39 lines
1.2 KiB
Agda
39 lines
1.2 KiB
Agda
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open import Algebra.Group
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open import Algebra.G-Set
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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 Level
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module Category.G-Sets {ℓ : Level} where
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open Category
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open G-Set-Morphism using () renaming (u to <_>)
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G-Sets : Group ℓ → Category (suc ℓ) (suc ℓ) ℓ
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G-Sets G .Obj = G-Set G
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G-Sets G ._⇒_ = G-Set-Morphism G
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G-Sets G ._≈_ f g = ∀ {x} → f.u x ≡ g.u x
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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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G-Sets G .id = record
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{ u = λ x → x
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; equivariance = ≡.refl
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}
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G-Sets G ._∘_ f g = record
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{ u = λ x → f.u (g.u x)
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; equivariance = ≡.trans (≡.cong < f > g.equivariance) f.equivariance
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}
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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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G-Sets G .assoc = ≡.refl
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G-Sets G .sym-assoc = ≡.refl
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G-Sets G .identityˡ = ≡.refl
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G-Sets G .identityʳ = ≡.refl
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G-Sets G .identity² = ≡.refl
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G-Sets G .equiv = record
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{ refl = ≡.refl
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; sym = λ eq → ≡.sym eq
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; trans = λ eq₁ eq₂ → ≡.trans eq₁ eq₂
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}
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G-Sets G .∘-resp-≈ {X} {Y} {Z} {f} {h} {g} {i} f≡h g≡i = ≡.trans f≡h (≡.cong < h > g≡i)
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