58 lines
2 KiB
Agda
58 lines
2 KiB
Agda
open import Algebra.Group
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open import Level
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open import Data.Product
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open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)
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module Algebra.G-Set {ℓ : Level} where
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open Group using () renaming (Carrier to ∣_∣)
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record G-Set (G : Group ℓ) : Set (suc ℓ) where
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open Group G using (ε; _∙_)
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field
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X : Set ℓ
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_⊳_ : ∣ G ∣ → X → X
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field
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ε⊳ : ∀ {x : X} → ε ⊳ x ≡ x
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∘⊳ : ∀ {g h : ∣ G ∣} {x : X} → (g ∙ h) ⊳ x ≡ (g ⊳ (h ⊳ x))
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-- orb : X → Set ℓ
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-- orb x = Σ[ y ∈ X ] Σ[ g ∈ ∣ G ∣ ] g ⊳ x ≡ y
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-- Orb : Set ℓ
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-- Orb = Σ[ x ∈ X ] orb x
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record orb (x : X) : Set ℓ where
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constructor orb[_,_,_]
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field
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y : X
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g : ∣ G ∣
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.eq : g ⊳ x ≡ y
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orb-intro : ∀ {x : X} {y z : X} {g h : ∣ G ∣} {eq₁ : g ⊳ x ≡ y} {eq₂ : h ⊳ x ≡ z} (p : y ≡ z) (q : g ≡ h)
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→ ≡.subst₂ (λ y g → g ⊳ x ≡ y) p q eq₁ ≡ eq₂
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→ orb[ y , g , eq₁ ] ≡ orb[ z , h , eq₂ ]
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orb-intro ≡.refl ≡.refl ≡.refl = ≡.refl
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record Orb : Set ℓ where
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constructor Orb[_,_]
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field
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x : X
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o : orb x
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Orb-intro : ∀ {x y : X} {o₁ : orb x} {o₂ : orb y} (p : x ≡ y)
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→ ≡.subst (λ x → orb x) p o₁ ≡ o₂
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→ Orb[ x , o₁ ] ≡ Orb[ y , o₂ ]
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Orb-intro ≡.refl ≡.refl = ≡.refl
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open G-Set using () renaming (X to ∣_∣)
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IsEquivariant : ∀ {G : Group ℓ} (X Y : G-Set G) (f : ∣ X ∣ → ∣ Y ∣) → Set ℓ
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IsEquivariant {G} X Y f = ∀ {g : ∣ G ∣} {x : ∣ X ∣} → f (g ⊳ˣ x) ≡ g ⊳ʸ (f x)
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where
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open G-Set X using () renaming (_⊳_ to _⊳ˣ_)
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open G-Set Y using () renaming (_⊳_ to _⊳ʸ_)
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record G-Set-Morphism (G : Group ℓ) (X Y : G-Set G) : Set (suc ℓ) where
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field
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u : ∣ X ∣ → ∣ Y ∣ -- u for underlying
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equivariance : IsEquivariant X Y u
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