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Some graphs stuff (#98)

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Patrick Stevens
2020-02-15 13:41:51 +00:00
committed by GitHub
parent 9de323c5e8
commit f41f5226b9
18 changed files with 275 additions and 39 deletions
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25
Categories/Examples.agda
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@@ -8,6 +8,11 @@ open import Vectors
open import Semirings.Definition
open import Categories.Definition
open import Groups.Definition
open import Setoids.Setoids
open import Groups.Homomorphisms.Definition
open import Groups.Homomorphisms.Examples
open import Groups.Homomorphisms.Lemmas
open import Functions
module Categories.Examples where
@@ -20,13 +25,13 @@ Category.rightId (SET {a}) = λ f → refl
Category.leftId (SET {a}) = λ f → refl
Category.compositionAssociative (SET {a}) = λ f g h → refl
GROUP : {a b : _} → Category {lsuc a ⊔ b} {a ⊔ b}
Category.objects (GROUP {a}) = Group {!!} {!!}
Category.arrows (GROUP {a}) = {!!}
Category.id (GROUP {a}) = {!!}
Category._∘_ (GROUP {a}) = {!!}
Category.rightId (GROUP {a}) = {!!}
Category.leftId (GROUP {a}) = {!!}
GROUP : {a b : _} → Category {lsuc a ⊔ lsuc b} {a ⊔ b}
Category.objects (GROUP {a} {b}) = Sg (Set a) (λ A → Sg (A → A → A) (λ _+_ → Sg (Setoid {a} {b} A) (λ S → Group S _+_)))
Category.arrows (GROUP {a}) (A , (_+A_ , (S , G))) (B , (_+B_ , (T , H))) = Sg (A → B) (GroupHom G H)
Category.id (GROUP {a}) (A , (_+A_ , (S , G))) = (λ i → i) , identityHom G
Category._∘_ (GROUP {a}) {A , (_+A_ , (S , G))} {B , (_+B_ , (T , H))} {C , (_+C_ , (U , I))} (f , fHom) (g , gHom) = (f ∘ g) , groupHomsCompose gHom fHom
Category.rightId (GROUP {a}) {A , (_+A_ , (S , G))} {B , (_+B_ , (T , H))} (f , fHom) = {!!}
Category.leftId (GROUP {a}) {A , (_+A_ , (S , G))} {B , (_+B_ , (T , H))} (f , fHom) = {!!}
Category.compositionAssociative (GROUP {a}) = {!!}
DISCRETE : {a : _} (X : Set a) → Category {a} {a}
@@ -34,6 +39,6 @@ Category.objects (DISCRETE X) = X
Category.arrows (DISCRETE X) = λ a b → a ≡ b
Category.id (DISCRETE X) = λ x → refl
Category._∘_ (DISCRETE X) = λ y=z x=y → transitivity x=y y=z
Category.rightId (DISCRETE X) = {!!}
Category.leftId (DISCRETE X) = {!!}
Category.compositionAssociative (DISCRETE X) = {!!}
Category.rightId (DISCRETE X) refl = refl
Category.leftId (DISCRETE X) refl = refl
Category.compositionAssociative (DISCRETE X) refl refl refl = refl