Implement homework
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11 changed files with 327 additions and 61 deletions
2
.gitignore
vendored
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.gitignore
vendored
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*.agdai
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_build/
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45
Algebra/G-Set.agda
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45
Algebra/G-Set.agda
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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
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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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-- 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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-- record Orb : Set ℓ where
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-- field
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-- x : X
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-- O : orb x
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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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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 Algebra.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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u : ∣ X ∣ → ∣ Y ∣ -- u for underlying
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isEqui : isEquivariant X Y u
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27
Algebra/Group.agda
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Algebra/Group.agda
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open import Level
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open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)
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module Algebra.Group where
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record IsGroup {ℓ} (A : Set ℓ) (_∙_ : A → A → A) (ε : A) (_⁻¹ : A → A) : Set (suc ℓ) where
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field
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assoc : ∀ {f g h : A} → (f ∙ g) ∙ h ≡ f ∙ (g ∙ h)
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idˡ : ∀ {g : A} → ε ∙ g ≡ g
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idʳ : ∀ {g : A} → g ∙ ε ≡ g
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invˡ : ∀ {g : A} → g ∙ (g ⁻¹) ≡ ε
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invʳ : ∀ {g : A} → (g ⁻¹) ∙ g ≡ ε
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record Group (ℓ : Level) : Set (suc ℓ) where
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infix 8 _⁻¹
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infixl 7 _∙_
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field
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Carrier : Set ℓ
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_∙_ : Carrier → Carrier → Carrier
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ε : Carrier
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_⁻¹ : Carrier → Carrier
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field
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isGroup : IsGroup Carrier _∙_ ε _⁻¹
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open IsGroup isGroup public
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∙-resp : ∀ {f h g i : Carrier} → f ≡ h → g ≡ i → f ∙ g ≡ h ∙ i
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∙-resp {f} {h} {g} {i} f≡h g≡i = ≡.trans (≡.cong (f ∙_) g≡i) (≡.cong (_∙ i) f≡h)
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27
Category/Construction/GroupAsCategory.agda
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Category/Construction/GroupAsCategory.agda
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open import Categories.Category.Core
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open import Algebra.Group
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open import Data.Unit.Polymorphic
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open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)
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module Category.Construction.GroupAsCategory {ℓ} (G : Group ℓ) where
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open Group G
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GroupAsCategory : Category ℓ ℓ ℓ
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GroupAsCategory = record
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{ Obj = ⊤
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; _⇒_ = λ _ _ → Carrier
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; _≈_ = _≡_
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; id = ε
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; _∘_ = _∙_
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; assoc = assoc
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; sym-assoc = ≡.sym assoc
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; identityˡ = idˡ
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; identityʳ = idʳ
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; identity² = idˡ
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; equiv = record
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{ refl = ≡.refl
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; sym = ≡.sym
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; trans = ≡.trans
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}
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; ∘-resp-≈ = ∙-resp
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}
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38
Category/G-Sets.agda
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Category/G-Sets.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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46
Category/G-Sets/Properties/A4.agda
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Category/G-Sets/Properties/A4.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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90
Category/G-Sets/Properties/Equivalence.agda
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90
Category/G-Sets/Properties/Equivalence.agda
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open import Category.G-Sets
