open import Cat.Diagram.Coequaliser
open import Cat.Diagram.Pullback
open import Cat.Prelude

import Cat.Reasoning

module Cat.Diagram.Coequaliser.RegularEpi where

module _ {o ℓ} (C : Precategory o ℓ) where
  open Cat.Reasoning C
  private variable a b : Ob

Regular epimorphisms🔗

Dually to regular monomorphisms, which behave as embeddings, regular epimorphisms behave like covers: A regular epimorphism $f:a\twoheadrightarrow b$ expresses $b$ as “a union of parts of $a\text{,}$ possibly glued together”.

The definition is also precisely dual: A map $f:a\to b$ is a regular epimorphism if it is the coequaliser of some pair of arrows $R\rightrightarrows a\text{.}$

  record is-regular-epi (f : Hom a b) : Type (o ⊔ ℓ) where
    no-eta-equality
    constructor reg-epi
    field
      {r}           : Ob
      {arr₁} {arr₂} : Hom r a
      has-is-coeq   : is-coequaliser C arr₁ arr₂ f
  
    open is-coequaliser has-is-coeq public

From the definition we can directly conclude that regular epis are in fact epic:

    is-regular-epi→is-epic : is-epic f
    is-regular-epi→is-epic = is-coequaliser→is-epic _ has-is-coeq
  
  open is-regular-epi using (is-regular-epi→is-epic) public

Effective epis🔗

Again by duality, we have a pair of canonical choices of maps which $f$ may coequalise: Its kernel pair, that is, the pullback of $f$ along itself. An epimorphism which coequalises its kernel pair is called an effective epi, and effective epis are immediately seen to be regular epis:

  record is-effective-epi (f : Hom a b) : Type (o ⊔ ℓ) where
    no-eta-equality
    field
      {kernel}       : Ob
      p₁ p₂          : Hom kernel a
      is-kernel-pair : is-pullback C p₁ f p₂ f
      has-is-coeq    : is-coequaliser C p₁ p₂ f
  
    is-effective-epi→is-regular-epi : is-regular-epi f
    is-effective-epi→is-regular-epi = re where
      open is-regular-epi hiding (has-is-coeq)
      re : is-regular-epi f
      re .r = _
      re .arr₁ = _
      re .arr₂ = _
      re .is-regular-epi.has-is-coeq = has-is-coeq

If a regular epimorphism (a coequaliser) has a kernel pair, then it is also the coequaliser of its kernel pair. That is: If the pullback of $a\stackrel{f}{\to }b\stackrel{f}{\leftarrow }a$ exists, then $f$ is also an effective epimorphism.

module _ {o ℓ} {C : Precategory o ℓ} where
  open Cat.Reasoning C
  private variable a b : Ob

  is-regular-epi→is-effective-epi
    : ∀ {a b} {f : Hom a b}
    → Pullback C f f
    → is-regular-epi C f
    → is-effective-epi C f
  is-regular-epi→is-effective-epi {f = f} kp reg = epi where
    module reg = is-regular-epi reg
    module kp = Pullback kp
  
    open is-effective-epi
    open is-coequaliser
    epi : is-effective-epi C f
    epi .kernel = kp.apex
    epi .p₁ = kp.p₁
    epi .p₂ = kp.p₂
    epi .is-kernel-pair = kp.has-is-pb
    epi .has-is-coeq .coequal = kp.square
    epi .has-is-coeq .universal {F = F} {e'} p = reg.universal q where
      q : e' ∘ reg.arr₁ ≡ e' ∘ reg.arr₂
      q =
        e' ∘ reg.arr₁                               ≡⟨ ap (e' ∘_) (sym kp.p₂∘universal) ⟩≡
        e' ∘ kp.p₂ ∘ kp.universal (sym reg.coequal)  ≡⟨ pulll (sym p) ⟩≡
        (e' ∘ kp.p₁) ∘ kp.universal _                ≡⟨ pullr kp.p₁∘universal ⟩≡
        e' ∘ reg.arr₂                               ∎
    epi .has-is-coeq .factors = reg.factors
    epi .has-is-coeq .unique = reg.unique