cat/src/Cat/Category/Properties.agda

{-# OPTIONS --allow-unsolved-metas --cubical #-}

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 {!!}
-}

  open import Cat.Categories.Cat
  open Exponential
  open HasExponentials CatHasExponentials

  Exp : Set {!!}
  Exp = Exponential (Cat {!!} {!!}) {{ℂHasProducts = {!!}}}
    Sets (Opposite {!!})

  -- _⇑_ : (A B : Catℓ .Object) → Catℓ .Object
  -- A ⇑ B = (exponent A B) .obj

  -- private
  --   :func*: : ℂ .Object → (Sets ⇑ Opposite ℂ) .Object
  --   :func*: x = {!!}

  -- yoneda : Functor ℂ (Sets ⇑ (Opposite ℂ))
  -- yoneda = record
  --   { func* = :func*:
  --   ; func→ = {!!}
  --   ; ident = {!!}
  --   ; distrib = {!!}
  --   }