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 β„“} (C : Precategory o β„“) where

Indexed coproductsπŸ”—

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

record is-indexed-coproduct (F : Idx β†’ C.Ob) (ΞΉ : βˆ€ i β†’ C.Hom (F i) S)
  : Type (o βŠ” β„“ βŠ” level-of Idx) where
  no-eta-equality
  field
    match   : βˆ€ {Y} β†’ (βˆ€ i β†’ C.Hom (F i) Y) β†’ C.Hom S Y
    commute : βˆ€ {i} {Y} {f : βˆ€ i β†’ C.Hom (F i) Y} β†’ match f C.∘ ΞΉ i ≑ f i
    unique  : βˆ€ {Y} {h : C.Hom S Y} (f : βˆ€ i β†’ C.Hom (F i) Y)
            β†’ (βˆ€ i β†’ h C.∘ ΞΉ i ≑ f i)
            β†’ h ≑ match f

  eta : βˆ€ {Y} (h : C.Hom S Y) β†’ h ≑ match (Ξ» i β†’ h C.∘ ΞΉ i)
  eta h = unique _ Ξ» _ β†’ refl

A category C\ca{C} admits indexed coproducts (of level ℓ\ell) if, for any type I:Type ℓI : \ty\ \ell and family F:I→CF : I \to \ca{C}, there is an indexed coproduct of FF.

record Indexed-coproduct (F : Idx β†’ C.Ob) : Type (o βŠ” β„“ βŠ” level-of Idx) where
  no-eta-equality
  field
    {Ξ£F}      : C.Ob
    ΞΉ         : βˆ€ i β†’ C.Hom (F i) Ξ£F
    has-is-ic : is-indexed-coproduct F ΞΉ
  open is-indexed-coproduct has-is-ic public

has-indexed-coproducts : βˆ€ β„“ β†’ Type _
has-indexed-coproducts β„“ = βˆ€ {I : Type β„“} (F : I β†’ C.Ob) β†’ Indexed-coproduct F

As colimitsπŸ”—

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

module _ {I : Type β„“'} (i-is-grpd : is-groupoid I) (F : I β†’ 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.Ξ£F
    thecolim .ψ = IC.ι
    thecolim .commutes {x} =
      J (Ξ» y p β†’ IC.ΞΉ y C.∘ subst (C.Hom (F x) βŠ™ F) p C.id ≑ IC.ΞΉ x)
        (C.elimr (transport-refl _))

    colim : Colimit _
    colim .bot = thecolim
    colim .hasβŠ₯ x .centre .hom = IC.match (x .ψ)
    colim .hasβŠ₯ x .centre .commutes o = IC.commute
    colim .hasβŠ₯ x .paths h = Cocone-hom-path _ (sym (IC.unique _ Ξ» i β†’ h .commutes _))

module _ {I : Type β„“'} (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

  Injβ†’Cocone : βˆ€ {Y} β†’ (βˆ€ i β†’ C.Hom (F.β‚€ i) Y) β†’ Cocone F
  Inj→Cocone f .coapex = _
  Injβ†’Cocone f .ψ = f
  Injβ†’Cocone f .commutes {x} = J (Ξ» y p β†’ 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 .Ξ£F = colim.coapex
    the-ic .ι  = colim.ψ
    the-ic .has-is-ic .match f = colim .hasβŠ₯ (Injβ†’Cocone f) .centre .hom
    the-ic .has-is-ic .commute = colim .hasβŠ₯ _ .centre .commutes _
    the-ic .has-is-ic .unique {h = h} f p i =
      colim .hasβŠ₯ (Injβ†’Cocone f) .paths hβ€² (~ i) .hom
      where
        hβ€² : Cocone-hom _ _ _
        hβ€² .hom = h
        hβ€² .commutes o = p _

Disjoint coproductsπŸ”—

An indexed coproduct βˆ‘F\sum F is said to be disjoint if every one of its inclusions Fiβ†’βˆ‘FF_i \to \sum F is monic, and, for unequal iβ‰ ji \ne j, the square below is a pullback with initial apex. Since the maps Fiβ†’βˆ‘F←FjF_i \to \sum F \ot F_j are monic, the pullback below computes the intersection of FiF_i and FjF_j as subobjects of βˆ‘F\sum F, hence the name disjoint coproduct: If βŠ₯\bot is an initial object, then Fi∩Fj=βˆ…F_i \cap F_j = \emptyset.

record is-disjoint-coproduct (F : Idx β†’ C.Ob) (ΞΉ : βˆ€ i β†’ C.Hom (F i) S)
  : Type (o βŠ” β„“ βŠ” level-of Idx) where
  field
    is-coproduct         : is-indexed-coproduct F ΞΉ
    injections-are-monic : βˆ€ i β†’ C.is-monic (ΞΉ i)
    summands-intersect   : βˆ€ i j β†’ Pullback C (ΞΉ i) (ΞΉ j)
    different-images-are-disjoint
      : βˆ€ i j β†’ (i ≑ j β†’ βŠ₯) β†’ is-initial C (summands-intersect i j .Pullback.apex)

Initial objects are disjointπŸ”—

We prove that if βŠ₯\bot is an initial object, then it is also an indexed coproduct β€” for any family βŠ₯β†’C\bot \to \ca{C} β€” and furthermore, it is a disjoint coproduct.

is-initial→is-disjoint-coproduct
  : βˆ€ {βˆ…} {F : βŠ₯ β†’ C.Ob} {i : βˆ€ i β†’ C.Hom (F i) βˆ…}
  β†’ is-initial C βˆ…
  β†’ 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