{-# OPTIONS --warning=error --safe --without-K #-}

open import LogicalFormulae
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Numbers.Naturals.Naturals
open import Numbers.Naturals.Order
open import Vectors
open import Semirings.Definition
open import Categories.Definition
open import Groups.Definition

module Categories.Examples where

SET : {a : _} → Category {lsuc a} {a}
Category.objects (SET {a}) = Set a
Category.arrows (SET {a}) = λ a b → (a → b)
Category.id (SET {a}) = λ x → λ y → y
Category._∘_ (SET {a}) = λ f g → λ x → f (g x)
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}) = {!!}
Category.compositionAssociative (GROUP {a}) = {!!}

DISCRETE : {a : _} (X : Set a) → Category {a} {a}
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) = {!!}