Algebraic theory of groups

Content created by Fredrik Bakke and Fernando Chu.

Created on 2023-03-20.
Last modified on 2023-06-25.

module universal-algebra.algebraic-theory-of-groups where
Imports 
open import elementary-number-theory.natural-numbers

open import foundation.dependent-pair-types
open import foundation.equality-dependent-pair-types
open import foundation.equivalences
open import foundation.function-extensionality
open import foundation.identity-types
open import foundation.propositions
open import foundation.universe-levels

open import group-theory.groups

open import linear-algebra.vectors

open import universal-algebra.algebraic-theories
open import universal-algebra.algebras-of-theories
open import universal-algebra.signatures
open import universal-algebra.terms-over-signatures

Idea

There is an algebraic theory of groups. They type of all such algebras is equivalent to the type of groups.

Definitions

The algebra of groups

data group-ops : UU lzero where
  unit-group-op mul-group-op inv-group-op : group-ops

group-signature : signature lzero
group-signature =
  pair
    group-ops
    ( λ { unit-group-op → 0 ;
          mul-group-op → 2 ;
          inv-group-op → 1})

data group-laws : UU lzero where
  associative-l-group-laws : group-laws
  invl-l-group-laws : group-laws
  invr-r-group-laws : group-laws
  idl-l-group-laws : group-laws
  idr-r-group-laws : group-laws

group-Theory : Theory group-signature lzero
group-Theory =
  pair
    ( group-laws)
    ( λ { associative-l-group-laws →
            pair
              ( op mul-group-op
                ( ( op mul-group-op
                  ( var 0 ∷ var 1 ∷ empty-vec)) ∷
                ( var 2) ∷ empty-vec))
              ( op mul-group-op
                ( var 0 ∷
                ( op mul-group-op (var 1 ∷ var 2 ∷ empty-vec)) ∷ empty-vec)) ;
          invl-l-group-laws →
            pair
              ( op mul-group-op
                ( op inv-group-op (var 0 ∷ empty-vec) ∷ var 0 ∷ empty-vec))
              (op unit-group-op empty-vec) ;
          invr-r-group-laws →
            pair
              ( op mul-group-op
                ( var 0 ∷ op inv-group-op (var 0 ∷ empty-vec) ∷ empty-vec))
              (op unit-group-op empty-vec) ;
          idl-l-group-laws →
            pair
              (op mul-group-op (op unit-group-op empty-vec ∷ var 0 ∷ empty-vec))
              (var 0) ;
          idr-r-group-laws →
            pair
              (op mul-group-op (var 0 ∷ op unit-group-op empty-vec ∷ empty-vec))
              (var 0)})
  where
  op = op-Term
  var = var-Term

group-Algebra : (l : Level) → UU (lsuc l)
group-Algebra l = Algebra group-signature group-Theory l

Properties

The algebra of groups is equivalent to the type of groups

group-Algebra-Group :
  {l : Level} →
  Algebra group-signature group-Theory l →
  Group l
group-Algebra-Group (((A , is-set-A) , models-A) , satisfies-A) =
  pair
    ( pair
      ( pair A is-set-A)
      ( pair
        ( λ x y →
          ( models-A mul-group-op (x ∷ y ∷ empty-vec)))
        ( λ x y z →
          ( satisfies-A associative-l-group-laws
            ( λ { zero-ℕ → x ;
                ( succ-ℕ zero-ℕ) → y ;
                ( succ-ℕ (succ-ℕ n)) → z})))))
    ( pair
      ( pair
        ( models-A unit-group-op empty-vec)
        ( pair
          ( λ x → satisfies-A idl-l-group-laws (λ _ → x))
          ( λ x → satisfies-A idr-r-group-laws (λ _ → x))))
      ( pair
        ( λ x → models-A inv-group-op (x ∷ empty-vec))
        ( pair
          ( λ x → satisfies-A invl-l-group-laws (λ _ → x))
          ( λ x → satisfies-A invr-r-group-laws (λ _ → x)))))

Group-group-Algebra :
  {l : Level} →
  Group l →
  Algebra group-signature group-Theory l
Group-group-Algebra G =
  pair
    ( pair
      ( ( set-Group G))
      ( λ { unit-group-op v → unit-Group G ;
            mul-group-op (x ∷ y ∷ empty-vec) → mul-Group G x y ;
            inv-group-op (x ∷ empty-vec) → inv-Group G x}))
    ( λ { associative-l-group-laws assign →
            associative-mul-Group G (assign 0) (assign 1) (assign 2) ;
          invl-l-group-laws assign →
            left-inverse-law-mul-Group G (assign 0) ;
          invr-r-group-laws assign →
            right-inverse-law-mul-Group G (assign 0) ;
          idl-l-group-laws assign →
            left-unit-law-mul-Group G (assign 0) ;
          idr-r-group-laws assign →
            right-unit-law-mul-Group G (assign 0)})

equiv-group-Algebra-Group :
  {l : Level} →
  Algebra group-signature group-Theory l ≃
  Group l
pr1 equiv-group-Algebra-Group = group-Algebra-Group
pr1 (pr1 (pr2 equiv-group-Algebra-Group)) = Group-group-Algebra
pr2 (pr1 (pr2 equiv-group-Algebra-Group)) G =
  eq-pair-Σ refl (eq-is-prop (is-prop-is-group (semigroup-Group G)))
pr1 (pr2 (pr2 equiv-group-Algebra-Group)) = Group-group-Algebra
pr2 (pr2 (pr2 equiv-group-Algebra-Group)) A =
  eq-pair-Σ
    ( eq-pair-Σ
      ( refl)
      ( eq-htpy
        ( λ { unit-group-op →
              ( eq-htpy λ {empty-vec → refl}) ;
              mul-group-op →
              ( eq-htpy λ { (x ∷ y ∷ empty-vec) → refl}) ;
              inv-group-op →
              ( eq-htpy λ { (x ∷ empty-vec) → refl})})))
    ( eq-is-prop
      ( is-prop-is-algebra
        ( group-signature) ( group-Theory)
        ( model-Algebra group-signature group-Theory A)))

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