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Module-ify

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Frederik Hanghøj Iversen 2018-02-25 14:27:37 +01:00
parent 12dddc2067
commit ce46e0ae7a
1 changed files with 26 additions and 25 deletions
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51
src/Cat/Category/Properties.agda
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@ -8,38 +8,39 @@ open import Cubical
open import Cat.Category open import Cat.Category
open import Cat.Category.Functor open import Cat.Category.Functor
open import Cat.Categories.Sets
open import Cat.Equality open import Cat.Equality
open Equality.Data.Product open Equality.Data.Product
module _ {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} { A B : Category.Object ℂ } {X : Category.Object ℂ} (f : Category.Arrow ℂ A B) where -- Maybe inline into RawCategory"
module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
open Category ℂ open Category ℂ
iso-is-epi : Isomorphism f → Epimorphism {X = X} f module _ {A B : Category.Object ℂ } {X : Category.Object ℂ} (f : Category.Arrow ℂ A B) where
iso-is-epi (f- , left-inv , right-inv) g₀ g₁ eq = begin iso-is-epi : Isomorphism f → Epimorphism {X = X} f
g₀ ≡⟨ sym (proj₁ isIdentity) ⟩ iso-is-epi (f- , left-inv , right-inv) g₀ g₁ eq = begin
g₀ ∘ 𝟙 ≡⟨ cong (_∘_ g₀) (sym right-inv) ⟩ g₀ ≡⟨ sym (proj₁ isIdentity) ⟩
g₀ ∘ (f ∘ f-) ≡⟨ isAssociative ⟩ g₀ ∘ 𝟙 ≡⟨ cong (_∘_ g₀) (sym right-inv) ⟩
(g₀ ∘ f) ∘ f- ≡⟨ cong (λ φ → φ ∘ f-) eq ⟩ g₀ ∘ (f ∘ f-) ≡⟨ isAssociative ⟩
(g₁ ∘ f) ∘ f- ≡⟨ sym isAssociative ⟩ (g₀ ∘ f) ∘ f- ≡⟨ cong (λ φ → φ ∘ f-) eq ⟩
g₁ ∘ (f ∘ f-) ≡⟨ cong (_∘_ g₁) right-inv ⟩ (g₁ ∘ f) ∘ f- ≡⟨ sym isAssociative ⟩
g₁ ∘ 𝟙 ≡⟨ proj₁ isIdentity ⟩ g₁ ∘ (f ∘ f-) ≡⟨ cong (_∘_ g₁) right-inv ⟩
g₁ ∎ g₁ ∘ 𝟙 ≡⟨ proj₁ isIdentity ⟩
g₁ ∎
iso-is-mono : Isomorphism f → Monomorphism {X = X} f iso-is-mono : Isomorphism f → Monomorphism {X = X} f
iso-is-mono (f- , (left-inv , right-inv)) g₀ g₁ eq = iso-is-mono (f- , (left-inv , right-inv)) g₀ g₁ eq =
begin begin
g₀ ≡⟨ sym (proj₂ isIdentity) ⟩ g₀ ≡⟨ sym (proj₂ isIdentity) ⟩
𝟙 ∘ g₀ ≡⟨ cong (λ φ → φ ∘ g₀) (sym left-inv) ⟩ 𝟙 ∘ g₀ ≡⟨ cong (λ φ → φ ∘ g₀) (sym left-inv) ⟩
(f- ∘ f) ∘ g₀ ≡⟨ sym isAssociative ⟩ (f- ∘ f) ∘ g₀ ≡⟨ sym isAssociative ⟩
f- ∘ (f ∘ g₀) ≡⟨ cong (_∘_ f-) eq ⟩ f- ∘ (f ∘ g₀) ≡⟨ cong (_∘_ f-) eq ⟩
f- ∘ (f ∘ g₁) ≡⟨ isAssociative ⟩ f- ∘ (f ∘ g₁) ≡⟨ isAssociative ⟩
(f- ∘ f) ∘ g₁ ≡⟨ cong (λ φ → φ ∘ g₁) left-inv ⟩ (f- ∘ f) ∘ g₁ ≡⟨ cong (λ φ → φ ∘ g₁) left-inv ⟩
𝟙 ∘ g₁ ≡⟨ proj₂ isIdentity ⟩ 𝟙 ∘ g₁ ≡⟨ proj₂ isIdentity ⟩
g₁ ∎ g₁ ∎
iso-is-epi-mono : Isomorphism f → Epimorphism {X = X} f × Monomorphism {X = X} f iso-is-epi-mono : Isomorphism f → Epimorphism {X = X} f × Monomorphism {X = X} f
iso-is-epi-mono iso = iso-is-epi iso , iso-is-mono iso iso-is-epi-mono iso = iso-is-epi iso , iso-is-mono iso
-- TODO: We want to avoid defining the yoneda embedding going through the -- TODO: We want to avoid defining the yoneda embedding going through the
-- category of categories (since it doesn't exist). -- category of categories (since it doesn't exist).