open import 1Lab.Prelude

open import Algebra.Magma.Unital hiding (idl ; idr)
open import Algebra.Semigroup
open import Algebra.Monoid hiding (idl ; idr)
open import Algebra.Magma

open import Cat.Instances.Delooping

import Cat.Reasoning

module Algebra.Group where

Groups🔗

A group is a monoid that has inverses for every element. The inverse for an element is necessarily, unique; Thus, to say that “$(G, \star)$ is a group” is a statement about $(G, \star)$ having a certain property (namely, being a group), not structure on $(G, \star)$.

Furthermore, since group homomorphisms automatically preserve this structure, we are justified in calling this property rather than property-like structure.

In particular, for a binary operator to be a group operator, it has to be a monoid, meaning it must have a unit.

record is-group {ℓ} {A : Type ℓ} (_*_ : A → A → A) : Type ℓ where
  no-eta-equality
  field
    unit : A
    has-is-monoid : is-monoid unit _*_

There is also a map which assigns to each element $x$ its inverse $x^{-1}$, and this inverse must multiply with $x$ to give the unit, both on the left and on the right:

    inverse : A → A

    inversel : {x : A} → inverse x * x ≡ unit
    inverser : {x : A} → x * inverse x ≡ unit

  open is-monoid has-is-monoid public

is-group is propositional🔗

Showing that is-group takes values in propositions is straightforward, but tedious. Suppose that $x, y$ are both witnesses of is-group for the same operator; We’ll build a path $x = y$.

is-group-is-prop : ∀ {ℓ} {A : Type ℓ} {_*_ : A → A → A}
                 → is-prop (is-group _*_)
is-group-is-prop {A = A} {_*_ = _*_} x y = path where
  open is-group

We begin by constructing a line showing that the underlying monoid structures are identical – but since these have different types, we must also show that the units are the same.

  same-unit : x .unit ≡ y .unit
  same-unit =
    identities-equal (x .unit) (y .unit)
      (is-monoid→is-unital-magma (x .has-is-monoid))
      (is-monoid→is-unital-magma (y .has-is-monoid))

We then use the fact that is-monoid is a proposition to conclude that the monoid structures underlying $x$ and $y$ are the same.

  same-monoid : PathP (λ i → is-monoid (same-unit i) _*_)
                      (x .has-is-monoid) (y .has-is-monoid)
  same-monoid =
    is-prop→pathp (λ i → hlevel {T = is-monoid (same-unit i) _*_} )
      (x .has-is-monoid) (y .has-is-monoid)

Since inverses in monoids are unique (when they exist), it follows that the inverse-assigning maps are pointwise equal; By extensionality, they are the same map.

  same-inverses : (e : A) → x .inverse e ≡ y .inverse e
  same-inverses e =
    monoid-inverse-unique (y .has-is-monoid) _ _ _
      (x .inversel ∙ same-unit) (y .inverser)

Since the underlying type of a group is a set, we have that any parallel paths are equal - even when the paths are dependent! This gives us the equations between the inversel and inverser fields of x and y.

  same-invl : (e : A) → Square _ _ _ _
  same-invl e =
    is-set→squarep (λ _ _ → x .has-is-monoid .has-is-set)
      (ap₂ _*_ (same-inverses e) refl) (x .inversel) (y .inversel) same-unit

  same-invr : (e : A) → Square _ _ _ _
  same-invr e =
    is-set→squarep (λ _ _ → x .has-is-monoid .has-is-set)
      (ap₂ _*_ refl (same-inverses e)) (x .inverser) (y .inverser) same-unit

Putting all of this together lets us conclude that x and y are identical.

  path : x ≡ y
  path i .unit         = same-unit i
  path i .has-is-monoid  = same-monoid i
  path i .inverse e    = same-inverses e i
  path i .inversel {e} = same-invl e i
  path i .inverser {e} = same-invr e i

instance
  H-Level-is-group
    : ∀ {ℓ} {G : Type ℓ} {_+_ : G → G → G} {n}
    → H-Level (is-group _+_) (suc n)
  H-Level-is-group = prop-instance is-group-is-prop

Group Homomorphisms🔗

In contrast with monoid homomorphisms, for group homomorphisms, it is not necessary for the underlying map to explicitly preserve the unit (and the inverses); It is sufficient for the map to preserve the multiplication.

As a stepping stone, we define what it means to equip a type with a group structure: a group structure on a type.

record Group-on {ℓ} (A : Type ℓ) : Type ℓ where
  field
    _⋆_ : A → A → A
    has-is-group : is-group _⋆_

  infixr  _⋆_
  infixl  _⁻¹

  _⁻¹ : A → A
  x ⁻¹ = has-is-group .is-group.inverse x

  open is-group has-is-group public

Group : (ℓ : Level) → Type (lsuc ℓ)
Group ℓ = Σ (Type ℓ) Group-on

We have that a map is a group homomorphism if it preserves the multiplication.

