module Cat.Abelian.Functor where

# Ab-enriched functorsπ

Since $Ab-categories$ are additionally equipped with the structure of abelian groups on their $Hom-sets,$ itβs natural that we ask that functors between $Ab-categories$ preserve this structure. In particular, since every functor $F:CβD$ has an action $F(β):Hom(a,b)βHom(Fa,Fb)$ which is a map of sets, when $C$ and $D$ are considered to be abelian groups, we should require that the action $F(β)$ be a group homomorphism.

module _ {o β o' β'} {C : Precategory o β} {D : Precategory o' β'} (A : Ab-category C) (B : Ab-category D) where private module A = Ab-category A module B = Ab-category B

record Ab-functor : Type (o β o' β β β β') where field functor : Functor C D open Functor functor public field F-+ : β {a b} (f g : A.Hom a b) β Fβ (f A.+ g) β‘ Fβ f B.+ Fβ g

In passing we note that, since the
$Hom-abelian-groups$
are *groups*, preservation of addition automatically implies
preservation of the zero morphism, preservation of inverses, and thus
preservation of subtraction.

F-hom : β {a b} β is-group-hom (AbelianβGroup-on (A.Abelian-group-on-hom a b)) (AbelianβGroup-on (B.Abelian-group-on-hom _ _)) Fβ F-hom .is-group-hom.pres-β = F-+ F-0m : β {a b} β Fβ {a} {b} A.0m β‘ B.0m F-0m = is-group-hom.pres-id F-hom F-diff : β {a b} (f g : A.Hom a b) β Fβ (f A.- g) β‘ Fβ f B.- Fβ g F-diff _ _ = is-group-hom.pres-diff F-hom F-inv : β {a b} (f : A.Hom a b) β Fβ (A.Hom.inverse f) β‘ B.Hom.inverse (Fβ f) F-inv _ = is-group-hom.pres-inv F-hom

Since the zero object
$β$
in an
$Ab-category$
is characterised as the unique object having
$id_{β}=0,$
and
$Ab-functors$
preserve both
$id$
and
$0,$
every
$Ab-functor$
preserves zero objects. We say that the zero object, considered as a
colimit, is **absolute**, i.e., preserved by every
(relevant) functor.

Ab-functor-pres-β : β {o β o' β'} {C : Precategory o β} {D : Precategory o' β'} {A : Ab-category C} {B : Ab-category D} β (F : Ab-functor A B) (β A : Zero C) β is-zero D (Ab-functor.Fβ F (Zero.β β A)) Ab-functor-pres-β {A = A} {B = B} F β A = id-zeroβzero B $ B.id β‘Λβ¨ F.F-id β©β‘Λ F.β A.id β‘β¨ ap F.β (is-contrβis-prop (Zero.hasβ€ β A (Zero.β β A)) _ _) β©β‘ F.β A.0m β‘β¨ F.F-0m β©β‘ B.0m β where module A = Ab-category A module B = Ab-category B module F = Ab-functor F