open import Algebra.Group
open import Level
open import Data.Product
open import Relation.Binary.PropositionalEquality

module Algebra.G-Set {ℓ : Level} where
  open Group using () renaming (Carrier to ∣_∣)
  record G-Set (G : Group ℓ) : Set (suc ℓ) where
    open Group G using (ε; _∙_)
    field
      X : Set ℓ
      _⊳_ : ∣ G ∣ → X → X
    field
      ε⊳ : ∀ {x : X} → ε ⊳ x ≡ x
      ∘⊳ : ∀ {g h : ∣ G ∣} {x : X} → (g ∙ h) ⊳ x ≡ (g ⊳ (h ⊳ x))

    orb : X → Set ℓ
    orb x = Σ[ y ∈ X ] Σ[ g ∈ ∣ G ∣ ] g ⊳ x ≡ y

    Orb : Set ℓ
    Orb = Σ[ x ∈ X ] orb x

    -- record orb (x : X) : Set ℓ where
    --   field
    --     y : X
    --     g : ∣ G ∣
    --     .eq : g ⊳ x ≡ y

    -- record Orb : Set ℓ where
    --   field
    --     x : X
    --     O : orb x

  open G-Set using () renaming (X to ∣_∣)

  IsEquivariant : ∀ {G : Group ℓ} (X Y : G-Set G) (f : ∣ X ∣ → ∣ Y ∣) → Set ℓ
  IsEquivariant {G} X Y f = ∀ {g : ∣ G ∣} {x : ∣ X ∣} → f (g ⊳ˣ x) ≡ g ⊳ʸ (f x)
    where
      open G-Set X using () renaming (_⊳_ to _⊳ˣ_)
      open G-Set Y using () renaming (_⊳_ to _⊳ʸ_)

  record G-Set-Morphism (G : Group ℓ) (X Y : G-Set G) : Set (suc ℓ) where
    field
      u : ∣ X ∣ → ∣ Y ∣ -- u for underlying
      equivariance : IsEquivariant X Y u