module Cat.Abelian.Instances.Functor
  where

Ab-enriched functor categories🔗

Recall that, for a pair of $Ab -categories$ $A$ and $B,$ we define the $Ab -functors$ between them to be the functors $F : A \to B$ which additionally preserve homwise addition¹. We can, mimicking the construction of the ordinary functor category, construct a category $[A, B]_{Ab}$ consisting only of the $Ab -functors,$ and prove that it is also an $Ab -category.$

  Ab-functors : Precategory _ _
  Ab-functors .Ob          = Ab-functor 𝒜 ℬ
  Ab-functors .Hom F G     = F .functor => G .functor
  Ab-functors .Hom-set _ _ = Nat-is-set
  Ab-functors .id    = Cat[ A , B ] .Precategory.id
  Ab-functors ._∘_   = Cat[ A , B ] .Precategory._∘_
  Ab-functors .idr   = Cat[ A , B ] .Precategory.idr
  Ab-functors .idl   = Cat[ A , B ] .Precategory.idl
  Ab-functors .assoc = Cat[ A , B ] .Precategory.assoc

We can calculate that the natural transformations $F \Rightarrow G$ between $Ab -functors$ have a pointwise abelian group structure. The most important thing to verify is that the pointwise sum of natural transformations is natural, which follows from multilinearity of the composition operation.

  [_,_]Ab : Ab-category Ab-functors
  [_,_]Ab .Abelian-group-on-hom F G = to-abelian-group-on grp where
    open make-abelian-group
    open Group-on
    module F = Ab-functor F
    module G = Ab-functor G

    grp : make-abelian-group (F .functor => G .functor)
    grp .mul f g .η x = f .η x B.+ g .η x
    grp .mul f g .is-natural x y h =
      (f .η y B.+ g .η y) B.∘ F.₁ h             ≡˘⟨ B.∘-linear-l _ _ _ ⟩
      (f .η y B.∘ F.₁ h) B.+ (g .η y B.∘ F.₁ h) ≡⟨ ap₂ B._+_ (f .is-natural x y h) (g .is-natural x y h) ⟩
      (G.₁ h B.∘ f .η x) B.+ (G.₁ h B.∘ g .η x) ≡⟨ B.∘-linear-r _ _ _ ⟩
      G.₁ h B.∘ (f .η x B.+ g .η x)             ∎

Specifically, given $(η_{x} + ε_{x}) F (h),$ we can distribute $F (h)$ into the composite, apply naturality to both summands, and distribute $G (h)$ out of the composite on the left. Similar computations establish that the pointwise inverse of natural transformations is natural.

    grp .1g .η x = B.0m
    grp .1g .is-natural x y f = B.∘-zero-l ∙ sym (B.∘-zero-r)

    grp .inv f .η x = B.Hom.inverse (f .η x)
    grp .inv f .is-natural x y g =
      B.Hom.inverse (f .η y) B.∘ F.₁ g   ≡˘⟨ B.neg-∘-l ⟩
      B.Hom.inverse ⌜ f .η y B.∘ F.₁ g ⌝ ≡⟨ ap! (f .is-natural x y g) ⟩
      B.Hom.inverse (G.₁ g B.∘ f .η x)   ≡⟨ B.neg-∘-r ⟩
      G.₁ g B.∘ B.Hom.inverse (f .η x)   ∎

    grp .assoc _ _ _ = ext λ _ → B.Hom.associative
    grp .idl _       = ext λ x → B.Hom.idl
    grp .invl _      = ext λ x → B.Hom.inversel
    grp .comm _ _    = ext λ x → B.Hom.commutes
    grp .ab-is-set   = Nat-is-set

  [_,_]Ab .∘-linear-l f g h = ext λ x → B.∘-linear-l _ _ _
  [_,_]Ab .∘-linear-r f g h = ext λ x → B.∘-linear-r _ _ _

i.e. those functors $F$ for which, for all $a, b,$ the $Hom -mapping$ $F_{a, b} : A (a, b) \to B (F (a), F (b))$ extends to a group homomorphism.↩︎