module Algebra.Group.Ab where
Abelian groups🔗
A very important class of groups (which includes most familiar examples of groups: the integers, all finite cyclic groups, etc) are those with a commutative group operation, that is, those for which Accordingly, these have a name reflecting their importance and ubiquity: They are called commutative groups. Just kidding! They’re named abelian groups, named after a person, because nothing can have self-explicative names in mathematics. It’s the law.
private variable ℓ : Level G : Type ℓ Group-on-is-abelian : Group-on G → Type _ Group-on-is-abelian G = ∀ x y → Group-on._⋆_ G x y ≡ Group-on._⋆_ G y x Group-on-is-abelian-is-prop : (g : Group-on G) → is-prop (Group-on-is-abelian g) Group-on-is-abelian-is-prop g = Π-is-hlevel² 1 λ _ _ → g .Group-on.has-is-set _ _
This module does the usual algebraic structure dance to set up the category of Abelian groups.
record is-abelian-group (_*_ : G → G → G) : Type (level-of G) where no-eta-equality field has-is-group : is-group _*_ commutes : ∀ {x y} → x * y ≡ y * x open is-group has-is-group renaming (unit to 1g) public
equal-sum→equal-diff : ∀ a b c d → a * b ≡ c * d → a — c ≡ d — b equal-sum→equal-diff a b c d p = commutes ∙ swizzle p inverser inversel
private unquoteDecl eqv = declare-record-iso eqv (quote is-abelian-group) instance H-Level-is-abelian-group : ∀ {n} {* : G → G → G} → H-Level (is-abelian-group *) (suc n) H-Level-is-abelian-group = prop-instance $ Iso→is-hlevel 1 eqv $ Σ-is-hlevel 1 (hlevel 1) λ x → Π-is-hlevel²' 1 λ _ _ → is-group.has-is-set x _ _
record Abelian-group-on (T : Type ℓ) : Type ℓ where no-eta-equality field _*_ : T → T → T has-is-ab : is-abelian-group _*_ open is-abelian-group has-is-ab renaming (inverse to infixl 30 _⁻¹) public
Abelian→Group-on : Group-on T Abelian→Group-on .Group-on._⋆_ = _*_ Abelian→Group-on .Group-on.has-is-group = has-is-group Abelian→Group-on-abelian : Group-on-is-abelian Abelian→Group-on Abelian→Group-on-abelian _ _ = commutes infixr 20 _*_ open Abelian-group-on using (Abelian→Group-on; Abelian→Group-on-abelian) public
Abelian-group-structure : ∀ ℓ → Thin-structure ℓ Abelian-group-on ∣ Abelian-group-structure ℓ .is-hom f G₁ G₂ ∣ = is-group-hom (Abelian→Group-on G₁) (Abelian→Group-on G₂) f Abelian-group-structure ℓ .is-hom f G₁ G₂ .is-tr = hlevel 1 Abelian-group-structure ℓ .id-is-hom .is-group-hom.pres-⋆ x y = refl Abelian-group-structure ℓ .∘-is-hom f g α β .is-group-hom.pres-⋆ x y = ap f (β .is-group-hom.pres-⋆ x y) ∙ α .is-group-hom.pres-⋆ (g x) (g y) Abelian-group-structure ℓ .id-hom-unique {s = s} {t} α _ = p where open Abelian-group-on p : s ≡ t p i ._*_ x y = α .is-group-hom.pres-⋆ x y i p i .has-is-ab = is-prop→pathp (λ i → hlevel {T = is-abelian-group (λ x y → p i ._*_ x y)} 1) (s .has-is-ab) (t .has-is-ab) i Ab : ∀ ℓ → Precategory (lsuc ℓ) ℓ Ab ℓ = Structured-objects (Abelian-group-structure ℓ) module Ab {ℓ} = Cat.Reasoning (Ab ℓ) instance Ab-equational : ∀ {ℓ} → is-equational (Abelian-group-structure ℓ) Ab-equational .is-equational.invert-id-hom = Groups-equational .is-equational.invert-id-hom
