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 groupsπ

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, weβ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
: β {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-β x y =
f # (x A.* y) B.* g # (x A.* y)          β‘β¨ apβ B._*_ (f .preserves .pres-β x y) (g .preserves .pres-β x y) β©β‘
(f # x B.* f # y) B.* (g # x B.* g # y)  β‘β¨ B.pullr (B.pulll refl)  β©β‘
f # x B.* (f # y B.* g # x) B.* g # y    β‘β¨ (Ξ» i β f # x B.* B.commutes {x = f # y} {y = g # x} i B.* (g # y)) β©β‘
f # x B.* (g # x B.* f # y) B.* g # y    β‘β¨ B.pushr (B.pushl refl) β©β‘
(f # x B.* g # x) B.* (f # y B.* g # y)  β

make-ab-on-hom .inv f .hom x = B._β»ΒΉ (f # x)
make-ab-on-hom .inv f .preserves .pres-β x y =
f # (x A.* y) B.β»ΒΉ            β‘β¨ ap B._β»ΒΉ (f .preserves .pres-β x y) β©β‘
(f # x B.* f # y) B.β»ΒΉ        β‘β¨ B.inv-comm β©β‘
(f # y B.β»ΒΉ) B.* (f # x B.β»ΒΉ) β‘β¨ B.commutes β©β‘
(f # x B.β»ΒΉ) B.* (f # y B.β»ΒΉ) β

make-ab-on-hom .1g .hom x = B.1g
make-ab-on-hom .1g .preserves .pres-β 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[_,_] : β {β} β Abelian-group β β Ab.Ob β Ab.Ob
β£ Ab[ A , B ] .fst β£ = _
Ab[ A , B ] .fst .is-tr = Ab.Hom-set A B
Ab[ A , B ] .snd = Abelian-group-on-hom A B


Itβs only a little more work to show that this extends to a functor

Ab-hom-functor : β {β} β Functor (Ab β ^op ΓαΆ Ab β) (Ab β)
Ab-hom-functor .Fβ (A , B) = Ab[ A , B ]
Ab-hom-functor .Fβ (f , g) .hom h = g Ab.β h Ab.β f
Ab-hom-functor .Fβ (f , g) .preserves .pres-β x y = ext Ξ» z β
g .preserves .pres-β _ _
Ab-hom-functor .F-id    = trivial!
Ab-hom-functor .F-β f g = trivial!