open import Cat.Functor.Equivalence.Properties
open import Cat.Functor.Adjoint.Continuous
open import Cat.Instances.Shape.Terminal
open import Cat.Diagram.Colimit.Base
open import Cat.Functor.Conservative
open import Cat.Instances.Coalgebras
open import Cat.Functor.Kan.Base
open import Cat.Diagram.Comonad
open import Cat.Displayed.Total
open import Cat.Prelude

import Cat.Reasoning
module Cat.Instances.Coalgebras.Colimits {o ℓ} {C : Precategory o ℓ} {F : Functor C C} (W : Comonad-on F) where
private
  module CoEM = Cat.Reasoning (Coalgebras W)
  module C = Cat.Reasoning C
  module W = Comonad-on W

open Coalgebra-on
open ∫Hom

Colimits in categories of coalgebras🔗

This module dualises the construction of limits in categories of algebras for a monad to colimits in categories of coalgebras for a comonad; see there for explanations.

module _ {jo jℓ} {J : Precategory jo jℓ} (F : Functor J (Coalgebras W)) where
  private
    module J = Precategory J
    module F = Functor F
    module FAlg j = Coalgebra-on (F.₀ j .snd)
    Forget-CoEM = πᶠ (Coalgebras-over W)
  open Functor
  open _=>_
  Forget-CoEM-preserves-colimits : preserves-colimit (πᶠ (Coalgebras-over W)) F
  Forget-CoEM-preserves-colimits = left-adjoint-is-cocontinuous (Forget⊣Cofree W)

  make-coalgebra-colimit
    : ∀ {K : Functor ⊤Cat C} {eta : Forget-CoEM F∘ F => K F∘ !F}
    → (colim : is-lan !F (Forget-CoEM F∘ F) K eta)
    → (rho : Coalgebra-on W (K .F₀ tt))
    → (∀ j → W.W₁ (is-colimit.ψ colim j) C.∘ FAlg.ρ j ≡ rho .ρ C.∘ is-colimit.ψ colim j)
    → make-is-colimit F (K .F₀ tt , rho)
  make-coalgebra-colimit colim apex-coalg comm = coem-colim where
    module colim = is-colimit colim
    open make-is-colimit
    module apex = Coalgebra-on apex-coalg
    open _=>_
    coem-colim : make-is-colimit F _
    coem-colim .ψ j .fst = colim.ψ j
    coem-colim .ψ j .snd = comm j
    coem-colim .commutes f    = ext (colim.commutes f)
    coem-colim .universal eta p .fst =
      colim.universal (λ j → eta j .fst) (λ f i → p f i .fst)
    coem-colim .factors eta p =
      ext (colim.factors _ _)
    coem-colim .unique eta p other q =
      ext (colim.unique _ _ _ λ j i → q j i .fst)
    coem-colim .universal eta p .snd = colim.unique₂ (λ j → W.W₁ (eta j .fst) C.∘ F.F₀ j .snd .ρ)
      (λ f → C.pullr (sym (F.F₁ f .snd))
           ∙ C.pulll (sym (W.W-∘ _ _) ∙ ap W.W₁ (ap fst (p f))))
      (λ j → C.pullr (sym (comm j))
           ∙ C.pulll (sym (W.W-∘ _ _) ∙ ap W.W₁ (colim.factors _ _)))
      (λ j → C.pullr (colim.factors _ _)
           ∙ sym (eta j .snd))

  Forget-lift-colimit : Colimit (Forget-CoEM F∘ F) → Colimit F
  Forget-lift-colimit colim-over = to-colimit $ to-is-colimit $ make-coalgebra-colimit
    (Colimit.has-colimit colim-over) coapex-coalgebra (λ j → sym (colim-over.factors _ _))
    where
    module colim-over = Colimit colim-over
    module colim = Lan colim-over

    coapex-coalgebra : Coalgebra-on W colim-over.coapex
    coapex-coalgebra .ρ =
      colim-over.universal (λ j → W.W₁ (colim-over.ψ j) C.∘ FAlg.ρ j) comm where abstract
      comm : ∀ {x y} (f : J.Hom x y)
           → (W.W₁ (colim-over.ψ y) C.∘ FAlg.ρ y) C.∘ F.₁ f .fst
          ≡ W.W₁ (colim-over.ψ x) C.∘ FAlg.ρ x
      comm {x} {y} f =
        (W.W₁ (colim-over.ψ y) C.∘ FAlg.ρ y) C.∘ F.₁ f .fst      ≡⟨ C.pullr (sym (F.₁ f .snd)) ⟩
        W.W₁ (colim-over.ψ y) C.∘ W.W₁ (F.₁ f .fst) C.∘ FAlg.ρ x ≡⟨ C.pulll (sym (W.W-∘ _ _)) ⟩
        W.W₁ (colim-over.ψ y C.∘ F.₁ f .fst) C.∘ FAlg.ρ x        ≡⟨ ap W.W₁ (colim-over.commutes f) C.⟩∘⟨refl ⟩
        W.W₁ (colim-over.ψ x) C.∘ FAlg.ρ x                       ∎
    coapex-coalgebra .ρ-counit = colim-over.unique₂ _ colim-over.commutes
      (λ j → C.pullr (colim-over.factors _ _)
          ∙∙ C.pulll (W.counit.is-natural _ _ _)
          ∙∙ C.cancelr (FAlg.ρ-counit j))
      (λ j → C.idl _)
    coapex-coalgebra .ρ-comult = colim-over.unique₂ _
      (λ f → C.pullr $ C.pullr (sym (F.₁ f .snd))
           ∙ C.pulll (sym (W.W-∘ _ _) ∙ ap W.W₁ (colim-over.commutes f)))
      (λ j → C.pullr (colim-over.factors _ _)
           ∙ sym (C.pulll (sym (W.W-∘ _ _) ∙ ap W.W₁ (colim-over.factors _ _) ∙ W.W-∘ _ _)
               ∙∙ C.extendr (sym (FAlg.ρ-comult j))
               ∙∙ ap (C._∘ FAlg.ρ j) (sym (W.comult.is-natural _ _ _))
                ∙ sym (C.assoc _ _ _)))
      (λ j → C.pullr (colim-over.factors _ _))
module _ {jo jℓ} {J : Precategory jo jℓ} where
  open lifts-colimit
  open creates-colimit
  Forget-CoEM-lifts-colimits : lifts-colimits-of J (πᶠ (Coalgebras-over W))
  Forget-CoEM-lifts-colimits colim .lifted = Forget-lift-colimit _ colim
  Forget-CoEM-lifts-colimits colim .preserved =
    Forget-CoEM-preserves-colimits _ (Lan.has-lan (Forget-lift-colimit _ colim))

  Forget-CoEM-creates-colimits : creates-colimits-of J (πᶠ (Coalgebras-over W))
  Forget-CoEM-creates-colimits = conservative+lifts→creates-colimits
    Forget-CoEM-is-conservative Forget-CoEM-lifts-colimits
Eilenberg-Moore-is-cocomplete
  : ∀ {a b} → is-cocomplete a b C → is-cocomplete a b (Coalgebras W)
Eilenberg-Moore-is-cocomplete = lifts-colimits→cocomplete
  (πᶠ (Coalgebras-over W)) Forget-CoEM-lifts-colimits