open import Cat.Functor.Hom.Representable
open import Cat.Instances.Shape.Terminal
open import Cat.Diagram.Colimit.Base
open import Cat.Instances.Functor
open import Cat.Functor.Kan.Base
open import Cat.Functor.Compose
open import Cat.Functor.Hom
open import Cat.Prelude

import Cat.Reasoning

module Cat.Diagram.Colimit.Representable where

Representability of colimits🔗

Since colimits are defined by universal property, we can also phrase the definition in terms of an equivalence between $\mathbf{Hom}\text{-functors.}$

module _
  {o ℓ}
  {J : Precategory ℓ ℓ} {C : Precategory o ℓ} {Dia : Functor J C}
  where
  private
    module C = Cat.Reasoning C
    open Functor
    open _=>_
    open Corepresentation
    open Colimit
    open is-lan

Let $\mathrm{Dia}:\mathcal{J}\to \mathcal{C}$ be some diagram in $\mathcal{C}\text{.}$ If $\mathrm{Dia}$ has a colimit $c\text{,}$ then that means that maps out of $c$ are in bijection with a product of maps ${\pi }_i\text{,}$ subject to some conditions.

  Lim[C[F-,=]] : Functor C (Sets ℓ)
  Lim[C[F-,=]] .F₀ c = el (Dia => Const c) Nat-is-set
  Lim[C[F-,=]] .F₁ f α = const-nt f ∘nt α
  Lim[C[F-,=]] .F-id = ext λ _ _ → C.idl _
  Lim[C[F-,=]] .F-∘ _ _ = ext λ _ _ → sym $ C.assoc _ _ _

  Hom-into-inj
    : ∀ {c : C.Ob} (eta : Dia => Const c)
    → Hom-from C c => Lim[C[F-,=]]
  Hom-into-inj eta .η x f = const-nt f ∘nt eta
  Hom-into-inj eta .is-natural x y f = ext λ g _ →
    sym $ C.assoc _ _ _

  represents→is-colimit
    : ∀ {c : C.Ob} {eta : Dia => Const c}
    → is-invertibleⁿ (Hom-into-inj eta)
    → is-colimit Dia c eta
  represents→is-colimit {c} {eta} nat-inv = colim where
    module nat-inv = is-invertibleⁿ nat-inv

    colim : is-colimit Dia c eta
    colim .σ {M} α =
      hom→⊤-natural-trans $ nat-inv.inv .η _ (idnat-constr ∘nt α)
    colim .σ-comm {M} {α} = ext λ j →
      unext nat-inv.invl _ _ j ∙ C.idl _
    colim .σ-uniq {M} {α} {σ'} q = ext λ j →
      nat-inv.inv .η _ (idnat-constr ∘nt ⌜ α ⌝)                               ≡⟨ ap! q ⟩≡
      nat-inv.inv .η _ ⌜ idnat-constr ∘nt (σ' ◂ !F) ∘nt cocone→unit Dia eta ⌝ ≡⟨ ap! (ext λ _ → C.idl _) ⟩≡
      nat-inv.inv .η (M .F₀ tt) (const-nt (σ' .η j) ∘nt eta)                  ≡⟨ unext nat-inv.invr _ _ ⟩≡
      σ' .η tt ∎