{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.Algebra.Group.Morphism where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Structure
open import Cubical.Algebra.Group.Base
open import Cubical.Data.Sigma
private
variable
ℓ ℓ' ℓ'' ℓ''' : Level
isGroupHom : (G : Group {ℓ}) (H : Group {ℓ'}) (f : ⟨ G ⟩ → ⟨ H ⟩) → Type _
isGroupHom G H f = (x y : ⟨ G ⟩) → f (x G.+ y) ≡ (f x H.+ f y) where
module G = GroupStr (snd G)
module H = GroupStr (snd H)
record GroupHom (G : Group {ℓ}) (H : Group {ℓ'}) : Type (ℓ-max ℓ ℓ') where
constructor grouphom
field
fun : ⟨ G ⟩ → ⟨ H ⟩
isHom : isGroupHom G H fun
record GroupEquiv (G : Group {ℓ}) (H : Group {ℓ'}) : Type (ℓ-max ℓ ℓ') where
constructor groupequiv
field
eq : ⟨ G ⟩ ≃ ⟨ H ⟩
isHom : isGroupHom G H (equivFun eq)
hom : GroupHom G H
hom = grouphom (equivFun eq) isHom
open GroupHom
open GroupStr
×hom : {A : Group {ℓ}} {B : Group {ℓ'}} {C : Group {ℓ''}} {D : Group {ℓ'''}}
→ GroupHom A C → GroupHom B D → GroupHom (dirProd A B) (dirProd C D)
fun (×hom mf1 mf2) = map-× (fun mf1) (fun mf2)
isHom (×hom mf1 mf2) a b = ≡-× (isHom mf1 _ _) (isHom mf2 _ _)
isInIm : (G : Group {ℓ}) (H : Group {ℓ'}) → GroupHom G H → ⟨ H ⟩ → Type (ℓ-max ℓ ℓ')
isInIm G H ϕ h = ∃[ g ∈ ⟨ G ⟩ ] ϕ .fun g ≡ h
isInKer : (G : Group {ℓ}) (H : Group {ℓ'}) → GroupHom G H → ⟨ G ⟩ → Type ℓ'
isInKer G H ϕ g = ϕ .fun g ≡ 0g (snd H)
Ker : {G : Group {ℓ}} {H : Group {ℓ'}} → GroupHom G H → Type _
Ker {G = G} {H = H} ϕ = Σ[ x ∈ ⟨ G ⟩ ] isInKer G H ϕ x
Im : {G : Group {ℓ}} {H : Group {ℓ'}} → GroupHom G H → Type _
Im {G = G} {H = H} ϕ = Σ[ x ∈ ⟨ H ⟩ ] isInIm G H ϕ x
isSurjective : (G : Group {ℓ}) (H : Group {ℓ'}) → GroupHom G H → Type (ℓ-max ℓ ℓ')
isSurjective G H ϕ = (x : ⟨ H ⟩) → isInIm G H ϕ x
isInjective : (G : Group {ℓ}) (H : Group {ℓ'}) → GroupHom G H → Type (ℓ-max ℓ ℓ')
isInjective G H ϕ = (x : ⟨ G ⟩) → isInKer G H ϕ x → x ≡ 0g (snd G)