Move lemma into IsCategory

This commit is contained in:
Frederik Hanghøj Iversen 2018-02-25 14:44:03 +01:00
parent d63ecc3a65
commit 2e7220567a
2 changed files with 26 additions and 31 deletions

View file

@ -107,6 +107,32 @@ record IsCategory {a b : Level} ( : RawCategory a b) : Set (lsuc
isIdentity : IsIdentity 𝟙 isIdentity : IsIdentity 𝟙
arrowsAreSets : ArrowsAreSets arrowsAreSets : ArrowsAreSets
univalent : Univalent isIdentity univalent : Univalent isIdentity
module _ {A B : Object} {X : Object} (f : Arrow A B) where
iso-is-epi : Isomorphism f Epimorphism {X = X} f
iso-is-epi (f- , left-inv , right-inv) g₀ g₁ eq = begin
g₀ ≡⟨ sym (fst isIdentity)
g₀ 𝟙 ≡⟨ cong (_∘_ g₀) (sym right-inv)
g₀ (f f-) ≡⟨ isAssociative
(g₀ f) f- ≡⟨ cong (λ φ φ f-) eq
(g₁ f) f- ≡⟨ sym isAssociative
g₁ (f f-) ≡⟨ cong (_∘_ g₁) right-inv
g₁ 𝟙 ≡⟨ fst isIdentity
g₁
iso-is-mono : Isomorphism f Monomorphism {X = X} f
iso-is-mono (f- , (left-inv , right-inv)) g₀ g₁ eq =
begin
g₀ ≡⟨ sym (snd isIdentity)
𝟙 g₀ ≡⟨ cong (λ φ φ g₀) (sym left-inv)
(f- f) g₀ ≡⟨ sym isAssociative
f- (f g₀) ≡⟨ cong (_∘_ f-) eq
f- (f g₁) ≡⟨ isAssociative
(f- f) g₁ ≡⟨ cong (λ φ φ g₁) left-inv
𝟙 g₁ ≡⟨ snd isIdentity
g₁
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
-- `IsCategory` is a mere proposition. -- `IsCategory` is a mere proposition.
module _ {a b : Level} {C : RawCategory a b} where module _ {a b : Level} {C : RawCategory a b} where

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@ -11,37 +11,6 @@ open import Cat.Category.Functor
open import Cat.Equality open import Cat.Equality
open Equality.Data.Product open Equality.Data.Product
-- Maybe inline into RawCategory"
module _ {a b : Level} ( : Category a b) where
open Category
module _ {A B : Object} {X : Object} (f : Arrow A B) where
iso-is-epi : Isomorphism f Epimorphism {X = X} f
iso-is-epi (f- , left-inv , right-inv) g₀ g₁ eq = begin
g₀ ≡⟨ sym (proj₁ isIdentity)
g₀ 𝟙 ≡⟨ cong (_∘_ g₀) (sym right-inv)
g₀ (f f-) ≡⟨ isAssociative
(g₀ f) f- ≡⟨ cong (λ φ φ f-) eq
(g₁ f) f- ≡⟨ sym isAssociative
g₁ (f f-) ≡⟨ cong (_∘_ g₁) right-inv
g₁ 𝟙 ≡⟨ proj₁ isIdentity
g₁
iso-is-mono : Isomorphism f Monomorphism {X = X} f
iso-is-mono (f- , (left-inv , right-inv)) g₀ g₁ eq =
begin
g₀ ≡⟨ sym (proj₂ isIdentity)
𝟙 g₀ ≡⟨ cong (λ φ φ g₀) (sym left-inv)
(f- f) g₀ ≡⟨ sym isAssociative
f- (f g₀) ≡⟨ cong (_∘_ f-) eq
f- (f g₁) ≡⟨ isAssociative
(f- f) g₁ ≡⟨ cong (λ φ φ g₁) left-inv
𝟙 g₁ ≡⟨ proj₂ isIdentity
g₁
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
-- 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).
open import Cat.Categories.Cat using (RawCat) open import Cat.Categories.Cat using (RawCat)