agda-gset/Algebra/GSet.agda

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

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

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

  isEquivariant : ∀ {G : Group c ℓ} (X Y : G-Set G) (f : ∣ X ∣ → ∣ Y ∣) → Set c
  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 c ℓ) (X Y : G-Set G) : Set (suc (c ⊔ ℓ)) where
    field
      u : ∣ X ∣ → ∣ Y ∣ -- u for underlying
      isEqui : isEquivariant X Y u