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work on gsets category

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Leon Vatthauer 2024-05-04 15:02:01 +02:00
parent 12e3092a17
commit e5cdeaee38
Signed by: leonv
SSH key fingerprint: SHA256:G4+ddwoZmhLPRB1agvXzZMXIzkVJ36dUYZXf5NxT+u8
2 changed files with 34 additions and 3 deletions
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6
Algebra/GSet.agda
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@ -3,7 +3,7 @@ open import Level
open import Data.Product
open import Relation.Binary.PropositionalEquality
module GSet {c ℓ : Level} where
module Algebra.GSet {c ℓ : Level} where
open Group using () renaming (Carrier to ∣_∣)
record G-Set (G : Group c ℓ) : Set (suc (c ⊔ ℓ)) where
open Group G using (ε; _∙_)
@ -24,5 +24,5 @@ module GSet {c ℓ : Level} where
record G-Set-Morphism (G : Group c ℓ) (X Y : G-Set G) : Set (suc (c ⊔ ℓ)) where
field
f : ∣ X ∣ → ∣ Y ∣
isEqui : isEquivariant X Y f
u : ∣ X ∣ → ∣ Y ∣ -- u for underlying
isEqui : isEquivariant X Y u

31
Category/GSets.agda
View file

@ -1,2 +1,33 @@
open import Algebra.Bundles
open import Algebra.GSet
open import Categories.Category.Core
open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)
open import Level
module Category.GSets {c ℓ : Level} where
open Category
G-Sets : Group c ℓ → Category (suc (c ⊔ ℓ)) (suc c ⊔ suc ℓ) c
G-Sets G .Obj = G-Set G
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 <_>)