Abelian-group : (ℓ : Level) → Type (lsuc ℓ) Abelian-group _ = Ab.Ob Abelian→Group : ∀ {ℓ} → Abelian-group ℓ → Group ℓ Abelian→Group G = G .fst , Abelian→Group-on (G .snd) record make-abelian-group (T : Type ℓ) : Type ℓ where no-eta-equality field ab-is-set : is-set T mul : T → T → T inv : T → T 1g : T idl : ∀ x → mul 1g x ≡ x assoc : ∀ x y z → mul x (mul y z) ≡ mul (mul x y) z invl : ∀ x → mul (inv x) x ≡ 1g comm : ∀ x y → mul x y ≡ mul y x make-abelian-group→make-group : make-group T make-abelian-group→make-group = mg where mg : make-group T mg .make-group.group-is-set = ab-is-set mg .make-group.unit = 1g mg .make-group.mul = mul mg .make-group.inv = inv mg .make-group.assoc = assoc mg .make-group.invl = invl mg .make-group.idl = idl to-is-abelian-group : is-abelian-group mul to-is-abelian-group .is-abelian-group.has-is-group = to-is-group make-abelian-group→make-group to-is-abelian-group .is-abelian-group.commutes = comm _ _ to-group-on-ab : Group-on T to-group-on-ab = to-group-on make-abelian-group→make-group to-abelian-group-on : Abelian-group-on T to-abelian-group-on .Abelian-group-on._*_ = mul to-abelian-group-on .Abelian-group-on.has-is-ab = to-is-abelian-group to-ab : Abelian-group ℓ ∣ to-ab .fst ∣ = T to-ab .fst .is-tr = ab-is-set to-ab .snd = to-abelian-group-on is-commutative-group : ∀ {ℓ} → Group ℓ → Type ℓ is-commutative-group G = Group-on-is-abelian (G .snd) from-commutative-group : ∀ {ℓ} (G : Group ℓ) → is-commutative-group G → Abelian-group ℓ from-commutative-group G comm .fst = G .fst from-commutative-group G comm .snd .Abelian-group-on._*_ = Group-on._⋆_ (G .snd) from-commutative-group G comm .snd .Abelian-group-on.has-is-ab .is-abelian-group.has-is-group = Group-on.has-is-group (G .snd) from-commutative-group G comm .snd .Abelian-group-on.has-is-ab .is-abelian-group.commutes = comm _ _ Grp→Ab→Grp : ∀ {ℓ} (G : Group ℓ) (c : is-commutative-group G) → Abelian→Group (from-commutative-group G c) ≡ G Grp→Ab→Grp G c = Σ-pathp refl go where go : Abelian→Group-on (from-commutative-group G c .snd) ≡ G .snd go i .Group-on._⋆_ = G .snd .Group-on._⋆_ go i .Group-on.has-is-group = G .snd .Group-on.has-is-group open make-abelian-group using (make-abelian-group→make-group ; to-group-on-ab ; to-is-abelian-group ; to-abelian-group-on ; to-ab) public Lift-ab : ∀ ℓ' → ⌞ Ab ℓ ⌟ → ⌞ Ab (ℓ ⊔ ℓ') ⌟ Lift-ab ℓ' G .fst = el! (Lift ℓ' ⌞ G ⌟) Lift-ab ℓ' G .snd = to-abelian-group-on record where module G = Abelian-group-on (G .snd) ab-is-set = hlevel 2 mul (lift x) (lift y) = lift (x G.* y) inv (lift x) = lift (G._⁻¹ x) 1g = lift G.1g idl x = ap lift G.idl assoc x y z = ap lift G.associative invl x = ap lift G.inversel comm x y = ap lift G.commutes open Functor Ab↪Grp : ∀ {ℓ} → Functor (Ab ℓ) (Groups ℓ) Ab↪Grp .F₀ = Abelian→Group Ab↪Grp .F₁ f = record { ∫Hom f } Ab↪Grp .F-id = ext λ _ → refl Ab↪Grp .F-∘ f g = ext λ _ → refl Ab↪Grp-is-ff : ∀ {ℓ} → is-fully-faithful (Ab↪Grp {ℓ}) Ab↪Grp-is-ff {x = A} {B} = is-iso→is-equiv $ iso (λ f → record { ∫Hom f }) (λ _ → ext λ _ → refl) (λ _ → ext λ _ → refl) Ab↪Sets : ∀ {ℓ} → Functor (Ab ℓ) (Sets ℓ) Ab↪Sets = Grp↪Sets F∘ Ab↪Grp
The fundamental example of an abelian group is the group of integers.
Given an abelian group we can define the negation automorphism which inverts every element: since the group operation is commutative, we have so this is a homomorphism.
negation : G Ab.≅ G negation = total-iso (_⁻¹ , is-involutive→is-equiv (λ _ → inv-inv)) (record { pres-⋆ = λ x y → inv-comm ∙ commutes })