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open import Algebra.Group
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open import Algebra.G-Set
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open import Category.Construction.GroupAsCategory
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open import Categories.Category.Construction.Functors renaming (Functors to [_,_])
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open import Categories.Category.Equivalence
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open import Categories.Category.Instance.Sets
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open import Categories.NaturalTransformation renaming (id to idN)
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open import Categories.NaturalTransformation.NaturalIsomorphism
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open import Categories.Functor.Core
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open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)
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open import Data.Unit.Polymorphic
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-- The category 'G-Sets' is equivalent to the category [ G , Sets ]
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module Category.G-Sets.Properties.Equivalence {ℓ} (G : Group ℓ) where
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open StrongEquivalence renaming (G to H)
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open WeakInverse renaming (F∘G≈id to F∘H≅id; G∘F≈id to H∘F≅id)
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open Functor
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open NaturalIsomorphism
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G-Sets≅[G,Sets] : StrongEquivalence (G-Sets G) [ GroupAsCategory G , Sets ℓ ]
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-- F : G-Sets → [ G , Sets ]
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G-Sets≅[G,Sets] .F .F₀ GS = record
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{ F₀ = λ _ → X
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; F₁ = λ f x → f ⊳ x
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; identity = ε⊳
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; homomorphism = ∘⊳
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; F-resp-≈ = λ f≡g {x} → ≡.cong (_⊳ x) f≡g
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}
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where open G-Set GS
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G-Sets≅[G,Sets] .F .F₁ {A} {B} f = ntHelper (record { η = λ X x → f.u x ; commute = λ g → f.equivariance })
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where
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module f = G-Set-Morphism f
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G-Sets≅[G,Sets] .F .identity = ≡.refl
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G-Sets≅[G,Sets] .F .homomorphism = ≡.refl
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G-Sets≅[G,Sets] .F .F-resp-≈ = λ x → x
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-- H : [ G , Sets ] → G-Sets
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G-Sets≅[G,Sets] .H .F₀ M = record
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{ X = M.₀ tt
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; _⊳_ = λ g x → (M.₁ g) x
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; ε⊳ = M.identity
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; ∘⊳ = M.homomorphism
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}
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where
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module M = Functor M
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G-Sets≅[G,Sets] .H .F₁ {M} {N} α = record
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{ u = α.η tt
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; equivariance = λ {g} {x} → α.commute g
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}
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where
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module α = NaturalTransformation α
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G-Sets≅[G,Sets] .H .identity = ≡.refl
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G-Sets≅[G,Sets] .H .homomorphism = ≡.refl
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G-Sets≅[G,Sets] .H .F-resp-≈ = λ x → x
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-- F∘H = Id
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G-Sets≅[G,Sets] .weak-inverse .F∘H≅id .F⇒G = ntHelper (record
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{ η = λ M → ntHelper (record
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{ η = λ _ x → x
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; commute = λ f {x} → ≡.refl
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})
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; commute = λ {M} {N} α {x} → ≡.refl
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})
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G-Sets≅[G,Sets] .weak-inverse .F∘H≅id .F⇐G = ntHelper (record
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{ η = λ M → ntHelper (record
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{ η = λ _ x → x
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; commute = λ f {x} → ≡.refl
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})
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; commute = λ {M} {N} α {x} → ≡.refl
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})
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G-Sets≅[G,Sets] .weak-inverse .F∘H≅id .iso M = record { isoˡ = ≡.refl ; isoʳ = ≡.refl }
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-- H∘F = Id
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G-Sets≅[G,Sets] .weak-inverse .H∘F≅id .F⇒G = ntHelper (record
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{ η = λ GS → record
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{ u = λ x → x
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; equivariance = ≡.refl
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}
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; commute = λ {GS} {GM} f {x} → ≡.refl
