29 lines
1 KiB
Agda
29 lines
1 KiB
Agda
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open import Algebra.Bundles
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open import Level
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open import Data.Product
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open import Relation.Binary.PropositionalEquality
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module GSet {c ℓ : Level} where
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open Group using () renaming (Carrier to ∣_∣)
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record G-Set (G : Group c ℓ) : Set (suc (c ⊔ ℓ)) where
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open Group G using (ε; _∙_)
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field
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X : Set c
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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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open G-Set using () renaming (X to ∣_∣)
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isEquivariant : ∀ {G : Group c ℓ} (X Y : G-Set G) (f : ∣ X ∣ → ∣ Y ∣) → Set c
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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 c ℓ) (X Y : G-Set G) : Set (suc (c ⊔ ℓ)) where
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field
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f : ∣ X ∣ → ∣ Y ∣
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isEqui : isEquivariant X Y f
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