47 lines
1.5 KiB
Agda
47 lines
1.5 KiB
Agda
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open import Algebra.G-Set
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open import Algebra.Group
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open import Category.G-Sets
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open import Categories.Category.Instance.Sets
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open import Categories.Functor.Core
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open import Categories.Category.Core
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open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)
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open import Relation.Binary.PropositionalEquality.WithK
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open import Axiom.UniquenessOfIdentityProofs.WithK
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open import Data.Product
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open import Data.Product.Properties renaming (Σ-≡,≡→≡ to peq)
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open import Level
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module Category.G-Sets.Properties.A4 {ℓ} (G : Group ℓ) where
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open Functor
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open G-Set-Morphism using () renaming (u to <_>)
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D : Functor (Sets ℓ) (G-Sets G)
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D .F₀ S = record
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{ X = S
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; _⊳_ = λ _ s → s
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; ε⊳ = ≡.refl
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; ∘⊳ = ≡.refl
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}
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D .F₁ {A} {B} f = record
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{ u = f
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; equivariance = ≡.refl
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}
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D .identity = ≡.refl
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D .homomorphism = ≡.refl
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D .F-resp-≈ f≡g = f≡g
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V : Functor (G-Sets G) (Sets ℓ)
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V .F₀ GS = Orb
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where
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open G-Set GS
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V .F₁ {A} {B} f (x , (y , (g , eq))) = (< f > x , (< f > y , (g , ≡.trans (≡.sym f.equivariance) (≡.cong < f > eq))))
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where
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open G-Set A
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module f = G-Set-Morphism f
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V .identity {GS} {(x , (y , (g , ≡.refl)))} = ≡.refl
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V .homomorphism {X} {Y} {Z} {f} {g} = {!!}
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V .F-resp-≈ {A} {B} {f} {g} eq {(x , (y , (h , ≡.refl)))} = peq (eq , {!!})
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where
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module f = G-Set-Morphism f
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module g = G-Set-Morphism g
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module A = G-Set A
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