record
  Group-hom {ℓ ℓ′} (A : Group ℓ) (B : Group ℓ′) (e : A .fst → B .fst) : Type (ℓ ⊔ ℓ′) where
  private
    module A = Group-on (A .snd)
    module B = Group-on (B .snd)

  field
    pres-⋆ : (x y : A .fst) → e (x A.⋆ y) ≡ e x B.⋆ e y

A tedious calculation shows that this is sufficient to preserve the identity:

  private
    1A = A.unit
    1B = B.unit

  pres-id : e 1A ≡ 1B
  pres-id =
    e 1A                       ≡⟨ sym B.idr ⟩≡
    e 1A B.⋆ ⌜ 1B ⌝            ≡˘⟨ ap¡ B.inverser ⟩≡˘
    e 1A B.⋆ (e 1A B.— e 1A)   ≡⟨ B.associative ⟩≡
    ⌜ e 1A B.⋆ e 1A ⌝ B.— e 1A ≡⟨ ap! (sym (pres-⋆ _ _) ∙ ap e A.idl) ⟩≡
    e 1A B.— e 1A              ≡⟨ B.inverser ⟩≡
    1B                         ∎

  pres-inv : ∀ {x} → e (A.inverse x) ≡ B.inverse (e x)
  pres-inv {x} =
    monoid-inverse-unique B.has-is-monoid (e x) _ _
      (sym (pres-⋆ _ _) ·· ap e A.inversel ·· pres-id)
      B.inverser

  pres-diff : ∀ {x y} → e (x A.— y) ≡ e x B.— e y
  pres-diff {x} {y} =
    e (x A.— y)                 ≡⟨ pres-⋆ _ _ ⟩≡
    e x B.⋆ ⌜ e (A.inverse y) ⌝ ≡⟨ ap! pres-inv ⟩≡
    e x B.— e y                 ∎

An equivalence is an equivalence of groups when its underlying map is a group homomorphism.

Group≃ : ∀ {ℓ} (A B : Group ℓ) (e : A .fst ≃ B .fst) → Type ℓ
Group≃ A B (f , _) = Group-hom A B f

Group[_⇒_] : ∀ {ℓ} (A B : Group ℓ) → Type ℓ
Group[ A ⇒ B ] = Σ (A .fst → B .fst) (Group-hom A B)

We automatically derive the proof that paths between groups are homomorphic equivalences:

Group-univalent : ∀ {ℓ} → is-univalent {ℓ = ℓ} (HomT→Str Group≃)
Group-univalent {ℓ = ℓ} =
  Derive-univalent-record (record-desc
    (Group-on {ℓ = ℓ}) Group≃
    (record:
      field[ _⋆_          by pres-⋆ ]
      axiom[ has-is-group by (λ _ → is-group-is-prop) ]))
  where
    open Group-on
    open Group-hom

Making groups🔗

Since the interface of Group-on is very deeply nested, we introduce a helper function for arranging the data of a group into a record.

record make-group {ℓ} (G : Type ℓ) : Type ℓ where
  no-eta-equality
  field
    group-is-set : is-set G
    unit : G
    mul  : G → G → G
    inv  : G → G

    assoc : ∀ x y z → mul (mul x y) z ≡ mul x (mul y z)
    invl  : ∀ x → mul (inv x) x ≡ unit
    invr  : ∀ x → mul x (inv x) ≡ unit
    idl   : ∀ x → mul unit x ≡ x

  to-group-on : Group-on G
  to-group-on .Group-on._⋆_ = mul
  to-group-on .Group-on.has-is-group .is-group.unit = unit
  to-group-on .Group-on.has-is-group .is-group.inverse = inv
  to-group-on .Group-on.has-is-group .is-group.inversel = invl _
  to-group-on .Group-on.has-is-group .is-group.inverser = invr _
  to-group-on .Group-on.has-is-group .is-group.has-is-monoid .is-monoid.idl {x} = idl x
  to-group-on .Group-on.has-is-group .is-group.has-is-monoid .is-monoid.idr {x} =
    mul x ⌜ unit ⌝           ≡˘⟨ ap¡ (invl x) ⟩≡˘
    mul x (mul (inv x) x)    ≡⟨ sym (assoc _ _ _) ⟩≡
    mul ⌜ mul x (inv x) ⌝ x  ≡⟨ ap! (invr x) ⟩≡
    mul unit x               ≡⟨ idl x ⟩≡
    x                        ∎
  to-group-on .Group-on.has-is-group .is-group.has-is-monoid .has-is-semigroup =
    record { has-is-magma = record { has-is-set = group-is-set }
           ; associative = λ {x y z} → sym (assoc x y z)
           }

open make-group using (to-group-on) public

Symmetric Groups🔗

If $X$ is a set, then the type of all bijections $X \simeq X$ is also a set, and it forms the carrier for a group: The symmetric group on $X$.

Sym : ∀ {ℓ} → Set ℓ → Group ℓ
Sym X .fst = ∣ X ∣ ≃ ∣ X ∣
Sym X .snd = to-group-on group-str where
  open make-group
  open n-Type X using (H-Level-n-type)
  group-str : make-group (∣ X ∣ ≃ ∣ X ∣)
  group-str .mul g f = f ∙e g

The group operation is composition of equivalences; The identity element is the identity equivalence.

  group-str .unit = id , id-equiv

This type is a set because $X \to X$ is a set (because $X$ is a set by assumption), and being an equivalence is a proposition.

  group-str .group-is-set =
    hlevel 

The associativity and identity laws hold definitionally.

  group-str .assoc _ _ _ = Σ-prop-path is-equiv-is-prop refl
  group-str .idl _ = Σ-prop-path is-equiv-is-prop refl

The inverse is given by the inverse equivalence, and the inverse equations hold by the fact that the inverse of an equivalence is both a section and a retraction.

  group-str .inv = _e⁻¹
  group-str .invl (f , eqv) =
    Σ-prop-path is-equiv-is-prop (funext (equiv→unit eqv))
  group-str .invr (f , eqv) =
    Σ-prop-path is-equiv-is-prop (funext (equiv→counit eqv))