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})
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G-Sets≅[G,Sets] .weak-inverse .H∘F≅id .F⇐G = ntHelper (record
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{ η = λ GS → record
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{ u = λ x → x
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; equivariance = ≡.refl
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}
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; commute = λ {GS} {GM} f {x} → ≡.refl
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})
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G-Sets≅[G,Sets] .weak-inverse .H∘F≅id .iso GS = record { isoˡ = ≡.refl ; isoʳ = ≡.refl }
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48
Category/G-Sets/Properties/FreeObject.agda
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48
Category/G-Sets/Properties/FreeObject.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.FreeObjects.Free
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open import Categories.Functor.Core
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open import Categories.Category.Instance.Sets
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open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)
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open import Data.Product
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-- The category G-Sets has free objects
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module Category.G-Sets.Properties.FreeObject {ℓ} (G : Group ℓ) where
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open Group using () renaming (Carrier to ∣_∣)
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open G-Set using () renaming (X to ∣_∣)
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open G-Set-Morphism using () renaming (u to <_>)
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open FreeObject
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open Group G hiding (Carrier)
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forgetfulF : Functor (G-Sets G) (Sets ℓ)
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forgetfulF = record
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{ F₀ = λ GS → ∣ GS ∣
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; F₁ = λ f → < f >
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; identity = ≡.refl
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; homomorphism = ≡.refl
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; F-resp-≈ = λ x → x
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}
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free-G-Set : ∀ {X : Set ℓ} → FreeObject forgetfulF X
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free-G-Set {X} .FX = record
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{ X = ∣ G ∣ × X
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; _⊳_ = λ g (h , x) → (g ∙ h , x)
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; ε⊳ = λ {(g , x)} → ≡.cong (_, x) idˡ
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; ∘⊳ = λ {g} {h} {(f , x)} → ≡.cong (_, x) assoc
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}
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free-G-Set .η x = (ε , x)
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free-G-Set {X} ._* {GS} f = record
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{ u = λ (g , x) → g ⊳ f x
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; equivariance = λ {g} {x} → ∘⊳
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}
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where open G-Set GS
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free-G-Set {X} .*-lift {GS} f {x} = ε⊳
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where open G-Set GS
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free-G-Set {X} .*-uniq {GS} f f' g-prop {(h , y)} = ≡.trans (≡.cong < f' > (≡.cong (_, y) (≡.sym idʳ)) )
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(≡.trans (f'.equivariance)
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(≡.cong (h ⊳_) g-prop))
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where
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open G-Set GS
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module f' = G-Set-Morphism f'
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open import Algebra.Bundles
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open import Algebra.GSet
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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.GSets {c ℓ : Level} where
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open Category
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G-Sets : Group c ℓ → Category (suc (c ⊔ ℓ)) (suc c ⊔ suc ℓ) c
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G-Sets G .Obj = G-Set G
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G-Sets G ._⇒_ = G-Set-Morphism G
|
||||
G-Sets G ._≈_ f g = ∀ {x} → f.u x ≡ g.u x
|
||||
where
|
||||
module f = G-Set-Morphism f
|
||||
module g = G-Set-Morphism g
|
||||
G-Sets G .id = record { u = λ x → x ; isEqui = ≡.refl }
|
||||
G-Sets G ._∘_ f g .G-Set-Morphism.u x = f.u (g.u x)
|
||||
where
|
||||
module f = G-Set-Morphism f
|
||||
module g = G-Set-Morphism g
|
||||
-- TODO without rewrite
|
||||
G-Sets G ._∘_ {X} {Y} {Z} f g .G-Set-Morphism.isEqui {h} {x} rewrite G-Set-Morphism.isEqui g {h} {x} | G-Set-Morphism.isEqui f {h} {G-Set-Morphism.u g x} = ≡.refl
|
||||
where
|
||||
module f = G-Set-Morphism f
|
||||
module g = G-Set-Morphism g
|
||||
G-Sets G .assoc = ≡.refl
|
||||
G-Sets G .sym-assoc = ≡.refl
|
||||
G-Sets G .identityˡ = ≡.refl
|
||||
G-Sets G .identityʳ = ≡.refl
|
||||
G-Sets G .identity² = ≡.refl
|
||||
G-Sets G .equiv = record { refl = ≡.refl ; sym = λ eq → ≡.sym eq ; trans = λ eq₁ eq₂ → ≡.trans eq₁ eq₂ }
|
||||
G-Sets G .∘-resp-≈ {X} {Y} {Z} {f} {h} {g} {i} f≈h g≈i = ≡.trans f≈h (≡.cong < h > g≈i)
|
||||
where open G-Set-Morphism using () renaming (u to <_>)
|
||||
4
Preliminaries.agda
Normal file
4
Preliminaries.agda
Normal file
|
|
@ -0,0 +1,4 @@
|
|||
module Preliminaries where
|
||||
open import Relation.Binary.PropositionalEquality as ≡ using (_≡_) public
|
||||
postulate
|
||||
FunExt : ∀ {ℓ} {X Y : Set ℓ} {f g : X → Y} → (∀ (x : X) → f x ≡ g x) → f ≡ g
|
||||
Loading…
Reference in a new issue