open import Cat.Diagram.Colimit.Base
open import Cat.Instances.Discrete
open import Cat.Diagram.Pullback
open import Cat.Diagram.Initial
open import Cat.Prelude

module Cat.Diagram.Coproduct.Indexed {o ℓ} (C : Precategory o ℓ) where

Indexed coproducts🔗

Indexed coproducts are the dual notion to indexed products, so see there for motivation and exposition.

record is-indexed-coproduct (F : Idx → C.Ob) (ι : ∀ i → C.Hom (F i) S)
  : Type (o ⊔ ℓ ⊔ level-of Idx) where
  no-eta-equality
  field
    match   : ∀ {Y} → (∀ i → C.Hom (F i) Y) → C.Hom S Y
    commute : ∀ {i} {Y} {f : ∀ i → C.Hom (F i) Y} → match f C.∘ ι i ≡ f i
    unique  : ∀ {Y} {h : C.Hom S Y} (f : ∀ i → C.Hom (F i) Y)
            → (∀ i → h C.∘ ι i ≡ f i)
            → h ≡ match f

  eta : ∀ {Y} (h : C.Hom S Y) → h ≡ match (λ i → h C.∘ ι i)
  eta h = unique _ λ _ → refl

  hom-iso : ∀ {Y} → C.Hom S Y ≃ (∀ i → C.Hom (F i) Y)
  hom-iso = (λ z i → z C.∘ ι i) , is-iso→is-equiv λ where
    .is-iso.inv → match
    .is-iso.rinv x → funext λ i → commute
    .is-iso.linv x → sym (unique _ λ _ → refl)

A category $\mathcal{C}$ admits indexed coproducts (of level $\ell$) if, for any type $I : {{\mathrm{Type}}}\ \ell$ and family $F : I \to \mathcal{C}$, there is an indexed coproduct of $F$.

record Indexed-coproduct (F : Idx → C.Ob) : Type (o ⊔ ℓ ⊔ level-of Idx) where
  no-eta-equality
  field
    {ΣF}      : C.Ob
    ι         : ∀ i → C.Hom (F i) ΣF
    has-is-ic : is-indexed-coproduct F ι
  open is-indexed-coproduct has-is-ic public

has-indexed-coproducts : ∀ ℓ → Type _
has-indexed-coproducts ℓ = ∀ {I : Type ℓ} (F : I → C.Ob) → Indexed-coproduct F

As colimits🔗

Similarly to the product case, when $I$ is a groupoid, indexed coproducts correspond to discrete diagrams of shape $I$.

module _ {I : Type ℓ'} (i-is-grpd : is-groupoid I) (F : I → C.Ob) where
  open Cocone-hom
  open Initial
  open Cocone

  IC→Colimit : Indexed-coproduct F → Colimit {C = C} (Disc-adjunct {iss = i-is-grpd} F)
  IC→Colimit IC = colim where
    module IC = Indexed-coproduct IC

    thecolim : Cocone _
    thecolim .coapex = IC.ΣF
    thecolim .ψ = IC.ι
    thecolim .commutes {x} =
      J (λ y p → IC.ι y C.∘ subst (C.Hom (F x) ⊙ F) p C.id ≡ IC.ι x)
        (C.elimr (transport-refl _))

    colim : Colimit _
    colim .bot = thecolim
    colim .has⊥ x .centre .hom = IC.match (x .ψ)
    colim .has⊥ x .centre .commutes o = IC.commute
    colim .has⊥ x .paths h = Cocone-hom-path _ (sym (IC.unique _ λ i → h .commutes _))

module _ {I : Type ℓ'} (isg : is-groupoid I) (F : Functor (Disc I isg) C) where
  private module F = Functor F
  open is-indexed-coproduct
  open Indexed-coproduct
  open Cocone-hom
  open Initial
  open Cocone

  Inj→Cocone : ∀ {Y} → (∀ i → C.Hom (F.₀ i) Y) → Cocone F
  Inj→Cocone f .coapex = _
  Inj→Cocone f .ψ = f
  Inj→Cocone f .commutes {x} = J (λ y p → f y C.∘ F.₁ p ≡ f x) (C.elimr F.F-id)

  Colimit→IC : Colimit {C = C} F → Indexed-coproduct F.₀
  Colimit→IC colim = the-ic where
    module colim = Cocone (colim .bot)

    the-ic : Indexed-coproduct _
    the-ic .ΣF = colim.coapex
    the-ic .ι  = colim.ψ
    the-ic .has-is-ic .match f = colim .has⊥ (Inj→Cocone f) .centre .hom
    the-ic .has-is-ic .commute = colim .has⊥ _ .centre .commutes _
    the-ic .has-is-ic .unique {h = h} f p i =
      colim .has⊥ (Inj→Cocone f) .paths h′ (~ i) .hom
      where
        h′ : Cocone-hom _ _ _
        h′ .hom = h
        h′ .commutes o = p _

Disjoint coproducts🔗

An indexed coproduct $\sum F$ is said to be disjoint if every one of its inclusions $F_i \to \sum F$ is monic, and, for unequal $i \ne j$, the square below is a pullback with initial apex. Since the maps $F_i \to \sum F {\leftarrow}F_j$ are monic, the pullback below computes the intersection of $F_i$ and $F_j$ as subobjects of $\sum F$, hence the name disjoint coproduct: If $\bot$ is an initial object, then $F_i \cap F_j = \emptyset$.

record is-disjoint-coproduct (F : Idx → C.Ob) (ι : ∀ i → C.Hom (F i) S)
  : Type (o ⊔ ℓ ⊔ level-of Idx) where
  field
    is-coproduct         : is-indexed-coproduct F ι
    injections-are-monic : ∀ i → C.is-monic (ι i)
    summands-intersect   : ∀ i j → Pullback C (ι i) (ι j)
    different-images-are-disjoint
      : ∀ i j → ¬ i ≡ j → is-initial C (summands-intersect i j .Pullback.apex)

Initial objects are disjoint🔗

We prove that if $\bot$ is an initial object, then it is also an indexed coproduct — for any family $\bot \to \mathcal{C}$ — and furthermore, it is a disjoint coproduct.

is-initial→is-disjoint-coproduct
  : ∀ {∅} {F : ⊥ → C.Ob} {i : ∀ i → C.Hom (F i) ∅}
  → is-initial C ∅
  → is-disjoint-coproduct F i
is-initial→is-disjoint-coproduct {F = F} {i = i} init = is-disjoint where
  open is-indexed-coproduct
  is-coprod : is-indexed-coproduct F i
  is-coprod .match _ = init _ .centre
  is-coprod .commute {i = i} = absurd i
  is-coprod .unique {h = h} f p i = init _ .paths h (~ i)

  open is-disjoint-coproduct
  is-disjoint : is-disjoint-coproduct F i
  is-disjoint .is-coproduct = is-coprod
  is-disjoint .injections-are-monic i = absurd i
  is-disjoint .summands-intersect i j = absurd i
  is-disjoint .different-images-are-disjoint i j p = absurd i