open import Algebra.Group.Ab
open import Algebra.Group

open import Cat.Displayed.Univalence.Thin
open import Cat.Instances.Product
open import Cat.Displayed.Total
open import Cat.Prelude
module Algebra.Group.Ab.Hom where

Maps between abelian groupsπŸ”—

open is-group-hom
open Total-hom

As groups are an algebraic theory, if is a group, we can equip the set of functions with the pointwise group structure. When considering a pair of groups however, we’re less interested in the functions and more interested in the homomorphisms Can these be equipped with a group structure?

It turns out that the answer is no: if you try to make into a functor on , equipping the pointwise group structure, you find out that the sum of group homomorphisms can not be shown to be a homomorphism. But when considering abelian groups, i.e.Β the category , this does work:

Abelian-group-on-hom
  : βˆ€ {β„“} (A B : Abelian-group β„“)
  β†’ Abelian-group-on (Ab.Hom A B)
Abelian-group-on-hom A B = to-abelian-group-on make-ab-on-hom module Hom-ab where
  open make-abelian-group
  private
    module B = Abelian-group-on (B .snd)
    module A = Abelian-group-on (A .snd)

  make-ab-on-hom : make-abelian-group (Ab.Hom A B)
  make-ab-on-hom .ab-is-set = Ab.Hom-set _ _
  make-ab-on-hom .mul f g .hom x = f # x B.* g # x
  make-ab-on-hom .mul f g .preserves .pres-⋆ x y =
    f # (x A.* y) B.* g # (x A.* y)          β‰‘βŸ¨ apβ‚‚ B._*_ (f .preserves .pres-⋆ x y) (g .preserves .pres-⋆ x y) ⟩
    (f # x B.* f # y) B.* (g # x B.* g # y)  β‰‘βŸ¨ B.pullr (B.pulll refl)  ⟩
    f # x B.* (f # y B.* g # x) B.* g # y    β‰‘βŸ¨ (Ξ» i β†’ f # x B.* B.commutes {x = f # y} {y = g # x} i B.* (g # y)) ⟩
    f # x B.* (g # x B.* f # y) B.* g # y    β‰‘βŸ¨ B.pushr (B.pushl refl) ⟩
    (f # x B.* g # x) B.* (f # y B.* g # y)  ∎

  make-ab-on-hom .inv f .hom x = B._⁻¹ (f # x)
  make-ab-on-hom .inv f .preserves .pres-⋆ x y =
    f # (x A.* y) B.⁻¹            β‰‘βŸ¨ ap B._⁻¹ (f .preserves .pres-⋆ x y) ⟩
    (f # x B.* f # y) B.⁻¹        β‰‘βŸ¨ B.inv-comm ⟩
    (f # y B.⁻¹) B.* (f # x B.⁻¹) β‰‘βŸ¨ B.commutes ⟩
    (f # x B.⁻¹) B.* (f # y B.⁻¹) ∎

  make-ab-on-hom .1g .hom x = B.1g
  make-ab-on-hom .1g .preserves .pres-⋆ x y = B.introl refl
  make-ab-on-hom .idl x       = ext Ξ» x β†’ B.idl
  make-ab-on-hom .assoc x y z = ext Ξ» _ β†’ B.associative
  make-ab-on-hom .invl x      = ext Ξ» x β†’ B.inversel
  make-ab-on-hom .comm x y    = ext Ξ» x β†’ B.commutes

open Functor

Ab[_,_] : βˆ€ {β„“} β†’ Abelian-group β„“ β†’ Ab.Ob β†’ Ab.Ob
∣ Ab[ A , B ] .fst ∣ = _
Ab[ A , B ] .fst .is-tr = Ab.Hom-set A B
Ab[ A , B ] .snd = Abelian-group-on-hom A B

It’s only a little more work to show that this extends to a functor 

Ab-hom-functor : βˆ€ {β„“} β†’ Functor (Ab β„“ ^op Γ—αΆœ Ab β„“) (Ab β„“)
Ab-hom-functor .Fβ‚€ (A , B) = Ab[ A , B ]
Ab-hom-functor .F₁ (f , g) .hom h = g Ab.∘ h Ab.∘ f
Ab-hom-functor .F₁ (f , g) .preserves .pres-⋆ x y = ext Ξ» z β†’
  g .preserves .pres-⋆ _ _
Ab-hom-functor .F-id    = trivial!
Ab-hom-functor .F-∘ f g = trivial!