≡ (a' , b')
eqpair eqa eqb i = eqa i , eqb i

open Functor
open Category
module _ {ℓ ℓ' : Level} {A B : Category ℓ ℓ'} where
  lift-eq-functors : {f g : Functor A B}
    → (eq* : f .func* ≡ g .func*)
    → (eq→ : PathP (λ i → ∀ {x y} → A .Arrow x y → B .Arrow (eq* i x) (eq* i y))
    (f .func→) (g .func→))
    --        → (eq→ : Functor.func→ f ≡ {!!}) -- Functor.func→ g)
    -- Use PathP
    -- directly to show heterogeneous equalities by using previous
    -- equalities (i.e. continuous paths) to create new continuous paths.
    → (eqI : PathP (λ i → ∀ {c : A .Object} → eq→ i (A .𝟙 {c}) ≡ B .𝟙 {eq* i c})
    (ident f) (ident g))
    → (eqD : PathP (λ i → { c c' c'' : A .Object} {a : A .Arrow c c'} {a' : A .Arrow c' c''}
      → eq→ i (A ._⊕_ a' a) ≡ B ._⊕_ (eq→ i a') (eq→ i a))
      (distrib f) (distrib g))
    → f ≡ g
  lift-eq-functors eq* eq→ eqI eqD i = record { func* = eq* i ; func→ = eq→ i ; ident = eqI i ; distrib = eqD i }

-- 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
      eq* : func* (h ∘f (g ∘f f)) ≡ func* ((h ∘f g) ∘f f)
      eq* = refl
      eq→ : PathP
        (λ i → {x y : A .Object} → A .Arrow x y → D .Arrow (eq* i x) (eq* i y))
        (func→ (h ∘f (g ∘f f))) (func→ ((h ∘f g) ∘f f))
      eq→ = refl
      id-l = (h ∘f (g ∘f f)) .ident -- = func→ (h ∘f (g ∘f f)) (𝟙 A) ≡ 𝟙 D
      id-r = ((h ∘f g) ∘f f) .ident -- = func→ ((h ∘f g) ∘f f) (𝟙 A) ≡ 𝟙 D
      postulate eqI : PathP
                 (λ i → ∀ {c : A .Object} → eq→ i (A .𝟙 {c}) ≡ D .𝟙 {eq* i c})
                 (ident ((h ∘f (g ∘f f))))
                 (ident ((h ∘f g) ∘f f))
      postulate eqD : PathP (λ i → { c c' c'' : A .Object} {a : A .Arrow c c'} {a' : A .Arrow c' c''}
                        → eq→ i (A ._⊕_ a' a) ≡ D ._⊕_ (eq→ i a') (eq→ i a))
                        (distrib (h ∘f (g ∘f f))) (distrib ((h ∘f g) ∘f f))
      -- eqD = {!!}

      assc : h ∘f (g ∘f f) ≡ (h ∘f g) ∘f f
      assc = lift-eq-functors eq* eq→ eqI eqD

    module _ {A B : Category ℓ ℓ'} {f : Functor A B} where
      lem : (func* f) ∘ (func* (identity {C = A})) ≡ func* f
      lem = refl
      -- lemmm : func→ {C = A} {D = B} (f ∘f identity) ≡ func→ f
      lemmm : PathP
        (λ i →
        {x y : Object A} → Arrow A x y → Arrow B (func* f x) (func* f y))
        (func→ (f ∘f identity)) (func→ f)
      lemmm = refl
      postulate lemz : PathP (λ i → {c : A .Object} → PathP (λ _ → Arrow B (func* f c) (func* f c)) (func→ f (A .𝟙)) (B .𝟙))
                  (ident (f ∘f identity)) (ident f)
      -- lemz = {!!}
      postulate ident-r : f ∘f identity ≡ f
      -- ident-r = lift-eq-functors lem lemmm {!lemz!} {!!}
      postulate ident-l : identity ∘f f ≡ f
      -- ident-l = lift-eq-functors lem lemmm {!refl!} {!!}

  Cat : Category (lsuc (ℓ ⊔ ℓ')) (ℓ ⊔ ℓ')
  Cat =
    record
      { Object = Category ℓ ℓ'
      ; Arrow = Functor
      ; 𝟙 = identity
      ; _⊕_ = _∘f_
      -- What gives here? Why can I not name the variables directly?
      ; isCategory = record
        { assoc = λ {_ _ _ _ f g h} → assc {f = f} {g = g} {h = h}
        ; ident = ident-r , ident-l
        }
      }

module _ {ℓ : Level} (C D : Category ℓ ℓ) where
  private
    :Object: = C .Object × D .Object
    :Arrow:  : :Object: → :Object: → Set ℓ
    :Arrow: (c , d) (c' , d') = Arrow C c c' × Arrow D d d'
    :𝟙: : {o : :Object:} → :Arrow: o o
    :𝟙: = C .𝟙 , D .𝟙
    _:⊕:_ :
      {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}

    instance
      :isCategory: : IsCategory :Object: :Arrow: :𝟙: _:⊕:_
      :isCategory: = record
        { assoc = eqpair C.assoc D.assoc
        ; ident
        = eqpair (fst C.ident) (fst D.ident)
        , eqpair (snd C.ident) (snd D.ident)
        }
        where
          open module C = IsCategory (C .isCategory)
          open module D = IsCategory (D .isCategory)

    :product: : Category ℓ ℓ
    :product: = record
      { Object = :Object:
      ; Arrow = :Arrow:
      ; 𝟙 = :𝟙:
      ; _⊕_ = _:⊕:_
      }

    proj₁ : Arrow Cat :product: C
    proj₁ = record { func* = fst ; func→ = fst ; ident = refl ; distrib = refl }

    proj₂ : Arrow Cat :product: D
    proj₂ = record { func* = snd ; func→ = snd ; ident = refl ; distrib = refl }

    module _ {X : Object (Cat {ℓ} {ℓ})} (x₁ : Arrow Cat X C) (x₂ : Arrow Cat X D) where
      open Functor

      -- ident' : {c : Object X} → ((func→ x₁) {dom = c} (𝟙 X) , (func→ x₂) {dom = c} (𝟙 X)) ≡ 𝟙 (catProduct C D)
      -- ident' {c = c} = lift-eq (ident x₁) (ident x₂)

      x : Functor X :product:
      x = record
        { func* = λ x → (func* x₁) x , (func* x₂) x
        ; func→ = λ x → func→ x₁ x , func→ x₂ x
        ; ident = lift-eq (ident x₁) (ident x₂)
        ; distrib = lift-eq (distrib x₁) (distrib x₂)
        }

      -- Need to "lift equality of functors"
      -- If I want to do this like I do it for pairs it's gonna be a pain.
      postulate isUniqL : (Cat ⊕ proj₁) x ≡ x₁
      -- isUniqL = lift-eq-functors refl refl {!!} {!!}

      postulate isUniqR : (Cat ⊕ proj₂) x ≡ x₂
      -- isUniqR = lift-eq-functors refl refl {!!} {!!}

      isUniq : (Cat ⊕ proj₁) x ≡ x₁ × (Cat ⊕ proj₂) x ≡ x₂
      isUniq = isUniqL , isUniqR

      uniq : ∃![ x ] ((Cat ⊕ proj₁) x ≡ x₁ × (Cat ⊕ proj₂) x ≡ x₂)
      uniq = x , isUniq

    instance
      isProduct : IsProduct Cat proj₁ proj₂
      isProduct = uniq

  product : Product {ℂ = Cat} C D
  product = record
    { obj = :product:
    ; proj₁ = proj₁
    ; proj₂ = proj₂
    }